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The resource theory of tensor networks
Matthias Christandl,1, ∗Vladimir Lysikov,2, †Vincent Steffan,1, ‡Albert H. Werner,1, §and Freek Witteveen1, ¶
1Department of Mathematical Sciences, University of Copenhagen,
Universitetsparken 5, 2100 Copenhagen, Denmark
2Faculty of Computer Science, Ruhr University Bochum,
Universit¨atsstraße 150, 44801 Bochum, Germany
(Dated: July 17, 2023)
Tensor networks provide succinct representations of quantum many-body states and are an im-
portant computational tool for strongly correlated quantum systems. Their expressive and com-
putational power is characterized by an underlying entanglement structure, on a lattice or more
generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated
to (hyper)edges. Changing this underlying entanglement structure into another can lead to both
theoretical and computational benefits. We study a natural resource theory which generalizes the
notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct
extension of resource theories of tensors studied in the context of multipartite entanglement and
algebraic complexity theory, allowing for the application of the sophisticated methods developed
in these fields to tensor networks. The resource theory of tensor networks concerns both the local
entanglement structure of a quantum many-body state and the (algebraic) complexity of tensor
network contractions using this entanglement structure. We show that there are transformations
between entanglement structures which go beyond edge-by-edge conversions, highlighting efficiency
gains of our resource theory that mirror those obtained in the search for better matrix multiplica-
tion algorithms. We also provide obstructions to the existence of such transformations by extending
a variety of methods originally developed in algebraic complexity theory for obtaining complexity
lower bounds.
I. INTRODUCTION
What is the structure of quantum many-body states?
Physically relevant states, such as ground states of local
Hamiltonians, typically have a very non-generic entan-
glement structure. Indeed, such states often exhibit en-
tanglement with a local character, expressed by an area
law for the entanglement entropy (as opposed to volume-
law entanglement entropy for generic states) [1]. This
observation has led to the Ansatz class of tensor network
states for representing quantum many-body states. Such
a tensor network state is created by first locally distribut-
ing states with bounded entanglement, and then apply-
ing local transformations. Here, the amount of initial
entanglement is captured by the bond dimension. Equiv-
alently, the state is constructed by taking a collection of
local tensors and contracting along a set of non-physical
indices. This encodes the global properties of the many-
body state into local entangled states together with a
local transformation.
Tensor network representations have become one of the
main theoretical and numerical tools for understanding
quantum many-body physics. The first examples, now
known as Matrix Product States (MPS), were discov-
ered in the study of spin chains (in particular the AKLT
∗christandl@math.ku.dk
†vladimir.lysikov@rub.de
‡sv@math.ku.dk
§werner@math.ku.dk
¶fw@math.ku.dk
model) as finitely correlated states [2]. Independently,
around the same time White invented the Density Ma-
trix Renormalization Group [3] as a numerical method,
which in hindsight is a method to optimize over MPS.
Since its conception, tensor network research has devel-
oped these two complementary perspectives, one strand
of research using tensor network states as a theoretical
tool to construct interesting many-body states and to
classify phases of matter, and another strand of research
developing sophisticated numerical methods to simulate
strongly interacting quantum many-body systems.
From a theoretical standpoint, tensor network states
approximately parametrize ground states of local Hamil-
tonians. Understanding phases of matter means that one
would like to understand the set of ground states of lo-
cal Hamiltonians under an appropriate equivalence rela-
tion. Using sets of suitable tensor network states as a
proxy for ground states allows one to reason more eas-
ily about phases of matter by reducing questions about
the global quantum state to questions involving only the
local tensors. In one spatial dimension it is rigorously
known that ground states of gapped Hamiltonians sat-
isfy an area law [4–6] and can be approximated by MPS
representations with bond dimensions growing polynomi-
ally with system size. In two or more spatial dimensions
it is widely believed that Projected Entangled Pair States
(PEPS) are good ground state approximations [7], with
area laws proven in special cases [8]. This has amongst
others been used to understand topological phases and
symmetries in such ground states, since the global prop-
erties of the many-body state are encoded in the local
tensors that make up the tensor network [9–16], see also
the reviews [17,18]. Tensor networks are also a pow-
arXiv:2307.07394v1 [quant-ph] 14 Jul 2023
2
erful numerical tool, since they hugely reduce the num-
ber of free parameters in the many-body state, allowing
for variational methods for ground state approximation.
The ground state correlation functions, energies or other
properties can then be extracted by tensor network con-
tractions. In one spatial dimension there exist rigorous
polynomial time algorithms for finding ground state ap-
proximations using MPS with polynomial bond dimen-
sion for gapped Hamiltonians [6] and in practice DMRG
provides an excellent simulation method for optimizing
MPS representations [3,19]. In two or more spatial di-
mensions, it is known that contraction of tensor networks
is computationally hard [20,21]. Nevertheless, there exist
numerical methods for (approximately) contracting and
optimizing tensor networks in two spatial dimensions [22–
24] and these have been successfully applied to strongly
interacting quantum systems [25,26].
Tensor networks have been developed mostly in the
context of condensed matter physics for the study of lat-
tice systems. However, they have found wide applica-
tion in other many-body physics problems, for instance
for simulating gauge theories and quantum field theories
[27,28], as (toy) models for holographic quantum gravity
[29–31] and in quantum chemistry [32–35]. Besides this,
tensor network methods can be used to simulate (small)
quantum computers [36–39]. The applications of tensor
networks extend beyond quantum many-body physics:
many mathematical and computational problems can be
phrased in terms of tensors, and tensor networks provide
general methods to decompose global tensors into a col-
lection of local tensors. A promising example is the use
of tensor networks as a tool for machine learning [40–49]
and graphical models [50–52]. Tensor networks also can
encode counting problems (and therefore tensor network
methods may be used for heuristic counting and opti-
mization algorithms [53,54]) and they are used in the
design and decoding of quantum error correcting codes
[55–57].
The essence of tensor network states is thus that they
are quantum states exhibiting ‘local entanglement’: they
are obtained by applying local transformations to net-
works of bipartite entangled states. There are both
theoretical and practical reasons, which we will review
in Section II, for allowing a more general entanglement
structure based on local multipartite entanglement [58].
Indeed, the standard way to construct tensor network
states is by placing maximally entangled pairs of dimen-
sion Don the edges of a graph G= (V, E ) with vertex
set Vand edge set E. This is the origin of the nomencla-
ture Projected Entangled Pair States (PEPS). Typically,
in the situation where we would like to simulate a con-
densed matter system, the graph will be a lattice. The
dimension Dof the maximally entangled states is the
bond dimension of the tensor network. The actual ten-
sor network state is now constructed by applying linear
maps Mvon each vertex of the graph. The resulting state
⇒Mv
Ne∈E|ϕ+
e⟩ |Ψ⟩
=|EPRD⟩=
D
P
i=1
|ii⟩
(a) A standard tensor network is built from EPR pairs.
=|ϕe⟩
(b) Graphical notation for
a plaquette state, here on
four parties.
(c) A rectangular lattice
with plaquette states on
four parties.
⇒
(d) A tensor network state constructed from an
entanglement structure with four-party states.
Figure 1: Graphical notation for entanglement
structures. Standard tensor networks are constructed
from an entanglement structure consisting of EPR pairs
of level D, as shown in 1a, corresponding to the special
case of 2-edges. In 1b we illustrate a state on a single
plaquette. We may then tile a lattice (or an arbitrary
hypergraph) with such plaquette states to obtain an
entanglement structure, as in 1c. Parties which belong
to the same node in the lattice are grouped together.
From the entanglement structure one may construct a
tensor network state by applying local linear maps, as
shown in 1d.
is given by
|Ψ⟩= O
v∈V
Mv!O
e∈E
|ϕ+
e⟩(1)
where |ϕ+
e⟩=PD
i=1 |ii⟩is an (unnormalized) maximally
entangled state, or EPR pair at edge e. This construction
3
is illustrated in Fig. 1a. One can think of the initial state
Ne∈E|ϕ+
e⟩as a resource for creating the state many-
body state |Ψ⟩: it is an entanglement structure for |Ψ⟩.
A natural generalization of Eq. (1) is to consider dif-
ferent local entanglement structures in this construction.
Here we are not restricted to only having states along
edges, but we may also tile the lattice with states shared
by more than two vertices. Formally speaking, we may
start with a hypergraph G= (V, E) where each (hy-
per)edge e∈Eis a subset of (possibly more than two)
vertices in V. Our main focus will be on two-dimensional
lattices of ‘plaquettes’ (we will use the terms plaquette
and (hyper)edge interchangeably).
We then consider again states of the form
|ϕ⟩G=O
e∈E
|ϕe⟩
as the entanglement structure, but now |ϕe⟩is a k-party
state if econsists of kvertices, visualized in Fig. 1c. For
instance, we could take a rectangular lattice of plaquettes
as depicted in Fig. 1and tile it with GHZ states of level
r
|GHZr⟩=
r
X
i=1
|iiii⟩
as a generalization of the usual maximally entangled
states. We then again obtain tensor network states by
applying maps at each vertex as in Eq. (1). This is illus-
trated in Fig. 1d. We will provide a more precise defi-
nition of this construction in Section II A. A key feature
is that for plaquettes containing more than two vertices
there is more freedom of choice of the entanglement struc-
ture than in the usual tensor network approach. In the
usual tensor network approach the bond dimension is the
only choice (as any other choice of state on the edge can
be absorbed into the tensor). This is not the case if the
plaquette has more than two parties: in that case there
are plenty of quantum states which are inequivalent un-
der applying local linear maps. In particular, there might
be various entanglement structures associated with the
same lattice yielding a given target quantum state |Ψ⟩.
In the usual PEPS picture of tensor network states,
the parameters that determine how expressive the class
of states is, is the set of bond dimensions on the edges.
Increasing the bond dimension allows one to represent a
larger class of states. In this work we study a resource
theory which allows one to compare different entangle-
ment structures.
Here, we will say that an entanglement structure |ϕ⟩G
with states |ϕe⟩on the edges is a stronger resource than
an entanglement structure |ψ⟩Gwhich has states |ϕe⟩on
the edges if there exist local transformations at the ver-
tices which map |ϕ⟩Gto |ψ⟩G. That is, there should exist
linear maps Mvat the vertices such that
O
v∈V
Mv!O
e∈E
|ϕe⟩=O
e∈E
|ψe⟩
N
v∈V
Mv|ϕ⟩G
=
|ψ⟩G
Figure 2: Relating different entanglement structures on
a lattice through local transformations.
as illustrated in Fig. 2. In other words, the entanglement
structure |ψ⟩Gcan be written as a tensor network state
using the resource |ϕ⟩Gand it is clear that this implies
that |ϕ⟩Gis a more powerful resource than |ψ⟩G. If there
exists a local transformation on every single edge efrom
|ϕe⟩to |ψe⟩then it is clear that we can just apply these
single-plaquette transformations in parallel. The main
question we will address in this work is the following:
In the resource theory of tensor networks, what
transformations between entanglement structures are
possible that are not given by single-plaquette
transformations?
The question of how to transform tensors through lo-
cal transformations has been extensively studied in the
theory of multipartite entanglement as well as in alge-
braic complexity theory. In complexity theory, finding
certain transformations between tensors is closely related
to finding faster algorithms for matrix multiplication. In
this context an extensive resource theory of tensors has
been created. One of the main goals of this work is to
introduce certain powerful techniques, developed to bet-
ter understand the complexity of matrix multiplication,
to the theory of tensor networks.
As our first main result, we show that there indeed
exist transformations of entanglement structures which
go beyond single-plaquette restrictions on lattices. That
is, we construct examples where one can not transform
|ϕe⟩into |ψe⟩, but once we place copies of these states
on a lattice the transformation becomes possible. There
are examples of such transformations where EPR pairs
are exchanged between different parties (and used as a re-
source). A plausible intuition may be that since in a two-
dimensional lattice the plaquettes are typically adjacent
in at most two vertices, any lattice transformation can be
performed by combining an exchange of EPR pairs with
single-plaquette transformations. We show that this is
not the case by giving an explicit example going beyond
such transformations. This demonstrates the richness of
the resource theory of entanglement structures.
On the other hand it is also important to be able to
show that there do not exist transformations between
different entanglement structures. As our second main
4
result we provide methods to find obstructions for the ex-
istence of entanglement structure transformations. The
fact that there exist transformations between entangle-
ment structures which go beyond single-plaquette trans-
formations implies that it does not suffice to prove ob-
structions on the level of individual plaquettes. We ex-
tend and apply various powerful methods from algebraic
complexity theory to prove obstructions in prototypical
examples. As a first example we use flattening ranks to
prove nontrivial bounds for transforming entanglement
structures with GHZ states to entanglement structures
using EPR pairs. Next, we use a version of the substitu-
tion method to prove that for the λ-state on the kagome
lattice (which is an entanglement structure for represent-
ing the Resonating Valence Bond state) there exists no
representation of bond dimension two, even though there
does exist an approximate representation of bond dimen-
sion two, answering one of the main open problems from
[58]. Finally, we study a class of asymptotic transfor-
mations where one has many copies of an entanglement
structure. Such transformations are characterized by the
asymptotic spectrum of tensors. We show that certain
points in this spectrum can be used as obstructions to
entanglement structure transformations.
Organization of the paper
We will provide background material on the relevance
of entanglement structures beyond maximally entangled
states in tensor network theory in Section II. There we
also provide an introduction to the resource theory of
tensors. We then introduce the resource theory of tensor
networks, and in particular of entanglement structures, in
Section III. We will discuss different types of local trans-
formations between entanglement structure that can be
considered. We relate this resource theory to the (alge-
braic) complexity of tensor network contractions, observ-
ing along the way that this is a V N P -complete problem.
After having introduced the resource theory of tensor
networks, we turn to the two main questions that can
be answered in this resource theory. Firstly, we provide
a number of explicit transformation which reduce one
entanglement structure to another in Section IV. Sec-
ondly, in Section V, we study the converse question and
give obstructions to the existence of transformations. We
present a number of general techniques for showing such
obstructions and apply them to concrete examples. In
Section VI we address structural questions relating to
symmetries and ranks of entanglement structures. We
end with a summary and conclusion in Section VII.
II. BACKGROUND
We will start by providing a detailed explanation of
the notion of an entanglement structure and motivate its
relevance for applications. Then, since readers may not
be familiar with the resource theory of tensors, we will
give a brief introduction to this resource theory. Finally,
we give a concise overview of previous work which is rel-
evant to the connection between the resource theory of
tensors and entanglement structures for tensor networks.
A. Entanglement structures
We start by defining the notion of a tensor network
and an entanglement structure on a hypergraph more
carefully. We start from some hypergraph G, which con-
sists of a set of vertices, which we denote by V, and a
set of hyperedges E. Each hyperedge e∈Econsists of
a subset of vertices, which are the vertices incident to e.
In many examples we will assume that the cardinality |e|
is a constant k, in which case we have a k-uniform hy-
pergraph. We will also assume that the vertices in any
edge e∈Eare all different. The degree of a vertex is
the number of edges esuch that v∈e. We allow double
edges. We consider Hilbert spaces of the form
HG=O
e∈EO
v∈e
He,v
where the He,v are Hilbert spaces. In other words, for
each vertex v∈Vwe have a Hilbert space for each edge
incident to v. We let
Hv=O
e:v∈e
He,v and He=O
v∈e
He,v
be the Hilbert spaces at some fixed vertex or edge.
A tensor network state is now constructed from the
following data: a collection of states |ϕe⟩∈He(not nec-
essarily normalized) for e∈Eand a collection of linear
maps
Mv:Hv→˜
Hvfor v∈V.
Here, ˜
Hv=Cdvis the physical Hilbert space and dvis
the physical dimension at v. Let
|ϕ⟩G=O
e∈E
|ϕe⟩.
We call |ϕ⟩Gan entanglement structure. In most exam-
ples the |ϕe⟩are copies of the same k-party state. We
will sometimes assume that the tensors |ϕe⟩are concise,
which means that the reduced density matrix on any sin-
gle party has full rank. While we have defined |ϕ⟩Gas a
tensor product over the edges, we can also regroup the
Hilbert spaces along the vertices and think of |ϕ⟩Gas a
state in Nv∈VHv. We then get a tensor network state
|Ψ⟩by applying the maps Mvto each vertex
|Ψ⟩= O
v∈V
Mv!|ϕ⟩G.(2)
5
⇒
D
P
i=1
Tjki Silm
j
k
i
l
m
T S
(a) Standard tensor contraction.
⇒
r
P
i=1
Tij SikUil
j
k
i
l
T
S U
|GHZr⟩
(b) Contraction with a GHZ state.
⇒T1jS0kU0l+T0jS1kU0l
+T0jS0kU1l
j
k
l
T
S U
|W⟩
(c) Contraction with a Wstate.
Figure 3: Illustration of how to interpret different
entanglement structures as tensor network contractions.
Tensor network states with entanglement structure |ϕ⟩G
are precisely the states which are restrictions of the en-
tanglement structure, where we have grouped according
to the vertices V. The usual version of tensor networks
is the case where the hypergraph is a graph (so for each
edge e∈Ewe have |e|= 2) and where we place (un-
normalized) EPR pairs |ϕe⟩=PD
i=1 |ii⟩=|EPRD⟩on
each edge. Here Dis known as the bond dimension of
the tensor network.
There are different perspectives on tensor network
states. Another perspective is that one takes a collection
of tensors, and contracts along edges in a graph. From
the perspective of tensor network states as contractions
of local tensors, we can also interpret different entangle-
ment structures as different contraction rules, see Fig. 3.
The idea to use entanglement structures different from
maximally entangled states has first come up in the con-
struction of concrete states [13] and its theory has been
developed more systematically in [58] (which introduced
the terminology of entanglement structures) and [59].
There are various reasons to study this generalization of
the usual notion of tensor network states. The main rea-
son to allow different entanglement structures is that this
can lead to representations of tensor network states with
smaller bond dimensions (i.e. an entanglement struc-
ture with lower Hilbert space dimensions). More efficient
representations with minimal bond dimension are crucial
for numerics in two or more spatial dimensions and de-
veloping the theory of entanglement structures beyond
maximally entangled states could lead to improved nu-
merical methods [60]. The special case with 3-tensors
on a kagome lattice has been proposed under the name
Projected Entangled Simplex States (PESS) and one can
extend PEPS optimization algorithms to this class of
states, achieving superior approximation in frustrated
lattice models with the appropriate entanglement struc-
ture, especially for spin liquids [61–63]. A prominent the-
oretical model for spin liquid behavior is the Resonating
Valence Bond (RVB) state and the closely related orthog-
onal dimer state, which is a superposition over dimers of
the lattice, where a dimer is a subset of edges such that
each edge is adjacent to exactly one edge in the dimer.
The state is then a uniform superposition over dimers,
where singlet states are distributed on the edges in the
dimer
|Ψ⟩=P
dimers
=|01⟩−|10⟩
In this case it is most interesting to study this on a frus-
trated lattice, such as the kagome lattice above. This
state can be obtained from an entanglement structure
placing a tensor |λ⟩at each plaquette
λ
|λ⟩=
2
P
i,j,k=0
ϵijk |ijk⟩+|222⟩
where ϵijk is the antisymmetric tensor, as shown in [64].
This perspective can be used to derive a PEPS repre-
sentation of the orthogonal dimer state, and therefore of
the RVB state, with reduced bond dimension [58]. One
should think of these examples as representing situations
where the local entanglement structure of the many-body
state is not accurately represented by pairwise entangle-
ment between neighboring sites, but rather by some lo-
cally shared multipartite entanglement.
Beyond this motivation there are also important the-
oretical reasons to study different entanglement struc-
tures. A central notion in the study of tensor network
states is that of injectivity, which means that the state
as constructed in Eq. (1) is with injective maps Mv. In
this case one can essentially invert the map Mvand the
properties of |Ψ⟩are very closely related to that of the
entanglement structure. Many theoretical results for ten-
sor networks are only valid for injective tensor networks
(or normal tensor networks, which become injective upon
blocking sites). For example, for an injective tensor net-
work state one can always find a local Hamiltonian (the
so-called parent Hamiltonian ) which has |Ψ⟩as its unique
ground state. There is a clear generalization of injectivity
to states where we allow different entanglement struc-
tures. This concept has been introduced in [59] as the
class of states where one considers tensor network states
with an arbitrary entanglement structure, and the maps
Mvare invertible (or more generally injective). A number
6
=|GHZ2(4)⟩
=|0000⟩+|1111⟩
=X⊗4Qi,j CZij
(a) An injective representation for the CZX model.
Here Xis the Pauli Xmatrix, and we act with
controlled Pauli Zoperators on the pairs of qubits.
=
=|01⟩ − |10⟩
= Π2
=
= Π4
(b) An injective representation of the AKLT model.
The operator Πnis the projector onto the symmetric
subspace on nqubits. Each party of an edge state
consists of two qubits; the parties share singlet states
to which one applies the projector Π2.
Figure 4: Examples of states which are injective when
using the right entanglement structure, see [59] for a
derivation.
of physically important models have injective PEPS rep-
resentations upon choosing the right entanglement struc-
ture, while they are not injective with respect to the
standard construction using maximally entangled states.
This is important in the classification of two-dimensional
symmetry protected topological phases [13,59]. A first
example is the CZX model [13], using GHZ states on a
square lattice an applying controlled Pauli Zoperations
as well as Pauli X. Another prominent example is the
AKLT model on the square lattice [65]. Its ground state
has an injective representation, illustrated in Fig. 4, by
using an entanglement structure with a 4-tensor given by
four singlet states where at each vertex one projects onto
the symmetric subspace (in other words, it is the one-
dimensional AKLT state on a periodic chain of length 4)
[59].
Finally, an especially natural example is using GHZ
states as entanglement structure: this simply corre-
sponds to contracting multiple indices at the same time
as illustrated in Fig. 3. This is relevant for tensor net-
work contractions in quantum circuits with controlled
gates [66]. Entanglement structures using GHZ states
for tensor networks have also been used in relating ten-
sor networks to graphical models [50] and in studying
random tensor networks [67].
B. Resource theory of tensors
Tensors are widely studied in mathematics, physics
and computer science. Two domains where tensors are a
central object of interest is in algebraic complexity theory
and quantum information theory. In both these fields
one of the main questions is when and how two tensors
can be transformed into one another using local opera-
tions. Throughout this work we will identify tensors with
quantum states shared between different parties where
we do not necessarily normalize the quantum states, so
ak-party quantum state is just a k-party tensor. As
an example of a transformation between quantum states,
suppose we have three parties Alice, Bob and Charlie,
and we have quantum states |ϕABC ⟩and |ψABC ⟩. Then
we can ask whether there exist linear maps MA,MBand
MCwhich Alice, Bob and Charlie can locally apply to
convert the states, i.e. such that
MA⊗MB⊗MC|ϕABC ⟩=|ψABC ⟩.(3)
In quantum information theory such transformations
can implemented using local operations and classical
communication, if we additionally allow postselection
on measurement outcomes and are therefore known as
Stochastic Local Operations and Classical Communica-
tion (SLOCC). This leads to a resource theory of entan-
glement in multiparty quantum states, under the class of
SLOCC operations. For instance, entanglement with re-
spect to SLOCC for three or four qubit systems has been
completely classified [68,69].
The same resource theory has been studied in the con-
text of theoretical computer science with the goal of
understanding the complexity of matrix multiplication.
Here the resource theory, as introduced by Strassen [70],
is formulated in terms of tensors rather than quantum
states. In this context, one says that |ϕ⟩restricts to |ψ⟩
if there exists a transformation as in Eq. (3) (then called
arestriction). More generally, if we have a collection of
kparties and
|ϕ⟩ ∈
k
O
i=1
Hi,|ψ⟩ ∈
k
O
i=1
H′
i
then |ϕ⟩restricts to |ψ⟩if there exist linear maps
Mi:Hi→ H′
i
such that
k
O
i=1
Mi!|ϕ⟩=|ψ⟩.(4)
We write |ϕ⟩≥|ψ⟩and this actually defines a partial
order on the set of k-party states. The interpretation
7
justifying the ≥sign is that |ϕ⟩as a resource is at least
as powerful as |ψ⟩.
The resource theory of tensors turns out to have inti-
mate connections to algebraic complexity theory. One of
the most important outstanding open problems in com-
puter science is to understand the computational com-
plexity of matrix multiplication. One would like to know
how many multiplication operations are required in or-
der to multiply n×nmatrices. The naive algorithm
uses n3multiplication operations. However, a surpris-
ing realization by Strassen was that one can multiply
two 2 ×2 matrices with only 7 (rather than 8) multi-
plications [71]. By recursively applying this construc-
tion one sees that asymptotically one can perform matrix
multiplication with only O(nα) multiplications where
α= log2(7) ≈2.81. The study of the complexity of
n×nmatrix multiplication can be recast as a problem
about tensors.
When computing C=AB for n×nmatrices Aand
B, from
Cij =
n
X
k=1
AikBk j
we see that matrix multiplication is closely related to the
tensor
|EPRn⟩△=
n
X
i,j,k=1
|ik⟩A|jk⟩B|ij⟩C
which is a 3-party tensor where each pair shares an EPR
pair (i.e. maximally entangled state) of dimension n(in
the algebraic complexity literature this is known as the
matrix multiplication tensor ⟨n, n, n⟩).
Every restriction of a 3-party GHZ state of rlevels
|GHZr(3)⟩=
r
X
i=1
|i⟩A|i⟩B|i⟩C
to |EPRn⟩△gives a method to perform n×nmatrix mul-
tiplication with rmultiplication operations. This moti-
vates one to compute the rank of a k-tensor |ϕ⟩as
R(|ϕ⟩) = min (r:|GHZr(k)⟩ ≥ |ϕ⟩)
where the GHZ state of level ron kparties is defined as
|GHZr(k)⟩=
r
X
i=1
|i⟩ ⊗ . . . ⊗ |i⟩
| {z }
ktimes
.
Equivalently, the rank R(|ϕ⟩) is the minimal number of
terms rneeded to write |ϕ⟩as a sum of product states:
|ϕ⟩=
r
X
i=1
|ei,1⟩ ⊗ . . . ⊗ |ei,k ⟩.(5)
Indeed, if we have a restriction |GHZr(k)⟩≥|ϕ⟩with
restriction maps Mj, for j= 1, . . . , k, then we may take
|ei,j ⟩=Mj|i⟩(and vice versa one can define the Mjfrom
the decomposition (5)). This definition is such that we
can do n×nmatrix multiplication using R(|EPRn⟩△)
multiplication operations. For example, the insight of
[71] is that R(|EPR2⟩△) = 7.
There is also an approximate version of restriction
known as degeneration. If |ϕ⟩and |ψ⟩are k-tensors, then
|ϕ⟩⊵|ψ⟩if there exist maps Mi(ε) for i= 1, . . . , k con-
tinuously depending on a parameter εsuch that
lim
ε→0 k
O
i=1
Mi(ε)!|ϕ⟩=|ψ⟩(6)
so for each ε > 0 we have a restriction, and its limit as
εgoes to zero is the target tensor |ψ⟩. Accordingly, one
may define the border rank
R(|ϕ⟩) = min (r:|GHZr(k)⟩⊵|ϕ⟩).
It turns out that if |ϕ⟩⊵|ψ⟩, there exist Ti(ε) which are
polynomial in εand a positive integer dsuch that
k
O
i=1
Ti(ε)!|ϕ⟩=εd|ψ⟩+
e
X
l=1
εd+l|ψl⟩(7)
for some degree eand tensors |ψl⟩. In this case, we will
write |ϕ⟩⊵e|ψ⟩to indicate the degree.
Complexity theory motivates another type of transfor-
mations. In complexity theory one is typically interested
in the asymptotic behavior as the instance size grows.
For this reason one may investigate the asymptotic rank
R
e(|ϕ⟩) = lim
n→∞ R(|ϕ⟩)1
n.
If we let ω= log2(R
e(|EPR2⟩△)) then the complexity of
matrix multiplication is given by O(nω+o(1)). The cur-
rent best upper bound on the matrix multiplication expo-
nent ωis approximately 2.37 [72,73] while it is possible
that ω= 2 (which coincides with the best known, and
trivial, lower bound). The asymptotic rank is closely re-
lated to asymptotic restriction; for tensors |ϕ⟩and |ψ⟩we
say that |ϕ⟩≳|ψ⟩if |ϕ⟩⊗(n+o(n)) ≥ |ψ⟩⊗nfor all n∈N.
In other words, for all nthere exist maps M(n)
iacting
on n+f(n) copies of the Hilbert space of |ϕ⟩for some
f(n) = o(n) such that
k
O
i=1
M(n)
i!|ϕ⟩⊗(n+f(n)) =|ψ⟩⊗n.(8)
Asymptotic restriction is also natural from the perspec-
tive of entanglement theory: it simply corresponds to
asymptotic SLOCC conversions! In this perspective,
log2(R
e(|ϕ⟩)) is the optimal rate at which one can convert
level-2 GHZ states to the state |ϕ⟩using SLOCC. Note
that when we write |ϕ⟩⊗nin this context for a k-tensor
|ϕ⟩, we really mean that we group together the ncopies
of the ksystems into a single party so we consider |ϕ⟩⊗n
8
as a k-tensor again [74]. This is of course the usual way
of thinking about asymptotics in quantum information
theory.
How are the three different transformations (restric-
tion, degeneration and asymptotic restriction, in equa-
tions (4), (6) and (8) respectively) and the corresponding
ranks related? By using polynomial interpolation [75,76]
we see that if in Eq. (7) the term with the highest degree
in εhas degree d+e, so |ϕ⟩⊵e|ψ⟩, we can get |ψ⟩from
restriction of a direct sum of e+ 1 copies of |ϕ⟩
⊵⇒e
L
i=0
≥
or equivalently
|ϕ⟩⊗|GHZe+1 (k)⟩≥|ψ⟩.
This can be used to show that |ϕ⟩⊵|ψ⟩implies |ϕ⟩≳|ψ⟩,
and hence R
e(|ϕ⟩)≤R(|ϕ⟩)≤R(|ϕ⟩).
The rank, asymptotic and border rank measure opti-
mal conversions of GHZ states to a tensor of interest.
One can also study the converse direction, and define
subrank as
Q(|ϕ⟩) = max (r:|ϕ⟩≥|GHZr(k)⟩),
the largest GHZ state which can be extracted by SLOCC
from |ϕ⟩. Similarly, we may define asymptotic subrank
as
Q
e
(|ϕ⟩) = lim
n→∞ Q(|ϕ⟩)1
n
and border subrank as
Q(|ϕ⟩)=(r:|ϕ⟩⊵|GHZr(k)⟩).
All these different notions of rank are related as
Q(|ϕ⟩)≤Q(|ϕ⟩)≤Q
e
(|ϕ⟩)≤R
e(|ϕ⟩)≤R(|ϕ⟩)≤R(|ϕ⟩).
While for 2-tensors all these notions collapse to the same
(standard) notion of rank, for k≥3 all inequalities can
be strict. In summary, understanding the computational
complexity of matrix multiplication naturally leads to a
resource theory of tensors, involving the notions of re-
striction,degeneration and asymptotic restriction. In al-
gebraic complexity theory one typically either shows by
some construction that a restriction (or degeneration, or
asymptotic restriction) exists, possibly leading to faster
algorithms; or one shows that there are obstructions to
the existence of a restriction, which corresponds to lower
bounds for certain algorithms.
See [77] for an accessible introduction to algebraic com-
plexity theory with a focus on the complexity of matrix
multiplication, as well as the reference work [78].
C. Prior work
As alluded to in Section II A, the idea of using entan-
glement structures beyond maximally entangled states
has been explored in various works, both for exactly
constructing interesting many-body ground states and
for numerical purposes. While the general theory of
entanglement structures has remained relatively under-
explored, we will here highlight some relevant previous
work.
The connection between tensor network states and the
resource theory of tensors was first studied in [58]. A first
observation is that a tensor network state is precisely a
restriction of an initial entanglement structure, by apply-
ing linear maps at all the vertices of the network. As a
demonstration of the applicability of the resource theory
of tensors to tensor networks, the authors show that de-
generations between plaquettes are a useful tool to get
lower bond dimension representations of tensor network
states. By an interpolation argument, one can prove that
if one has a degeneration |ϕ⟩⊵|ψ⟩and one considers the
entanglement structures |ϕ⟩Gand |ψ⟩Gwhich have these
states on each edge of some hypergraph G, then one can
compute observables for any tensor network state |Ψ⟩
constructed from the entanglement structure |ψ⟩Gfrom
observables in O(|E|) tensor network states using |ϕ⟩as
entanglement structure. For instance, given a degenera-
tion on a single plaquette from an entanglement structure
of level-DEPR pairs
⊵
this gives a representation using O(|E|) bond dimension
Dtensor network states
≥
L
O(|E|)
Importantly, the overhead from the interpolation which
turns the degeneration into a restriction (i.e. a proper
tensor network state) is only linear in the system size,
while contraction algorithms for tensor networks typi-
cally scale in the bond dimension as O(Dm) where m
grows with the system size. In other words, when com-
puting tensor network observables, the potential savings
in bond dimension from a plaquette degeneration are
much more significant than the overhead from the in-
terpolation. As an example of the techniques in [58],
the border subrank of |EPRn⟩△gives rise to low bond
dimension representations of entanglement structures of
|GHZr(3)⟩, by computing observables in a linear number
(in the system size) of tensor network states. Another ap-
plication is that there is a degeneration |EPR2⟩△≥ |λ⟩,
where |λ⟩is the tensor which can be used to construct
the RVB state. Therefore, one can compute expectation
values for the RVB state using a linear number (in the
system size) of tensor network states with bond dimen-
sion 2.
9
One can also investigate states which are the limit of
tensor network states. In the language of tensors, these
are states which are degenerations of an entanglement
structure, an idea which has been explored in [60].
Another systematic study of entanglement structures
investigated entanglement structures on two-dimensional
rectangular lattices [59]. The results of this work apply
to the translation invariant case with periodic boundary
conditions. The main results concern the question when
two entanglement structures are equivalent, i.e. they are
related by an invertible transformation Mvon each of
the vertices, moreover assuming that each Mvis equal.
Under these assumptions, an invertible transformation
between two entanglement structures exists for some lat-
tice of size n×mfor n, m ≥3 if and only if it exists
for all sizes. Moreover, in that case the map Mvcan be
taken to be a composition of maps only acting pairwise:
=
invertible ⇒=
These results are then used to show that the clas-
sification of symmetry protected topological phases us-
ing group cohomology is also valid for injective tensor
networks using translation invariant entanglement struc-
tures.
While the resource theory of entanglement structures
on lattices and other hypergraphs have not been stud-
ied in generality, there are various known results in the
study of tensors which are closely related. There are a
number of important results in the resource theory of ten-
sors which can be formulated as computing tensor ranks
and subranks of entanglement structures of hypergraphs.
These results provided evidence that the resource the-
ory of tensor networks is highly nontrivial. To make this
concrete, consider the Wstate
|W⟩=|100⟩+|010⟩+|001⟩.
It is known that it has tensor rank R(|W⟩) = 3, so there
exists a restriction from |GHZ3(3)⟩
≥
3W
and there does not exist such a restriction for |GHZ2(3)⟩
(see also Section V C). However, if we take two copies and
take this again as a 3-tensor (i.e. we take the Kronecker
product of two copies of |W⟩), the resulting object only
has rank 7 rather than the naive 9 = 32[79]. In other
words, the tensor rank is not multiplicative under the
Kronecker product, a fact already observed in the con-
text of matrix multiplication [77]. One can think of this
situation as placing two copies of the Wstate on top of
each other
≥
7W
We can also think of the following scenario: we place the
two Wstates on the completely disconnected hypergraph
consisting of two 3-edges (i.e. this is the usual tensor
product and we now have six parties). It is known that
this state has tensor rank 8 [76,80], so there exists a
restriction from a six-party |GHZ8(6)⟩state
≥8W W
and the tensor rank is also not multiplicative for the
tensor product. Similar strict submultiplicativity un-
der the tensor product is known for the border rank
[81]. The (asymptotic) tensor rank of other entanglement
structures, especially for states which are EPR pairs dis-
tributed over a graph [82,83], have also been studied.
One can pose the converse question as well where one
starts with some arbitrary multiparty tensor and tries
to transform to a global GHZ state. That is, one tries
to compute the subrank. This question first came up in
[84], lower bounding the border subrank Q(|EPRn⟩△).
For general hypergraphs with GHZ states, the asymp-
totic subrank was determined in [85].
The picture that emerges is that understanding the
tensor (sub)rank of entanglement structures is a rich sub-
ject which does not reduce to the ranks of the individual
edge states, suggesting that the resource theory of entan-
glement structures could be similarly nontrivial. How-
ever, the above works do not study the resource theory
of tensor networks, as the resource state (or target state)
is a global GHZ state. We will rather consider scenar-
ios where both the resource and the target state are a
tensor product of GHZ or other states over the edges of
the hypergraph. This resource theory has so far not been
studied.
There is a number of other works studying tensor net-
works using tools and ideas from algebraic complexity
theory. For example, [86] defines the G-rank of a tensor
with respect to a graph Gas the minimal bond dimen-
sions required to write the tensor as a tensor network
state on the graph G. This is in similar spirit as our
work, but it does not take into account entanglement
structures beyond maximally entangled states and does
not formulate a resource theory of tensor networks. We
also mention that there is a line of works, inspired by
algebraic complexity theory, studying sets of tensor net-
work states as algebraic varieties [87–90].
10
III. THE RESOURCE THEORY OF TENSOR
NETWORKS
We proceed to define our object of study: the resource
theory of tensor networks and entanglement structures.
This resource theory is the natural extension of the re-
source theory of tensors in algebraic complexity and the
theory of entanglement under SLOCC transformations.
We will introduce the resource theory from the entangle-
ment perspective, where we define tensor network states
as quantum states arising from SLOCC transformations
of strictly local networks of quantum states. However,
tensor networks are also a computational tool, and we
show that the resource theory of tensor networks relates
directly to the algebraic complexity of tensor network
contraction. Additionally we investigate the computa-
tional complexity of tensor network contractions from
the algebraic perspective and observe that tensor net-
work contraction is a V N P -complete problem.
A. The resource theory of tensor networks
It is clear that the formalism of tensor network states
closely aligns with the notions of tensor restriction in
Section II B. Indeed, a tensor network state is nothing
else than a restriction of a distribution of EPR pairs,
where the parties are the vertices of the graph. When we
consider the order induced by restrictions (i.e. SLOCC
transformations on the vertices), a state |Ψ⟩has a repre-
sentation as a tensor network state using an entanglement
structure |ϕ⟩Gas in Eq. (2) if and only if |ϕ⟩G≥ |Ψ⟩.
We will use this perspective to compare different entan-
glement structures. If we are given a lattice (or more gen-
erally some hypergraph) G= (V, E ), then |ϕ⟩G≥ |ψ⟩Gif
|ψ⟩Ghas a tensor network representation using |ϕ⟩Gas
entanglement structure. Concretely, |ϕ⟩G≥ |ψ⟩Gif there
exist maps Mvon each of the vertices of Gsuch that
O
v∈V
Mv!O
e∈E
|ϕe⟩=O
e∈E
|ψe⟩.
If we have some many-body state |Ψ⟩with a tensor
network representation by some entanglement structure
|ψ⟩G, then the existence of a restriction |ϕ⟩G≥ |ψ⟩Gdi-
rectly implies that we also obtain a tensor network rep-
resentation for |Ψ⟩using |ϕ⟩Gas our initial entanglement
structure:
physical
state ≤
current
entanglement
structure
≤
desired
entanglement
structure
For instance, if we have an injective representation of a
many-body state using GHZ states on each plaquette,
the question of finding a minimal bond dimension repre-
sentation of the state is equivalent to finding an optimal
restriction from the entanglement structure which has
EPR pairs along all the edges.
It is now clear that this defines a resource theory for
entanglement structures. This is the resource theory
of entanglement structures induced by SLOCC (in in-
formation theory terminology), or restrictions (in alge-
braic complexity theory terminology). Note that while in
quantum information theory the notion of SLOCC as the
class of allowed operations for a theory of multipartite en-
tanglement is not completely natural (operationally one
would prefer LOCC, but this is too complicated to clas-
sify in practice), for tensor networks it is the natural class
of allowed operations.
Based on the three different notions of tensor trans-
formations (restrictions, degenerations, asymptotic de-
generations) we can study different transformations of
entanglement structures.
1. Given two entanglement structures we can ask
whether we can transform one into another on each
individual plaquette. Whether this is possible then
can be formulated as asking whether there is a re-
striction |ϕe⟩≥|ψe⟩for each edge [58]. We will call
such transformations single-plaquette restrictions.
2. We can also ask whether on the plaquette level
there exists a degeneration, so |ϕe⟩⊵|ψe⟩for
all e∈E. The consequences of the existence
of both single-plaquette restrictions and degener-
ations have been investigated in [58]. Given single-
plaquette degenerations, one can represent the ten-
sor network state with only linear overhead in the
system size as explained in Section II C.
3. We know that there exist transformations of tensors
which are only possible asymptotically. Motivated
by this fact we can ask whether there exists a re-
striction on the global entanglement structure, i.e.
|ϕ⟩G≥ |ψ⟩G, or a global degeneration |ϕ⟩G⊵|ψ⟩G.
4. Also motivated by asymptotic restrictions we can
ask, given some entanglement structure |ϕ⟩Gand a
tensor network target state |Ψ⟩, at what rate we can
produce copies of |Ψ⟩from copies of |ϕ⟩G? More
generally, given |ϕ⟩⊗n
G, what is the optimal number
msuch that |ϕ⟩⊗n
G≥ |Ψ⟩⊗m?
It is the third and fourth questions on which we focus
on in this work, in order to develop a comprehensive re-
source theory of tensor networks, in particular of the un-
derlying entanglement structures. While in principle one
can study restrictions, degenerations and asymptotic re-
strictions between arbitrary many-body states, from the
perspective of tensor networks the main interest is the
situation where we look for restrictions
|ϕ⟩G≥ |Ψ⟩
so we have a restriction from an entanglement structure
(which means that |Ψ⟩is by definition a tensor network
11
state). The focus of our work is the situation where both
many-body states are entanglement structures, that is,
we investigate restrictions
|ϕ⟩G≥ |ψ⟩G.
The existence of such a restriction means that the class of
tensor network states using the entanglement structure
|ϕ⟩Gencompasses the class of tensor network states using
the entanglement structure |ψ⟩G, justifying the terminol-
ogy of a resource theory. Moreover, this encompasses the
important class of injective tensor network states (which
are by definition the states which are equivalent to an en-
tanglement structure). Similarly, we study this resource
theory with respect to degenerations, so
|ϕ⟩G⊵|ψ⟩G.
This relation is less standard in the tensor network lit-
erature, but we emphasize that (as argued in [58]) if the
degeneration has low degree, tensor network computa-
tions for |ψ⟩Greduce to tensor network computations
with |ϕ⟩Gwith only small overhead. An important rea-
son to consider degenerations is that they can allow for
significant savings in bond dimensions and degenerations
are in practice often easier to find (and again, the small
overhead is not relevant for the asymptotic scaling of the
complexity of the algorithm); for these reasons degener-
ations play a crucial role in the search for faster matrix
multiplication algorithms.
We may for instance take a rectangular lattice and con-
sider 4-party states |ϕ⟩and |ψ⟩and investigate whether
there exist maps at the vertices such that
O
v∈V
Mv!|ϕ⟩G=|ψ⟩G
as illustrated in Fig. 2. On first sight, one might think
that since the initial and final states are tensor product
states over the edges, the best one can do is perform
transformations on each plaquette. However, it is well
known that there exist examples for which |ϕ⟩≱|ψ⟩but
|ϕ⟩⊗n≥ |ψ⟩⊗nfor some n. One can think of this as the
case where the hypergraph is such that the edges are all
stacked on top of each other, so we may have
while≱≥
The question is whether such phenomena still occur when
we use a lattice hypergraph (which is a much sparser
structure), which is the main question we address in this
work.
We will indeed construct examples where |ϕ⟩G≥ |ψ⟩G
while |ϕ⟩≱|ψ⟩in Section IV. There are obvious examples
of such transformations where EPR pairs are exchanged
between different parties (and used as a resource). A
basic example is shown in Fig. 5where we have tensors
|ϕ⟩and |ψ⟩consisting of two EPR pairs which are clearly
|ϕ⟩ |ψ⟩
⇒
Figure 5: An example of two states which can not be
transformed into one another on a single plaquette, but
which give the same entanglement structure.
not equivalent (or related by a restriction) on a single
plaquette, but when placed on a periodic lattice they give
rise to the same state, as already observed in [59]. Given
this example, a plausible intuition is that since in a two-
dimensional lattice the plaquettes are typically adjacent
in at most two vertices, any lattice transformation can be
performed by combining an exchange of EPR pairs with
single plaquette transformations. This is however not the
case, as we will show in Theorem 2. This demonstrates
that the resource theory of tensors is highly nontrivial.
On the other hand, we will also provide obstructions for
the existence of entanglement structure transformations
in Section V, applying various methods from algebraic
complexity for proving complexity lower bounds.
The physically most interesting case for tensor net-
works are two- and higher-dimensional lattices. In one
spatial dimension, using multipartite entanglement is less
natural, although one can in principle consider entangle-
ment structures on a ‘strip’ of plaquettes. However, our
concepts and methods apply to arbitrary hypergraphs.
From a mathematical perspective this poses new inter-
esting questions in the theory of tensors. We know that
in general |ϕ⟩≳|ψ⟩does not imply |ϕ⟩≥|ψ⟩. We can
think of the asymptotic restriction as closely related to
restrictions on the hypergraph where we stack all edges
on top of each other. This leads to the general question
how connected the hypergraph Ghas to be in order for
a restriction |ϕ⟩G≥ |ψ⟩Gto be possible. In the special
case where the hypergraph is acyclic, we show in Corol-
lary 6that any global transformation reduces to a local
one, acting on each plaquette separately, so |ϕ⟩G≥ |ψ⟩G
implies |ϕ⟩≥|ψ⟩.
B. Algebraic complexity theory of tensor networks
One of the main motivations for studying the resource
theory of tensors under restrictions, degenerations and
asymptotic restrictions was to understand the complex-
ity of matrix multiplication. A similar natural question
is to study the algebraic complexity of tensor network
contractions, and we will now explain how the resource
theory of tensor networks relates to the (algebraic) com-
plexity of tensor network contractions. This discussion
is not needed to understand the remainder of this pa-
per. The question we would like to understand is the
12
following: given an entanglement structure |ϕ⟩G, what
is the number of operations needed to compute a coef-
ficient of a tensor network state using the entanglement
structure |ϕ⟩G? Here the model of complexity is given by
arithmetic circuits, which perform addition, scalar mul-
tiplication, multiplication and division. To measure the
complexity of computing a certain function, one assigns
weights to these different operations. A common choice
(when studying for instance the complexity of matrix
multiplication) is to make addition and scalar multiplica-
tion ‘free’ and only count the number multiplications and
divisions. This is a different model than the usual Turing
machine model for computation. In particular, we work
directly over some field of numbers (such as the complex
numbers C) and we do not use a binary representation
of elements in the field. In the case of matrix multipli-
cation, this is the complexity which is closely related to
the tensor rank.
Given an entanglement structure |ϕ⟩G, we may obtain
a tensor network state by applying local linear maps Mv
for each vertex v∈V. Choose a basis |iv⟩, for the phys-
ical Hilbert space at vertex v, and let T(iv)
v=⟨iv|Mv.
Then we may expand the tensor network state as
X
{iv}v∈V O
v∈V
T(iv)
v!|ϕ⟩G!O
v∈V
|iv⟩
The terms
O
v∈V
T(iv)
v!|ϕ⟩G
are therefore the coefficients of the state when expanded
in the chosen product basis. In general, we define the
following map, which assigns a value to any collection of
linear maps {Tv}v∈V,Tv:Hv→C
f|ϕ⟩G:M
v∈V
H∗
v→C
(Tv)v∈V7→ O
v∈V
Tv!|ϕ⟩G.
(9)
We call this map a tensor network coefficient. We will
think of f|ϕ⟩Gas a polynomial in the entries of the Tv(so
it has Qvdim(Hv) variables). We will also refer to the
task of computing the value of this polynomial on some
input (i.e. computing a coefficient of some choice of ten-
sors) as tensor network contraction and it is the basic
computational primitive in any tensor network based al-
gorithm. For instance, when performing variational op-
timization over the class of tensor network states, the
energy expectation value reduces to computing a tensor
network coefficient. We will now investigate the hardness
of computing f|ϕ⟩Gas a polynomial in the arithmetic cir-
cuit model. Given an entanglement structure |ϕ⟩G, we
define C(|ϕ⟩G, G) to be the minimal size of an arithmetic
circuit computing f|ϕ⟩G. The resource theory for tensor
networks we have defined now relates back to algebraic
complexity theory by the following simple observation.
Theorem 1. 1. If |ϕ⟩G≥ |ψ⟩G, it holds that
C(|ψ⟩G, G)≤C(|ϕ⟩G, G).
2. Similary, if |ϕ⟩G⊵e|ψ⟩G, it holds that
C(|ψ⟩G, G)≤(e+ 1)C(|ϕ⟩G, G)
Proof. Suppose that we have an arithmetic circuit that
computes f|ϕ⟩G, and we have maps Mvsuch that
O
v∈V
Mv!|ϕ⟩G=|ψ⟩G.
Then
f|ψ⟩G((Tv)v∈V) = f|ϕ⟩G((TvMv)v∈V)
so we can compute f|ψ⟩G((Tv)v∈V) by first computing
(Tv)v∈V7→ (TvMv)v∈Vand then use the circuit for f|ϕ⟩G.
The cost of the first step is zero in the model we consider
(as it only requires multiplications with fixed numbers
and additions, which are free), proving assertion 1. We
conclude that the cost of evaluating f|ψ⟩G((Tv)v∈V) is
at most C(|ϕ⟩G, G). For 2we note that by a standard
polynomial interpolation argument [58,76,91] we can
compute f|ψ⟩G((Tv)v∈V) from e+ 1 different restrictions
of |ϕ⟩G, giving the desired result.
How hard are tensor network computations? The stan-
dard algorithms contract along edges one by one. Re-
stricting to such computations on a graph (and an en-
tanglement structure with EPR pairs), one finds that the
complexity is exponential in the treewidth of the graph
[92]. Finding optimal contraction orders is NP-hard in
general, so one typically resorts to heuristics to finding
good contraction orders [66,93,94]. It is well-known
that in general tensor network contraction is indeed a
hard problem. For instance, one can show that tensor
network contraction is a #P-hard problem [20,21], see
[95,96] for further complexity-theoretic investigations of
tensor networks. One can also study the hardness of ten-
sor network contraction in the arithmetic circuit model.
Recall that in this model of computation, the goal is to
compute some polynomial through an arithmetic circuit.
The analog of the class Pis the class V P , which con-
sists of families of polynomials fnfor which there is a
family of polynomial-sized circuit computing the desired
polynomials. The arithmetic analog of N P is the class
V NP , which is, informally speaking, the class of fami-
lies of polynomials fnof polynomial degree which is such
that for each monomial one can determine its coefficient
in fnby a polynomial-sized circuit. In Appendix Cwe
prove that on arbitrary hypergraphs the computation of
tensor network coefficients is in V NP and tensor network
13
contraction on a two-dimensional square lattice with con-
stant bond dimension D= 2 is V N P -hard. This is not
very surprising (given the corresponding #P-hardness)
but we are not aware of previous work explicitly making
this observation.
Theorem. The problem of computing tensor network
contraction coefficients, given by the polynomial f|ϕ⟩Gas
in Eq. (9)on hypergraphs with nedges and constant de-
gree is in V N P . The problem of computing tensor net-
work contraction coefficients with bond dimension D= 2
on an n×nsquare lattice is V N P -hard.
In Appendix Cwe state this result more formally
as Theorem 25 and provide the proof. While for two-
dimensional lattices tensor network contraction is V NP -
hard, it is easy to see that for acyclic hypergraphs the
problem of computing f|ϕ⟩Gis in V P . Finally, we men-
tion that from a complexity theory perspective, tensor
network contractions are closely related to an approach to
counting problems known as Holant problems, see [97,98]
for a discussion relating to quantum states and multi-
party entanglement. The resource theory of tensor net-
works should therefore also be relevant to such Holant
problems.
IV. CONSTRUCTIONS
Recall that the central question of our work is to study
what transformations of entanglement structures are pos-
sible beyond transformations that act on the individual
plaquettes. In this section we will give explicit construc-
tions to demonstrate that there indeed exist transforma-
tions of entanglement structures which go beyond plaque-
tte by plaquette restrictions. This can be studied on arbi-
trary hypergraphs, but we will focus on examples which
are relevant to lattices, as these are of primary interest
for many-body physics applications. We will consider pe-
riodic lattices, to avoid boundary terms (but this is not
crucial). We will focus on two-dimensional structures,
since these are important in tensor network applications,
and in one spatial dimension entanglement structures be-
yond EPR pairs are somewhat artificial (although one
can study ‘strips’ of plaquettes to obtain nontrivial ex-
amples). We also provide classes of examples in higher
spatial dimensions and we end by discussing the asymp-
totic resource theory of tensor networks.
A. Two-dimensional entanglement structures
The first and most basic class of examples are examples
where we just distribute EPR pairs of some dimension
over the plaquettes. When arranged on the lattice, we
may move EPR pairs to neighboring plaquettes
⇒
which is an invertible transformation. For example, as
we already saw
|ϕ⟩ |ψ⟩⇒
which is an example where on the lattice the entangle-
ment structures are equivalent while there are no restric-
tions (or degenerations) between |ϕe⟩and |ψe⟩for single
plaquettes e.
This principle can be used to construct many more
examples! One way is to use the EPR pairs for tele-
portation. Here is an example on a cyclic graph of four
vertices
≥
In general, in a lattice one can move an EPR pair to
a neighboring plaquette, and there use it as a resource.
For example, consider plaquettes
≱
|ϕe⟩ |ψe⟩
where we place a GHZr(3) state and a level-pEPR pair
as plaquette state |ϕe⟩and |ψe⟩is given by an arbitrary 3-
party state |ψ⟩distributed over three of the four parties.
When we place these on a lattice
≥
we see there exists a restriction if the restriction
≥r
p
exists. In fact, the existence of such a restriction is (by
definition) the statement that the p-aided rank of |ψ⟩is
at most r. See [99] for a study of the aided rank, and
example computations of the aided rank of tensors.
Next, for a slightly more involved example we consider
a triangular lattice. Let us suppose that we start with
|GHZr(3)⟩on a sublattice as entanglement structure |ϕ⟩G
and take |ψ⟩Gto be the entanglement structure where we
place |GHZn(3)⟩at each edge, and we look for a degen-
eration |ϕ⟩G⊵|ψ⟩G:
14
⊵
rn
For which values of rcan we perform such a transfor-
mation? It is clear that this is not an edge-wise trans-
formation as in |ϕ⟩Gwe only have entangled states on a
sublattice and the other sublattice is ‘empty’ (i.e. has
only product states). A good strategy to perform this
conversion is as follows. First, convert each
GHZr(3) ⊵GHZn(3) ⊗ |EPRD⟩△
in |ϕ⟩G. This is certainly possible if we take rto be the
border rank times the remaining level n, so
r=nR(|EPRD⟩△).
Next, for each of the empty edges, we may now take EPR
pairs from the surrounding edges, so we get an |EPRD⟩△
state at this edge. From this, we can now extract a
GHZm(3) state, where the optimal m(by definition) is
given by the border subrank m= Q(|EPRD⟩△).
⊵ ⊵
r n
D
It is known that
Q(|EPRD⟩△) = 3D2
4(10)
by a construction in [70] and a lower bound in [100]. So,
we can achieve the transformation with
r=nR3n2
4.(11)
An interesting question is whether this transformation
is actually optimal, or whether a smaller rwould also
suffice.
While nontrivial, the above example still had as a cru-
cial ingredient that we somehow ‘moved’ EPR pairs from
one edge to another. In the lattice hypergraphs we con-
sider, two edges have at most two overlapping vertices.
This suggests the intuition that all ‘interaction’ between
different edges may be mediated by two-party entangle-
ment, i.e. by moving an EPR pair from one edge to
another. We will now show that this intuition is not cor-
rect and that the resource theory of tensor networks has
a richer variety of possible transformations.
We will show that there exists a degeneration on a hy-
pergraph with three 3-edges, where we place GHZ5(3)
states on the three edges, and end up with an entangle-
ment structure with EPR pairs.
Theorem 2. There exists a degeneration
5 5
5
222
2
2
3
⊵
Note that there is a one level-3 EPR pair. The key
point is that the final structure consists just of EPR pairs,
but there is no way to distribute them over the three pla-
quettes such that we have an edge-to-edge degeneration.
Indeed, if we distribute the EPR pairs over the edges, at
least one of them will have both a level-3 and a level-2
EPR pair, and there is no possible degeneration
⊵/3
2(12)
The impossibility of this degeneration follows from an
easy flattening rank bound (see Section V).
Nevertheless, the degeneration in Theorem 2does ex-
ist. The main ingredient is the three-party Bini tensor
|β⟩∈HA⊗ HB⊗HCwith each Hilbert space equal to
(C2)⊗2, which is given by taking three level-2 EPR pairs,
that is the tripartite state |EPR2⟩△, projecting out |11⟩
on C, so
β=
A
B
C
=
1
− |11⟩⟨11|
We may write this out in the standard basis
|βABC ⟩=|00⟩A|00⟩B|00⟩C+|00⟩A|01⟩B|10⟩C
+|01⟩A|10⟩B|00⟩C+|10⟩A|00⟩B|01⟩C
+|01⟩A|11⟩B|10⟩C+|11⟩A|10⟩B|01⟩C
so it is clear that R(β)≤6. In [101] it was shown that
there exists a degeneration |GHZ5(3)⟩⊵|β⟩(so we have
border rank R(|β⟩)≤5), given by
|01⟩+ε|11⟩|10⟩|01⟩+ε|00⟩
+|00⟩|00⟩+ε|01⟩|10⟩+ε|00⟩
− |01⟩(|00⟩+|10⟩+ε|11⟩)|01⟩
−|00⟩+|01⟩+ε|10⟩|00⟩ |10⟩
+|01⟩+ε|10⟩|00⟩+ε|11⟩|01⟩+|10⟩
=ε|β⟩+O(ε2)
The authors of [101] were motivated by the complexity
of matrix multiplication, as |EPR2⟩△is the 2 ×2 matrix
multiplication tensor, and they used a direct sum of two
copies of |β⟩to perform 2 ×3 matrix multiplication.
Proof of Theorem 2.We start by using the degeneration
|GHZ5(3)⟩⊵|β⟩on all three |GHZ5(3)⟩states
15
⊵5 5
5
β β
β
Let us call the three outer parties A,Band C, and the
interior party D. With the Bini tensors, on the outer
edges we now have level-2 EPR pairs (between AB,BC
and AC), and it remains to extract EPR pairs between
Dand A,Band Crespectively:
=β β
βA
B
C
D
We divide Dinto parties A′,B′and C′, and apply a
projection operator on D
A
B
C
A′
B′
C′=P
where Pprojects onto the span of the states
n|0a⟩A′|0b⟩B′|0c⟩C′oa,b,c∈{0,1}
together with
n|10⟩A′|b0⟩B′|c0⟩C′ob,c∈{0,1}.
Note that all of these states are in the support of the
tensor on D, indeed, the only elements which are not in
the support are
|a1a2⟩A′|b1b2⟩B′|c1c2⟩C′
with either a2=b1= 1, or b2=c1= 1 or c2=a1= 1.
Finally, we apply the linear map
|0⟩ ⟨00|+|1⟩ ⟨01|+|2⟩ ⟨10|
on Aand A′, and on B,B′,Cand C′we apply the map
|0⟩ ⟨00|+|1⟩⟨01|+⟨10|
which results in the transformation
A
B
C
⇒
3
2
2
which gives the target state of Theorem 2.
We can also put this degeneration on a lattice, see
Fig. 6a. An interesting open question is whether this is
optimal for this lattice, or whether one can do better.
We give one more example. This time we will see a
restriction which is only possible on the lattice, which is
such that it extracts a global GHZ state. We consider the
square lattice, and consider the entanglement structure
where we place the state
|ϕe⟩= = +
+ +
A
B
D
C
(13)
at each plaquette. Here, the four parties of the pla-
quette each have Hilbert spaces (Cn+1)⊗2, with basis
|0⟩,...,|n⟩. Unconnected dots mean that the two par-
ties have a product state |0⟩ |0⟩, connected dots mean
that they share a level-nEPR pair Pn
i=1 |i⟩ |i⟩. For ex-
ample, the first term on the right-hand side in Eq. (13)
is given by
n
X
i,j=1
|0i⟩A|ij⟩B|j0⟩C|00⟩D.
We will show that there exists a restriction to the en-
tanglement structure where we have level-nEPR pairs
on the square lattice (i.e. the entanglement structure
for a standard bond dimension ntensor network state)
together with a global level-4 GHZ state shared by all
parties v∈V, as illustrated in Fig. 6b.
Theorem 3. Let Gbe the square lattice, and let |ϕ⟩G
be the entanglement structure with edge states given by
Eq. (13). Let |ψ⟩Gbe the entanglement structure with
level-nEPR pairs on the square lattice as in Fig. 5. Then
|ϕ⟩G≥ |ψ⟩G⊗GHZ4(|V|).
Proof. We will now argue that we can apply a restriction
to get the entanglement structure with level-nEPR pairs
on the edges with additionally a level-4 GHZ state shared
by all vertices. We apply the projection given by P=
P1+P2+P3+P4
P= = +
+ +
A
B
D
C
where the figure should be interpreted similar to Eq. (13);
for instance the first term is the projection onto the sub-
space spanned by
{|i0⟩A|jk⟩B|0l⟩C|00⟩D;i, j, k, l = 1, . . . , n}
16
⊵
234
5
(a) One can place the degeneration of Theorem 2on a
lattice.
≥
|GHZ4(|V|)⟩
(b) There is a restriction from the entanglement
structure using the state in (13) to an entanglement
structure with level-nEPR pairs and a global level-4
GHZ state.
Figure 6: Transformations of lattice entanglement
structures beyond exchanging EPR pairs.
It is clear that each of the four terms in this projection
gives a lattice of EPR pairs, and the images of the Piare
orthogonal so we extract a global GHZ4(|V|) state.
Note that such a global GHZ state may in turn be used
for the purpose of transforming degenerations of entan-
glement structures into restrictions [58] as explained in
Section II C. Finally, an interesting open question is the
following: do there exist entanglement structures (with
the same state at each plaquette) where the existence of
a transformation depends on the size of the lattice? So
far, our lattice examples exist for any lattice size, while
there may exist transformations that only become possi-
ble when the size of the lattice is sufficiently large.
B. Higher dimensional entanglement structures
The construction principles which are demonstrated in
Section IV A may be applied in similar fashion for higher
dimensional lattices. The most basic constructions in two
spatial dimensions used that two adjacent plaquettes can
exchange EPR pairs (even though we show that, perhaps
surprisingly, there exist transformations that go beyond
this mechanism). In higher spatial dimensions it is signif-
icantly easier to construct nontrivial examples of lattice
transformations since adjacent plaquettes now may be
neighboring in more than two parties. This means that
plaquettes can exchange multipartite entangled states.
It allows for a new construction principle, which does
not yet show up in two spatial dimensions. This is based
on the fact that there exist states |ϕ⟩,|ψ⟩on more than
two parties which are such that
|ϕ⟩≱|ψ⟩while |ϕ⟩⊗2≥ |ψ⟩⊗2
and the same for degenerations. Let us see in a con-
crete example how this can be used. We will again use
that we know the precise value of the border subrank
Q(|EPRD⟩△) as in Eq. (10). Now, as plaquettes we
take tetrahedra (so with four parties), where for |ϕe⟩we
place level-D2EPR-pairs on the 2-edges of the tetrahe-
dron, which corresponds to placing |EPRD⟩△states on
the faces of the tetrahedron. For |ψe⟩we place level-r
GHZ states on the faces of the tetrahedron:
⇒
r
D|ϕe⟩
|ψe⟩
Then, on an individual plaquette, we can use a
|EPRD⟩∆state on each of the faces to perform a degen-
eration to a |GHZr(3)⟩state for r=⌈3D2
4⌉, so for this
value of rwe have a degeneration |ϕe⟩⊵|ψe⟩. Now, if
we consider a three-dimensional lattice where the tetra-
hedra have adjacent faces, we can group together EPR
pairs from adjacent plaquettes and apply a degeneration
(and at the end redistribute the resulting GHZ states to
the two plaquettes). This allows a transformation for
r=jrl3D4
4mk≥Q(|EPRD⟩△).
To give a concrete example, for D= 4 the single plaque-
tte transformation is possible for r= 12, while on the
lattice we can achieve r= 13.
C. Asymptotic conversions of tensor network states
We now address a question of a different nature, where
we consider many copies of an entanglement structure
to see how it can be used asymptotically as a resource.
Let |ϕ⟩G=Ne∈E|ϕe⟩be an entanglement structure on
a hypergraph G= (V, E ). Then we let |ϕ⟩⊗n
Gbe the
entanglement structure on the same hypergraph Gwhere
we place |ϕe⟩⊗nat edge e
|ϕ⟩⊗n
G=O
e∈E
|ϕe⟩⊗n.
So, at each plaquette we place ncopies of the state |ϕe⟩:
17
Given a target state |Ψ⟩(which could be another en-
tanglement structure, but in principle any state) we
would like to know how useful |ϕ⟩Gis as a resource for
|Ψ⟩. More precisely, we would like to know for which m
and nit is true that
|ϕ⟩⊗n
G≥ |Ψ⟩⊗m.
To study the asymptotics, we say that |ϕ⟩G≳|Ψ⟩if
|ϕ⟩⊗(n+o(n))
G≥ |Ψ⟩⊗n.
In the particular case where |Ψ⟩=|ψ⟩Gis some entan-
glement structure
≳
means that we have restrictions
≥
}n+o(n)
}n
for n→ ∞.
To find an example, it suffices to have k-party states
|ϕ⟩and |ψ⟩such that on the one hand if we consider the
entanglement structures where we place these states on
each hyperedge, there is no restriction, so |ϕ⟩G≱|ψ⟩G,
but we do have |ϕ⟩≳|ψ⟩. It is clear that by applying
the asymptotic restrictions plaquette-wise in that case
we obtain |ϕ⟩G≳|ψ⟩G. There are many examples of
tensors for which |ϕ⟩≳|ψ⟩but |ϕ⟩≱|ψ⟩. However, it
is nontrivial to show that |ϕ⟩G≱|ψ⟩G. In Section V C
we investigate the case where on the kagome lattice we
have |ϕ⟩=|EPR2⟩△and |ψ⟩=|λ⟩, the tensor from
which we may obtain the RVB state. It is known that
|EPR2⟩△⊵|λ⟩and hence |EPR2⟩△≳|λ⟩. We prove that
on the kagome lattice |ϕ⟩G≱|ψ⟩G[58]. Note that this
example also directly implies that for the RVB state |Ψ⟩
on the kagome lattice we have |EPR2⟩△≳|Ψ⟩(although
we do not prove that |EPR2⟩△≱|Ψ⟩). By a similar
method we prove that when we take |ϕ⟩=|GHZ2(3)⟩
and |ψ⟩=|W(3)⟩on the kagome lattice there is no re-
striction |ϕ⟩G≥ |ψ⟩G, while already on the level of a
single plaquette |GHZ2(3)⟩≳|W(3)⟩.
For a final example, the value of the asymptotic sub-
rank of |EPRD⟩is given by [70]
Q
e
(|EPRD⟩△) = D2
≳
D
D
D
D2
This means that we have an asymptotic restriction
≳
DD
On the other hand, a simple flattening rank bound (see
Section V) shows that this is optimal. Determining the
optimal bond dimension for finite nis an interesting open
problem.
V. OBSTRUCTIONS
In Section IV we saw explicit examples where the con-
version between entanglement structures was possible
only by putting the tensors on the plaquettes of a lattice.
This clearly demonstrates that in order to show that no
conversion can exist we must take into account the en-
tire lattice. That is, if one wants to show |ϕ⟩G≱|ψ⟩Git
does not suffice to show that for the edges |ϕe⟩≱|ψe⟩.
Whereas previous work proved the optimality of certain
single-plaquette transformations [58], our work provides
for the first time general tools to show the impossibility
of the conversion of entanglement structures.
The fundamental insight is that we may coarse-grain
(or ‘fold’) the network, and that any restriction on the
initial graph should also be a restriction on the folded
graph. This allows one to reduce the problem to show-
ing that there does not exist a restriction on the folded
graph. We will explain two important methods from al-
gebraic complexity theory for obtaining obstructions for
the existence of restrictions, namely (generalized) flat-
tenings and the substitution method and we adapt them
to prove the nonexistence of a restriction in interesting
and nontrivial examples.
Next, we will discuss the question of asymptotic con-
version of entanglement structures. Asymptotic restric-
tion is completely characterized by a compact topological
space of functionals (the so-called asymptotic spectrum).
We will explain how to use this in the context of entan-
glement structures.
A. Folding the tensor network
If Gis a hypergraph, we can obtain a new hypergraph
by grouping some vertices together to a single vertex. We
call this procedure folding, and if His obtained in such a
way from Gthen His a folding of G. To be precise, Hhas
vertex set V′and we have a surjective map f:V→V′.
The edge set E′of His given by edges
f(e) = {v′such that v′=f(v) for v∈e}
for each edge e={v1, . . . , vk} ∈ E. Note that |f(e)|
may be smaller than |e|since it is possible that fsends
vertices in the same edge to the same image. If we have
an entanglement structure |ϕ⟩G=Ne∈E|ϕe⟩on G, then
18
this can naturally also be interpreted as an entanglement
structure on H, by grouping the Hilbert spaces as
Hv′=O
v:f(v)=v′
Hvand Hf(e)=He
and simply reinterpreting the state as
|ϕ⟩H=O
e′∈E′
|ϕe′⟩.
The only thing we have done here is that we have grouped
parties (vertices) together into a single parties. It is clear
that folding preserves restrictions.
Lemma 4. Let G= (V, E )be a hypergraph and H=
(W, F )a folding of G. Let |ϕ⟩Gand |ψ⟩Gbe entanglement
structures for G. Then |ϕ⟩G≥ |ψ⟩Gimplies |ϕ⟩H≥
|ψ⟩H. Similarly, |ϕ⟩G⊵|ψ⟩Gimplies |ϕ⟩H⊵|ψ⟩H.
Proof. If the restriction is given by maps Mvfor v∈V,
then it is clear that Nv:f(v)=v′Mvfor v′∈V′defines a
restriction on H.
While Lemma 4is obvious, it is the key to proving
obstructions! An important special case is where we fold
to a graph with only two vertices. For 2-tensors |ϕ⟩and
|ψ⟩we know that |ϕ⟩ ≥ |ψ⟩(and in fact |ϕ⟩⊵|ψ⟩) if
and only if |ϕ⟩has Schmidt rank at least as large as |ψ⟩.
For a 2-tensor |ϕAB ⟩ ∈ HAB we denote by rk(|ϕAB⟩) its
Schmidt rank (and we will just call this the rank ), which
is equal to the rank of its reduced density matrix on either
Aor B, or the rank of |ϕAB ⟩as a linear map HA→ H∗
B.
The theory of restrictions (or SLOCC conversion) and
degenerations for 2-tensors is completely determined by
the rank.
With Lemma 4this directly yields the following:
Corollary 5. Let G= (V, E )be a hypergraph and let
|ϕ⟩Gand |ψ⟩Gbe entanglement structures for G. If
|ϕ⟩G⊵|ψ⟩G, then we must have that for any biparti-
tioning of the vertices V=A⊔Bthe rank along Aand
Bof |ϕ⟩Gis at least that of |ψ⟩G.
As an example, if we have the matrix multiplication
tensor for n×nmatrix multiplication (i.e. level-nEPR
pairs shared between three parties) and make the bipar-
titioning
A
B
C
n
n
n
then this has Schmidt rank n2between Aand BC. If
we compare with the 3-party GHZ state of level rwe get
Schmidt rank r, and therefore |GHZr(3)⟩ ≥ |EPRn⟩△
implies r≥n2, so
R(|EPRn⟩△)≥n2.(14)
This gives a lower bound on the matrix multiplication
exponent of ω≥2. While this lower bound is obvious,
it is highly nontrivial to find lower bounds improving
Eq. (14)!
Another example is the impossibility of degeneration
in Eq. (12), where we see that if we flatten, the GHZ5(3)
tensor has rank 5, while the two EPR pairs together have
rank 6.
In the algebraic complexity literature, grouping parties
such that we get a 2-tensor is often called a flattening,
since the resulting 2-tensor can be thought of as a matrix
(which is a two-dimensional array, as opposed to a k-
tensor, which is a k-dimensional array).
Let us now look at an example of an entanglement
structure on a lattice. We take a rectangular lattice of
size n1×n2with n1, n2even, with periodic boundary
conditions and rectangular plaquettes as in Fig. 1c. For
|ϕ⟩Gwe tile plaquettes with GHZ states on four parties,
so |ϕe⟩=|GHZr(4)⟩for each edge. For |ψ⟩Gwe place
maximally entangled pairs at the boundary of the pla-
quette, so |ψe⟩=|EPRD⟩□for each edge. We now take
the following bipartitioning:
A
B
It is easy to see that this gives Schmidt rank of |E|r
for |ϕ⟩Gand |E|D4for |ψ⟩G. Therefore, by Corollary 5,
|ϕ⟩G≥ |ψ⟩Gimplies that r≥D4. On the other hand,
it is easy to see that on the level of a single plaquette,
for r≥D4we have |GHZr(4)⟩ ≥ |EPRD⟩□. The lower
bound is sharp, and there is no gain from placing the
states on the lattice (i.e. in this case |ϕ⟩G≥ |ψ⟩Gis only
possible when |ϕe⟩ ≥ |ψe⟩).
A next observation is that if the folding fis such that
for some edge e=v1, . . . , vkall vertices are mapped to
the same vertex v′, then we may remove this edge for the
purpose of finding obstructions. To see this, let ˜
Hbe the
hypergraph with the same vertex set as H, and with the
edge f(e) removed from the edge set. An entanglement
structure |ϕ⟩Hthen defines an entanglement structure
|ϕ⟩˜
Hon ˜
Hby leaving out the state |ϕe⟩on the edge e
that gets removed. It holds that
|ϕ⟩H≥ |ψ⟩Hif and only if |ϕ⟩˜
H≥ |ψ⟩˜
H
and the same statement holds with degenerations rather
than restrictions. This is the case since |ϕe⟩is at a sin-
gle party v′, and we have the equivalences |ϕ⟩H∼
=|ϕ⟩˜
H
and |ψ⟩H∼
=|ψ⟩˜
H(since we can locally prepare the states
|ϕe⟩and |ψe⟩at v′). This is often helpful to reduce a
problem on a large hypergraph such as a lattice, to a
problem involving only a small number of edges. As a
19
special case one can show that if the hypergraph has no
cycles, |ϕ⟩G≥ |ψ⟩Himplies |ϕe⟩≥|ψe⟩for each edge (so
the graph structure does not allow additional transforma-
tions). Here a cycle is any path of vertices v1, v2, . . . , vk
for k≥2 such that vk=v1and for each i= 1, . . . , k = 1,
viand vi+1 are not equal and are connected by an edge,
so there is an edge ewith vi∈eand vi+1 ∈e. A hy-
pergraph is acyclic if it has no cycles (this notion is also
known as Berge-acyclic [102]). Note that in this defini-
tion if a graph is acyclic, any two different edges share at
most one vertex. If Gis an acyclic hypergraph, then it is
easy to see that for any edge ethere is a folding from G
to the hypergraph which consists just of the single edge
e. Thus, Lemma 4directly implies the following result.
Corollary 6. Let Gbe an acyclic hypergraph and let
|ϕ⟩Gand |ψ⟩Gbe entanglement structures for G. Then
|ϕ⟩G≥ |ψ⟩Gif and only if |ϕe⟩≥|ψe⟩for each edge
e∈E. Similarly, |ϕ⟩G⊵|ψ⟩Gif and only if |ϕe⟩⊵|ψe⟩
for each edge e∈E.
Finally, we consider the situation where we have a k-
uniform hypergraph Gwhich is such that there exists a
folding onto kvertices {v′
1, . . . , v′
k}, such that each edge e
gets mapped to {v′
1, . . . , v′
k}. This means that edges are
all folded on top of each other (and this is often possible
for a lattice). We will say that the hypergraph is foldable
in this case.
⇓
Consider entanglement structures |ϕ⟩Gand |ψ⟩Gwhere
we place the same k-party states |ϕ⟩and |ψ⟩at each
edge, and where we moreover assume that |ϕ⟩and |ψ⟩
are invariant under permutation of the kparties. Then
we directly see by folding that if |ϕ⟩G≥ |ψ⟩Gthen we
also find a restriction |ϕ⟩⊗|E|≥ |ψ⟩⊗|E|. In particular,
this implies the following result.
Theorem 7. Suppose Gis a foldable k-uniform hyper-
graph, and |ϕ⟩Gand |ψ⟩Gare entanglement structures
using permutation invariant k-party states |ϕ⟩and |ψ⟩.
Then if |ϕ⟩G⊵|ψ⟩Git holds that
|ϕ⟩≳|ψ⟩.
B. Folding and generalized flattenings
In this section we describe another useful way of group-
ing vertices in a hypergraph by folding some vertices onto
each other, leaving other vertices separate, leading to a
fan-like structure. Using this grouping, we can get ob-
structions for the conversion of entanglement structures
using the ideas of [76] about multiplicativity of rank lower
bounds obtained via generalized flattenings of tensors.
Let us first explain the idea of using generalized flat-
tenings of tensors. For convenience we will restrict to
the case of 3-tensors (although this is not crucial). Let
|ϕ⟩∈HA⊗ HB⊗ HC. A ‘usual’ flattening bound would
be to group two out of A,Band Ctogether and compute
the rank of the resulting 2-tensor (i.e. matrix) and use
this as a lower bound for the tensor rank. However, we
may also try to ‘split’ by some arbitrary linear map. Let
|ϕ⟩∈HA⊗ HB⊗ HCand let
P:HA⊗ HB⊗ HC→ HX⊗ HY
be a linear map. Then we say that the 2-tensor
|ϕ(P)⟩=P|ϕ⟩ ∈ HX⊗ HY
is a generalized flattening of |ϕ⟩.
We need to take into account the map Pwhen we using
this relation to bound the rank. To this end we define
the commutative rank to be the maximum rank in the
image of Pof any rank-1 tensor
CR(P) = max
R(|ψABC ⟩)=1 rk(P(|ψABC ⟩))
so we maximize over
|ψABC ⟩=|ψA⟩ |ψB⟩ |ψC⟩ ∈ HA⊗ HB⊗ HC.
It is then easy to see that one can bound the (border)
rank as follows (see e.g. [76]):
R(|ϕ⟩)≥rk(|ϕ(P)⟩)
CR(P).
Lower bound methods of this type have been studied
in [103,104] where it has been shown that the power
of such bounds when applied to the border rank of the
matrix multiplication tensor is limited.
An important special case is where one applies a linear
map PC:HC→ HX′⊗ HY′only to the C-system, and
let P=
1
AB ⊗PCand group together AX′as Xand
BY ′as Y, so
|ϕ(P)⟩= (
1
AB ⊗PC)|ϕ⟩.
⇒
A
B
PC
X′
Y′
A
B
X′
Y′
|ϕ(P)⟩
The most useful generalized flattenings for (border)
rank bounds are the Koszul flattenings arising from
the antisymmetrization map viewed as a linear map
PC:HC→ΛpH∗
C⊗Λp+1HC. In this case CR(P) = c−1
p
where c= dim HC. There are also the more general
Young flattenings, based on other representations of the
20
symmetric group. See [105] for extensive discussion of
Young flattenings and associated lower bounds.
Here, we generalize this to the case where the system
Cis replaced by multiple systems, each of which we split
up. Let
|ϕ⟩ ∈ HA⊗ HB⊗
m
O
i=1
HCi
and let Pi:HCi→ HXi⊗ HYibe linear maps. Then the
2-tensor obtained by grouping together AX1. . . Xmand
BY1. . . Ymfrom
|ϕ(P)⟩= (
1
AB ⊗P1⊗. . . ⊗Pm)|ϕ⟩
| {z }
∈HA⊗HB⊗Nm
i=1 HXi⊗HYi
is a generalized multiflattening of |ϕ⟩.
Lemma 8.
R(|ϕ⟩)≥rk(|ϕ(P)⟩)
Qm
i=1 CR(Pi).
Proof. Suppose that
|ϕ⟩=
r
X
i=1
|ai⟩ |bi⟩ |c1,i⟩. . . |cm,i ⟩
|{z }
:=|ψi⟩
so R(|ϕ⟩)≤r. Then |ϕ(P)⟩can be written as a sum of
terms |ψ(P)
i⟩. Now, it is clear that since |ψi⟩is a product
tensor
rk(|ψ(P)
i⟩)≤
m
Y
i=1
CR(Pi)
and therefore
rk(|ϕ(P)⟩)≤
r
X
i=1
rk(|ψ(P)
i⟩)
≤r
m
Y
i=1
CR(Pi)
so we get
R(|ϕ⟩)≥rk(|ϕ(P)⟩)
Qm
i=1 CR(Pi).
By semicontinuity of the rank the same inequality holds
for tensors |ϕ⟩with R(|ϕ⟩) = r.
Next, we will combine this idea with a folding. Con-
sider some hypergraph G, which we assume to be 3-
uniform. We assume that we can divide the vertices
together into subsets A,Band a collection of sub-
sets Ciin such a way that the resulting folded hyper-
graph has a fan structure, meaning that there are edges
{A, B, Ci}, but no edges involving multiple Ci. We
will denote by fan(n) the fan hypergraph, with vertices
V={A, B, C1, . . . , Cm}and edges ei={A, B , Ci}.
A
B
C1
C2
C3
C4
The fan hypergraph is useful for two reasons. First
of all, many interesting lattice graphs can be folded to
a fan. Secondly, the generalized multiflattenings behave
well with respect to the fan structure!
Lemma 9. Let |ϕ⟩fan(m)be an entanglement structure
with state |ϕi⟩on edge {A, B, Ci}. Then for any gener-
alized flattening maps Pi:HC→ HC⊗ HYwe have
R(|ϕ⟩G)≥
m
Y
i=1
rk(|ϕ(P)
i⟩)
CR(Pi)
Proof. If |ψ⟩=|ϕ⟩fan(m)we see that |ψ(P)⟩is a tensor
product of the states |ϕPi
i⟩, so
rk(|ϕ⟩(P)
fan(m)) =
m
Y
i=1
rk(|ϕ(P)
i⟩).
We conclude by Lemma 8.
This means that once we have folded to a fan, we can
apply generalized flattening bounds on individual pla-
quettes. Two important examples where we can fold to a
fan are the triangular lattice and the kagome lattice, see
Fig. 7. We denote by kagome(n) and tri(n) the kagome
and triangular lattice with nedges.
From these foldings and Lemma 4we see that
Lemma 10. Suppose that |ϕ⟩kagome(2n)≥ |ψ⟩kagome(2n),
then
|ϕ⊗2⟩fan(n)≥ |ψ⊗2⟩fan(n).
If |ϕ⟩tri(6n)≥ |ψ⟩tri(6n), then
|ϕ⊗6⟩fan(n)≥ |ψ⊗6⟩fan(n).
The same statements are true using degenerations or
asymptotic restrictions.
We can use this to prove obstructions to entanglement
structure transformations, where we consider transfor-
mations from an entanglement structure where we place
|GHZr(3)⟩states on each plaquette to an entanglement
structure with some 3-party state |ϕ⟩at each plaque-
tte. To make the question more concrete, let us take the
kagome lattice, and study the conversion of |GHZr(3)⟩
states to |EPRD(3)⟩states. The ‘naive’ flattening bound,
by just taking the rank at a single vertex, gives
r2≥D4.(15)
Asymptotic lower bounds do not perform better: if we
use Theorem 7, we see that we need
|GHZr(3)⟩≳|EPRD⟩△,
but since the best known lower bound for the matrix
multiplication exponent is ω≥2, we again get r≥D2.
21
A
B
=
A
B
Ci
.
.
.
A
B
C1
C2
C3
C4
(a) Folding a kagome lattice into a fan.
A
B
=
A
B
Ci
.
.
.
A
B
C1
C2
C3
C4
(b) Folding a triangular lattice into a fan.
Figure 7: Foldings of two lattices with 3-edges. For the
kagome lattice, the fan has plaquettes which consist of
two of the original plaquettes, whereas for the
triangular lattice each fan plaquette consists of six
copies of the original plaquette.
Theorem 11. Suppose that
|GHZr(3)⟩kagome(n)⊵|ϕ⟩kagome(n),
then for any generalized flattening we must have
r2≥rk((|ϕ⟩⊗2)(P))
CR(P).
In particular, if |ϕ⟩=|EPRD⟩△we must have
r2≥2D4−D2.
Similarly, if
|GHZr(3)⟩tri(n)⊵|ϕ⟩tri(n),
then for any generalized flattening we must have
r6⊵rk((|ϕ⟩⊗6)(P))
CR(P).
If |ϕ⟩=|EPRD⟩△we must have r6≥2D12 −D6.
Proof. First of all, note that if we have nedges on an
arbitrary 3-regular hypergraph G
R(|GHZr(3)⟩G)≤rn
and |ψ⟩G⊵|ϕ⟩Gimplies R(|ψ⟩G)≥R(|ϕ⟩G). On the
other hand, by Lemma 8and Lemma 9, for any general-
ized flattening P
R(|ϕ⟩fan(n))≥ rk((|ϕ⟩⊗2)(P))
CR(P)!n
.
The statement about a general 3-party state |ϕ⟩now fol-
lows from Lemma 10.
For the special case where |ϕ⟩=|EPRD⟩△, using
Koszul flattenings one obtains [106] (see also [105])
|GHZr(3)⟩⊵|EPRD⟩△⇒r≥2D2−D.
Since this bound is obtained by a generalized flattening,
this implies the desired result.
For instance, on the kagome lattice, if D= 2 this re-
quires r≥6 while Eq. (15) only gives r≥4, and for
D= 3 this gives r≥13, while Eq. (15) only requires
r≥9. On the triangular lattice, if D= 2 this requires
r≥5 and for D= 3 this requires r≥11 (while again,
Eq. (15) only gives r≥4 and r≥9 respectively).
C. Folding and the substitution method
Another important method for proving obstructions
(i.e. lower bounds in algebraic complexity) is the sub-
stitution method. We start by explaining the idea of the
substitution method, and for convenience we again re-
strict to 3-tensors. Suppose we have states |ϕABC ⟩and
|ψABC ⟩on three parties A,Band Cand suppose there
exists a restriction
(MA⊗MB⊗MC)|ϕABC ⟩=|ψABC ⟩.
In the substitution method we observe that we may ‘sub-
stitute’ any |x⟩∈HAin the first factor and get a restric-
tion
(⟨x|MA⊗MB⊗MC)|ϕABC ⟩= (⟨x| ⊗
1
BC )|ψABC ⟩.
Note that if we think of the tensor as a map from HA
to HB⊗ HCthis corresponds to substituting some value
in the map. This is typically used for the case where
|ϕABC ⟩is a GHZ state and we want to compute the rank
R(|ψ⟩).
As a concrete example, let us consider the Wstate on
three parties
|W(3)⟩=|100⟩+|010⟩+|001⟩
and bound its rank. It is clear that there exists a restric-
tion |GHZ3(3)⟩≥|W(3)⟩, since |W(3)⟩is defined as a
22
sum of three product tensors. So, R(|W(3)⟩)≤3. Can
we do better? The substitution method shows us that
this is not the case. Let |x⟩=|0⟩+x|1⟩be nonzero, with
x∈C, then applying the substitution we find that
|W(x)(3)⟩=|01⟩+|10⟩+x|00⟩.
Note that R(|W(x)(3)⟩) = rk(|W(x)(3)⟩)≥2 for all
choices of |x⟩. On the other hand, suppose that we have
a restriction |GHZ2(3)⟩≥|W(3)⟩, then we have
|W(3)⟩=X
i=0,1
|ai⟩ |bi⟩ |ci⟩.
At least one of the |ai⟩must not be proportional to |0⟩,
so we can choose |x⟩such that ⟨x|ai⟩= 0. But this would
imply that |W(x)(3)⟩is a product state, which is a con-
tradiction. Therefore, there does not exist a restriction
|GHZ2(3)⟩≥|W(3)⟩and R(W(3)) = 3.
There does exist a degeneration |GHZ2(3)⟩⊵|W(3)⟩,
given by
(|0⟩+ε|1⟩)⊗3− |0⟩⊗3=ε|W(3)⟩+O(ε2)
so R(|W(3)⟩) = 2 and we see that the substitution
method is able to distinguish degenerations from restric-
tions (which is not the case for flattening ranks).
Here we will apply a more complicated version of the
substitution method to prove that on the kagome lattice
or a triangular lattice there is no restriction from the en-
tanglement structure using |GHZ2(3)⟩states to the en-
tanglement structure using |W(3)⟩states. This is not
a direct consequence of the substitution method above,
as it could be the case that the transformation is not
possible on a single plaquette, but is possible on the lat-
tice. Note that we have |GHZ2(3)⟩⊵|W(3)⟩and hence
|GHZ2(3)⟩≳|W(3)⟩.
We will use a folding, where we choose one specific
plaquette (because of translation invariance of the lattice
it does not matter which one). Let us denote the three
vertices adjacent to this plaquette a,band c. In the
folding we choose we map ato A,bto Band all other
vertices to C. We may apply a similar folding to a (half-
filled) triangular lattice, see Fig. 8. For the remainder of
this section we let Geither be this lattice or the kagome
lattice.
For the structure with |GHZ2(3)⟩this gives a state
equivalent to the three parties sharing a |GHZ2(3)⟩
state, and level-2 EPR pairs between Aand C, and be-
tween Band C, a state we denote by |ϕABC ⟩the state
|GHZ2(3)⟩⊗|EPR2⟩∧. Similarly, for the entanglement
structure using the |W(3)⟩state we get a state equiva-
lent to the three parties sharing a Wstate, and level-2
EPR pairs between Aand C, and between Band C,
which we denote by |ψABC ⟩=|W(3)⟩ ⊗ |EPR2⟩∧. We
label the parties as follows:
A1
A2
B1
B2
C2C1C3
A1
A2
B1
B2
C2C1C3
2W
|ϕ⟩ |ψ⟩
A
B
C
(a) Folding a kagome lattice to a single plaquette.
A
B
C
(b) Folding a triangular lattice to a single plaquette.
A
B
C
(c) Folding a kagome lattice with boundary (so here
the plaquette neighboring Aand Bis at the boundary
of the lattice).
Figure 8: Foldings of two lattices with 3-edges. For the
kagome lattice, the fan has plaquettes which consist of
two of the original plaquettes, whereas for the
triangular lattice each fan plaquette consists of six
copies of the original plaquette.
Theorem 12. There is no restriction from |ϕABC ⟩to
|ψABC ⟩.
By the folding argument Theorem 12 has as a direct
consequence that the entanglement structure with the W
state cannot be obtained from the entanglement struc-
ture with GHZ2(3) states on the triangular or kagome
lattice G.
Corollary 13.
|GHZ2(3)⟩G≱|W(3)⟩G.
Proof of Theorem 12.We assume that there exists a re-
striction
(MA⊗MB⊗MC)|ϕABC ⟩=|ψABC ⟩(16)
and derive a contradiction. In this case, the tensors are
all concise (their reduced density matrices have full rank)
23
and since the systems on both sides are of equal dimen-
sion, the maps MA,MBand MCmust be invertible. We
will use a version of the substitution method to show that
this is not possible.
For |x⟩ ∈ (C2)⊗2, which is the Hilbert space of the
A=A1A2system for both |ϕABC ⟩and |ψAB C ⟩, let
|ϕ(x)⟩= (⟨x| ⊗
1
BC )|ϕABC ⟩
2
x
|ϕ(x)⟩
and similarly for |ψ(x)
ABC ⟩. These are 2-tensors on systems
Band C. We then define
Xϕ={|x⟩such that rk(|ϕ(x)⟩)≤2}
Xψ={|x⟩such that rk(|ψ(x)⟩)≤2}.
Since the maps MA,MBand MCare invertible, it is easy
to see that
|x⟩ ∈ Xψ⇔ |y⟩=M†
A|x⟩ ∈ Xϕ.
Indeed,
|ψ(x)⟩= (⟨x| ⊗
1
BC )(MA⊗MB⊗MC)|ϕ⟩
= (MB⊗MC)|ϕ(y)⟩
so rk(|ψ(x)⟩) = rk(|ϕ(y)⟩). So, we see that (if a restriction
exists) the sets Xϕand Xψmust be related by a linear
transformation. Now, we note that for any |x⟩,|ϕ(x)⟩and
|ψ(x)⟩still share an EPR pair between Band C. Denote
by |˜
ϕ⟩and |˜
ψ⟩the states where we have left out the EPR
pair between Band C, so
A1
A2
B1
C2C1
A1
A2
B1
C2C1
2W
|˜
ϕ⟩ | ˜
ψ⟩
Now, for x∈Xϕwe let
|˜
ϕ(x)⟩= (⟨x| ⊗
1
BC )|˜
ϕ⟩
which is just |ϕ(x)⟩without the EPR pair between Band
C, and therefore has rank rk(|ϕ(x)⟩) = 2 rk(|˜
ϕ(x)⟩)≤2 so
rk(|˜
ϕ(x)⟩)≤1. The state |x⟩has either rank 1 or 2 as a
state on A1A2, so we can write a Schmidt decomposition
|x⟩=
r
X
i=1
|fi⟩A1|ei⟩A2
with r= 1 or r= 2. Then,
|˜
ϕ(x)⟩=
r
X
i=1 ⟨fi|0⟩ |0⟩B1|0⟩C1+⟨fi|1⟩ |1⟩B1|1⟩C1|ei⟩C2
which has rank at least r. So, rk(|˜
ϕ(x)⟩)≤1 implies
that |x⟩as a state on A1A2must have rank 1. So, write
|x⟩=|f⟩A1|e⟩A2with |f⟩=f0|0⟩+f1|1⟩. Then
|˜
ϕ(x)⟩=¯
f0|0⟩B1|0⟩C1+¯
f1|1⟩B1|1⟩C1|e⟩C2
This has rank 1 if and only f0= 0 or f1= 0, so if |x⟩
can be written as
|x⟩=|0⟩ |e⟩or |x⟩=|1⟩ |e⟩
for |e⟩ ∈ C2. In other words, Xϕis a union of two complex
planes. For |ψ⟩the same reasoning holds, but with the
difference that |˜
ψ(x)⟩is given by
¯
f0(|0⟩B1|1⟩C1+|1⟩B1|0⟩C1) + ¯
f1|0⟩B1|0⟩C1|e⟩C2
which has rank 1 if and only if f0= 0, and Xψconsists
of all vectors
|x⟩=|1⟩ |e⟩
for |e⟩ ∈ C2. As Xϕis a union of two planes, and Xψis
a single plane, they can not be related by an invertible
linear map MA, and we conclude that no restriction as
in (16) exists.
In this case, the proof was relatively straightforward
due to the fact that the restriction had to be invertible
(since the tensors were concise on Hilbert spaces of the
same dimension). However, the proof is instructive since
the same strategy can be used in a more general setting,
where the map MAneed not be invertible.
We use this more general approach to show that there
is no restriction from the entanglement structure using
EPR-pairs (so placing |EPR2⟩△, the 2 ×2 matrix multi-
plication tensor on each plaquette) to the entanglement
structure using the λ-tensor
|λ⟩=
2
X
i,j,k=0
ϵijk |ijk⟩+|222⟩
where ϵijk is the antisymmetric tensor. It is already
known [58] that on the level of a single plaquette there is
no restriction
|EPR2⟩△≱|λ⟩(17)
while there does exist a degeneration
|EPR2⟩△⊵|λ⟩.
Hence, there are also asymptotic restrictions |EPR2⟩△≳
|λ⟩and a lattice degeneration |EPR2⟩G⊵|λ⟩G. Since
we need to be able to separate a degeneration from a
restriction, the substitution method is again a natural
choice in this case.
We let Gbe the kagome lattice (or the half-filled trian-
gular lattice), but without periodic boundary conditions.
24
We perform the folding at the boundary of the lattice;
see Fig. 8c
In this case, for |λ⟩Gwe get a state equivalent to
the three parties sharing one copy of |λ⟩and a level-3
EPR pair between Band C. We denote this state by
|λ⟩⊗|EPR3⟩BC . For the state |EPR2⟩Gwe get a state
where there are level-2 EPR pairs between Aand Band
between Aand Cand a level-8 EPR pair between Band
C. One can think of this as |EPR2⟩△⊗ |EPR4⟩BC So,
we would like to show
≱
A B
C
A B
C
λ
2 8
2
3
Theorem 14. There does not exist a restriction from
|EPR2⟩△⊗ |EPR4⟩BC to |λ⟩⊗|EPR3⟩BC .
The proof (based on the substitution method and gen-
eralizing the ideas in Theorem 12) can be found in Ap-
pendix A. The folding argument implies that Theorem 14
has as a direct consequence that the entanglement struc-
ture with the λ-tensor can not be represented with bond
dimension 2 on any kagome lattice Gwith a boundary,
solving a main open problem from [58].
Corollary 15.
|EPR2⟩G≱|λ⟩G
We conjecture that this should also not be possible
with periodic boundary conditions, and explain how our
methods could be used for this in Appendix A.
D. Asymptotic restrictions and the asymptotic
spectrum
We already saw that at least for foldable hypergraphs
(and permutation-invariant states), the non-existence of
asymptotic restrictions is an obstruction to the exis-
tence of entanglement structure conversions. We will
now study in more depth the asymptotic conversion of
entanglement structures |ϕ⟩G≳|ψ⟩Gas in Section IV C.
There is a well-developed theory for asymptotic conver-
sion of tensors. It turns out that whether or not there ex-
ists an asymptotic conversion between two tensors can be
decided by evaluating a compact set of appropriate func-
tionals on the tensors, known as the asymptotic spectrum
of tensors. Suppose we consider k-tensors. Then we de-
note by X(k) the set of all functions on non-normalized
quantum states (i.e. tensors)
f:{k-party quantum states} → R≥0
which are such that they are
1. monotone under restriction, so |ϕ⟩ ≥ |ψ⟩implies
f(|ϕ⟩)≥f(|ψ⟩)
2. multiplicative under tensor products, so f(|ϕ⟩ ⊗
|ψ⟩) = f(|ϕ⟩)f(|ψ⟩)
3. additive under direct sums, so f(|ϕ⟩⊕|ψ⟩) =
f(|ϕ⟩) + f(|ψ⟩)
4. normalized by f(|GHZr(k)⟩) = r.
This collection of functionals X(k) can be given the struc-
ture of a compact topological space and is known as the
asymptotic spectrum of tensors. A function f∈ X (k) is
apoint in the asymptotic spectrum. A cornerstone result
of the algebraic theory of tensors is the following theorem
by Strassen [84,107].
Theorem 16. Let |ϕ⟩and |ψ⟩be k-tensors. We have an
asymptotic restriction |ϕ⟩≳|ψ⟩if and only if f(|ϕ⟩)≥
f(|ψ⟩)for all f∈ X (k).
This gives a complete characterization of asymptotic
restriction. However, we do not have complete knowledge
of the asymptotic spectrum and only know a few points
(note that if one could evaluate all f∈ X (3) this would
allow one directly to compute the matrix multiplication
exponent ω). We will relate the asymptotic spectrum to
asymptotic tensor network conversions.
Given f∈ X (n) we can restrict to functionals which
only act on a subset of the nparties. So, we assume we
have nparties X={x1, . . . , xn}. Let k < n and let Ybe
a subset of Xof size |Y|=k. Given a state |ϕY⟩on the
kparties Yon a Hilbert space HY=Nx∈YHx, define
|ϕY⟩X=|ϕY⟩ ⊗ O
x∈X\Y
|0⟩.
For any f∈ X (n) we define a nonnegative function fY
on kparties by
fY(|ϕY⟩) = f(|ϕY⟩X).
Lemma 17. The functions fYare monotone under re-
striction and multiplicative under tensor products. More-
over,
fY(|ϕ⟩⊕|ϕ⟩) = αfY(|ϕ⟩)
for
α=fY(|GHZ2(k)Y⟩).
Proof. First of all, fYis monotone under restriction and
multiplicative under tensor products. Indeed, a restric-
tion |ϕY⟩≥|ψY⟩implies |ϕY⟩X≥ |ψY⟩X, so using the
monotonicity of fwe have fY(|ϕY⟩)≥fY(|ψY⟩). Simi-
larly, (|ϕY⟩⊗|ψY⟩)X=|ϕY⟩X⊗ |ψY⟩X, so
fY(|ϕY⟩⊗|ψY⟩) = f(|ϕY⟩X⊗ |ψY⟩X)
=f(|ϕY⟩X)f(|ψY⟩X)
=fY(|ϕY⟩)fY(|ψY⟩).
25
In general, fYneed not be additive or normalized. For
any |ϕY⟩we have
(|ϕY⟩⊕|ϕY⟩)X∼
=(|ϕY⟩⊗|GHZ2(k)Y⟩)X
and hence, by multiplicativity
fY(|ϕY⟩⊕|ϕY⟩) = fY(|ϕY⟩)fY(|GHZ2(k)Y⟩).
While we do not necessarily get additivity, for the fY,
multiplicativity and monotonicity are in some sense the
most important properties of the points in the asymptotic
spectrum, since these properties directly relate to the
notion of asymptotic restriction. We will apply this to
the case where we have a graph G, so we may consider
the asymptotic spectrum X(n) where n=|V|. Then as
in Lemma 17 we get functions fefor each edge e.
Theorem 18. Suppose Gis a hypergraph with |V|=n
and |ϕ⟩Gand |ψ⟩Gare entanglement structures. Then
|ϕ⟩G≳|ψ⟩Gif and only if for all f∈ X (n)
Y
e∈E
fe(|ϕe⟩)≥Y
e∈E
fe(|ψe⟩)
Proof. By Theorem 16 we have |ϕ⟩G≳|ψ⟩Gif and only
if
f O
e∈E
|ϕe⟩!≥f O
e∈E
|ϕe⟩!
for all f∈ X (n). By definition, f∈ X (n) is multiplica-
tive under tensor products. This gives
f O
e∈E
|ϕe⟩!=Y
e∈E
fe(|ϕe⟩)
and similarly for |ψ⟩G.
Note that in Theorem 18 the functions feare not inde-
pendent, as they have to derive from a global f∈ X (n).
So, the condition in Theorem 18 is not equivalent to
fe(|ϕe⟩)≥fe(|ψe⟩) for all edges. This makes sense, as
in general |ϕ⟩G≳|ψ⟩Gdoes not imply conversion on the
level of individual plaquettes, as in the basic example
|ϕ⟩ |ψ⟩⇒
In this case |ϕ⟩Gand |ψ⟩Gare actually equivalent
(and therefore asymptotically equivalent), so we have
f(|ϕ⟩G) = f(|ψ⟩G) for all f∈ X (n). On the other hand,
an appropriate rank functional clearly distinguishes sin-
gle plaquettes |ϕe⟩and |ψe⟩.
To be able to make use of Theorem 18, we need to
know concrete points in the asymptotic spectrum. Easy
examples are given by ranks across bipartitions of the
parties. This of course is equivalent to previous obstruc-
tions obtained by folding. However, there exists a broad
class of examples (in fact encompassing all known ex-
amples), which are the so-called quantum functionals as
introduced in [108]. These are defined as follows. For a k-
particle quantum state |ϕ⟩we denote by ρ(ϕ)
ithe (mixed)
quantum state resulting from tracing out all but the i’th
system of the normalized state
ρ(ϕ)=1
⟨ϕ|ϕ⟩|ϕ⟩⟨ϕ|.
The von Neumann entropy of the reduced state ρ(ϕ)
iis
given by
H(ρ(ϕ)
i) = −trhρ(ϕ)
ilog ρ(ϕ)
ii
For a probability distribution θ={θ1, . . . , θn}the cor-
responding quantum functional is defined as Fθ= 2Eθ
with
Eθ(|ϕ⟩) = sup
|ϕ⟩⊵|ψ⟩
n
X
i=1
θiH(ρ(ψ)
i).
The set that one optimizes over is closely related to the
entanglement polytope of |ϕ⟩, which is the set of all spec-
tra of ρ(ψ)
ithat can be obtained from a degeneration
|ϕ⟩⊵|ψ⟩[109]. A dual characterization of the entan-
glement polytope is crucial in showing that the quantum
functionals are multiplicative.
As an example application, consider again the Wstate
on three parties. The Wstate does not asymptotically
restrict to the level-2 GHZ state [108] which can be seen
by computing the quantum functional using the uniform
distribution θ={1
3,1
3,1
3}, in which case
Fθ(|GHZ2(3)⟩)=2> Fθ(|W(3)⟩)=2H({1
3,2
3})≈1.89.
The quantum functionals behave especially nice when
applied as in Theorem 18. In this case, choose a distri-
bution θ={θv:v∈V}. We now see that (by multi-
plicativity)
Eθ(|ϕ⟩) = X
e∈EX
v∈V
Eθe(|ϕe⟩)
=X
e∈E
ΘeEθ(e)(|ϕe⟩)
where
Θe=X
v∈e
θvand θ(e)={θ(e)
v=θv
Θe
:v∈e}.
So, Eθ(|ϕ⟩) is a convex combination of quantum func-
tionals Eθ(e)(|ϕe⟩). In particular, if in Theorem 18 we
26
take f=Fθ, then the normalized restriction to egiven
by fΘ−1
e
eis again a quantum functional. This is stronger
than Lemma 17, as it implies that (after normalization)
we also have additivity of the restricted functional.
If we have a graph (so a 2-uniform hypergraph) with
nvertices and we place maximally entangled states of
bond dimension Deon each edge giving an entanglement
structure |ϕ⟩G, we find that taking the uniform distribu-
tion θwith θv=1
nfor v∈V, the quantum functional
gives
Fθ(|ϕ⟩G) = Y
e∈E
D2
n
e
and more generally for an arbitrary distribution θwe find
Fθ(|ϕ⟩G) = Y
e=(vw)∈E
Dθv+θw
e.
This can be used to lower bound the asymptotic bond
dimension required for some entanglement structure of
interest.
As an example application, the quantum functionals
allow one to conclude that on any hypergraph (so it does
not have to be foldable), we can not have an (asymptotic)
conversion of an entanglement structure using Wstates
to one using GHZ2(3) states. A small example of such a
graph which can not be folded is
as can be easily seen.
Theorem 19. Let Gbe any 3-uniform hypergraph, and
let |W⟩Gand |GHZ2(3)⟩Gbe the entanglement struc-
tures where each edge is assigned a Wstate or GHZ2(3)
state respectively. Then there is no asymptotic restric-
tion |W⟩G≳|GHZ2(3)⟩G.
In particular, there is also no restriction, so |W⟩G≱
|GHZ2(3)⟩G.
Proof. For the uniform distribution over the vertices, we
find that each θ(e)is a uniform distribution, and therefore
Fθ(|GHZ2(3)⟩G)> Fθ(|W⟩G)
which implies there is no asymptotic restriction.
VI. SYMMETRIES AND RANKS OF
ENTANGLEMENT STRUCTURES
We end our investigation of the resource theory of ten-
sor networks by studying some structural properties of
entanglement structures. An important feature of tensor
networks is the so-called gauge symmetry which implies
that the same quantum state can have equal bond di-
mension representations with different tensors applied.
This is a crucial ingredient in relating tensor network
states to phases of matter. Here, we will introduce this
gauge symmetry for arbitrary entanglement structures.
This gauge symmetry has rather different properties than
in the standard tensor network formalism. For one, the
gauge group is potentially a finite group. Secondly, we
show that there are entanglement structures which have
gauge symmetries which act on multiple edges. We show
that this does not happen for acyclic hypergraphs. After
this we turn to the tensor rank of entanglement struc-
tures. We work out a nontrivial example, solving an open
question in [80].
A. Stabilizers of entanglement structures
We will now turn our attention to the stabilizer group,
or gauge group, of entanglement structures. Given an
entanglement structure with Hilbert spaces Hvof dimen-
sion dvat vertex v∈V, we have the group Qv∈VGL(dv)
which acts on |ϕ⟩Gby
(gv)v∈V· |ϕ⟩G= O
v∈V
gv!|ϕ⟩G.
Then we define the stabilizer of the entanglement
structure as
Stab(|ϕ⟩G) ={(gv∈GL(dv))v∈V
such that (gv)v∈V· |ϕ⟩G=|ϕ⟩G}.
A well-known example is the situation where we have
maximally entangled states along edges. If we have a
maximally entangled state of level Dat some edge, we
may act with g∈GL(D) on one side of the edge and
with g−T= (g−1)Ton the other side.
g g−T
This gives an element in the stabilizer since
(g⊗g−T)
D
X
i=1
|ii⟩=
D
X
i=1
|ii⟩.
This gauge symmetry is widely studied in the context
of tensor networks and is an essential ingredient in the
classification of phases and symmetries using the tensor
network formalism [18]. Note that it implies that for a
tensor network state with tensors Tv, there exist transfor-
mations on the tensors Tvwhich change the local tensors
but do not change the resulting tensor network state.
More generally, given an arbitrary entanglement struc-
ture we get elements in the stabilizer by looking at the
stabilizers for single edges, that is, if e={v1, . . . , vk}we
may look at ge,vi∈GL(He,v) such that
(ge,v1⊗. . . ⊗ge,vk)|ϕe⟩=|ϕe⟩.
27
Such ge,viis called an edge stabilizer. Then it is clear
that
(gv)v∈V, gv=O
e:v∈e
ge,v
is an element of Stab(|ϕ⟩G). In the special case of an EPR
pair, this group was a continuous group, but in general
this group can be finite [110].
An interesting question is whether all symmetries of
the entanglement structure arise as a symmetry of the
edges states. In other words, the question is whether
every element of Stab(|ϕ⟩G) arises as a product of edge
stabilizers. The answer is no as the following example
shows. Consider the hypergraph Gconsisting of two 3-
edges with two common vertices. As entanglement struc-
ture we place a Wstate on both edges. For q∈C, let gq
be defined by
g(q):(C2)⊗2→(C2)⊗2
g(q) =
1
+q|00⟩ ⟨11|.(18)
This is clearly not a tensor product operator for q= 0.
We now apply g(q) and g(−q) to the vertices of degree 2
g(q)
g(−q)
W W
This stabilizes |W⟩G, but is not a product of edge stabi-
lizers.
On the other hand, for acyclic hypergraphs we will
show that the stabilizer is just the product of the edge
stabilizers.
Lemma 20. Suppose Ghas two edges e1and e2, sharing
a single vertex v. Let |ϕ⟩G=|ϕe1⟩ |ϕe2⟩and |ψ⟩G=
|ψe1⟩ |ψe2⟩be an entanglement structure with |ϕ1⟩and
|ϕ2⟩concise, and suppose that Mvis such that
(Mv⊗
1
V\v)|ϕ⟩G=|ψ⟩G.
Then there exist Miacting only on Hei,v such that M=
M1⊗M2.
So, if we have
=
M
ϕe1ϕe2ψe1ψe2
then
M
=
M1M2
Proof. We group together all vertices in respectively e1
and e2into parties Aand B. Then we consider Schmidt
decompositions
|ϕ1⟩=
r1
X
i=1
si|ai⟩ |ei⟩ |ϕ2⟩=
r1
X
i=1
ti|bi⟩ |fi⟩
where the |ei⟩and |fi⟩are orthonormal bases of He1,v and
He1,v (by conciseness). Now, the map Mvis completely
defined by
Mv|ei⟩ |fj⟩=
1
v⊗ ⟨ai| ⟨bj|(Mv⊗
1
V\v)|ϕ⟩G
=
1
v⊗ ⟨ai| ⟨bj||ψ⟩G
=(
1
v⊗ ⟨ai|)|ψ1⟩(
1
v⊗ ⟨bj|)|ψ2⟩.
This is again a tensor product state, so we see that we
may take
M1=X
i
1
v⊗ ⟨ai||ψ1⟩
M2=X
j
1
v⊗ ⟨bj||ψ2⟩
Theorem 21. If Gis an acyclic hypergraph and each
|ϕe⟩is concise, all elements of Stab(|ϕ⟩G)are products
of edge stabilizers.
Proof. Take (gv)v∈V∈Stab(|ϕ⟩G). Pick an arbitrary
vertex v∈V, and divide V\ {v}into two groups V1
and V2such that there are no edges between V1and V2
(this is possible since the hypergraph is acyclic). Let
Eibe the set of edges efor which e⊂Ei∪ {v}. The
disjoint union of E1and E2is the full edge set E. Now,
define a hypergraph Hwith vertex set Vand edge set
F={f1, f2}, where fi=Vi∪ {v}. Define |ϕ⟩Hby
|ϕfi⟩=O
e∈Ei
|ϕe⟩.
That is, we coarse-grain the hypergraph ss
v
f1f2
Let
|ψ⟩H=
1
v⊗
O
v∈V\{v}
gv
|ϕ⟩H.
Note that |ψ⟩His again an entanglement structure on H.
Then, (gv⊗
1
V\{v})|ϕ⟩H=|ψ⟩H, and by Lemma 20 there
28
exist gv,1, gv,2such that gv=gv,1⊗gv,2. Since we showed
this for an arbitrary vertex and arbitrary assignment of
edges, we conclude that each gvcan be written as a tensor
product over the adjacent edges
gv=O
e:v∈e
ge,v.
This decomposition is unique up to a choice of phases.
After an appropriate choice of phases, in order for
(gv)v∈Vto be a stabilizer of |ϕ⟩Gwe have
O
v∈e
ge,v!|ϕe⟩
and we conclude that (gv)v∈V∈Stab(|ϕ⟩G) is a product
of edge stabilizers.
This result supplements Corollary 6, which shows that
on an acyclic graph |ϕ⟩G≥ |ψ⟩Gimplies |ϕe⟩≥|ψe⟩for
each edge e∈E.
B. Ranks of entanglement structures
We will end by making some observations regarding
the tensor rank of entanglement structures. Recall that
for a quantum state |ϕ⟩on kparties we call
R(|ϕ⟩) = min (r:|ϕ⟩=
r
X
i=1
|ui1⟩. . . |uik⟩, uij ∈Uj)
the rank of |ϕ⟩. In other words, it is the minimal num-
ber of product states whose linear span contains |ϕ⟩, or
it is the minimal rsuch that there exists a restriction
GHZr(k)≥ |ϕ⟩. In the case of 3-tensors it is closely
related to the complexity of computing bilinear opera-
tions (e.g. the complexity of matrix multiplication is
related to the tensor rank of |EPRn⟩△as discussed in
Section II B). The rank of a quantum state is a natural
measure of multipartite entanglement (as it is the entan-
glement cost from GHZ states under SLOCC) [111], and
it is interesting to study this measure for entanglement
structures.
Given a hypergraph Gand entanglement structure
|ϕ⟩Gwith state of |ϕe⟩on the plaquettes of Git is inter-
esting to compare the ranks of the individual |ϕe⟩with
the rank of the entanglement structure |ϕ⟩G. It is imme-
diate that
R(|ϕ⟩G)≤Y
e∈E
R(|ϕe⟩).(19)
This inequality can be an equality but is not so in general
[76]. The same is true for the border rank [81]. The
tensor rank R(|ϕ⟩G) is well-studied in the special case
where Gis a graph and the |ϕe⟩are two-party states
(and then we may assume without loss of generality they
are EPR pairs). For example, when Ghas no cycles, one
has equality in Eq. (19). More generally, it is easy to
see that we have equality if the graph is bipartite [83],
since we get a sharp flattening rank lower bound from
the bipartitioning of the graph. This for instance allows
one to compute the rank of an entanglement structure
with EPR pairs on a square lattice.
On the other hand, for example for any cycle of odd
length one has strict inequality in Eq. (19) (with the cy-
cle of length 3 corresponding to matrix multiplication)
[83]. The asymptotic version (computing R
e(|ϕ⟩G)) of this
question has been studied in [82] for complete graphs.
For hypergraphs with edges with |e|>2, in general we
can even have strict inequality in Eq. (19) if the hyper-
graph is completely disconnected [76]. This occurs for
instance when one takes two copies of the Wstate, as
already discussed in Section II C. All together this leads
to the following question:
Given ncopies of a k-party state |ϕ⟩, and a k-regular
hypergraph Gwith nedges, how does R(|ϕ⟩G)depend on
the structure of G?
In this work we only briefly touch on this question.
As an example, we completely work out the (nontrivial)
answer in the case of two Wstates. We have the following
four possible graphs
G6G5
G4G4
By folding we see that
R(|W⟩G6)≥R(|W⟩G5)≥R(|W⟩G4)≥R(|W⟩G3).(20)
In [79] it was shown that
R(|W⟩G3)=7
(i.e. the Kronecker product of two copies of |W⟩has
rank 7). On the other hand, in [76] it was shown that
R(|W⟩G6)≤8 and in [80] that R(|W⟩G6)≥8 yielding
that the tensor product of two copies of |W⟩has
R(|W⟩G6) = 8.
Thus, with Eq. (20) we know
8 = R(|W⟩G6)≥R(|W⟩G5)
≥R(|W⟩G4)
≥R(|W⟩G3)=7.
In Appendix Bwe prove
Theorem 22.
R(|W⟩G4) = 8.
29
This solves an open problem in [80], and completes the
characterization of the tensor rank of two copies of the
Wstate on different hypergraphs
R(|W⟩G6) = R(|W⟩G5) = R(|W⟩G4)=8
R(|W⟩G3)=7.
In general, for acyclic hypergraphs one might expect
that since their stabilizer group equals that of the sep-
arate edge, as we have shown in Theorem 21, the rank
of an entanglement structure on the acyclic hypergraph
equals that of the rank one has when one just takes the
tensor product. That is, given an entanglement structure
|ϕ⟩Gon an acyclic hypergraph G, one may consider the
completely disconnected graph Hon the same number of
edges (and we define |ϕ⟩Hby just placing the same edge
states |ϕe⟩on these disconnected edges). We then have
the open question whether for any acyclic hypergraph
R(|ϕ⟩G)?
= R(|ϕ⟩H).
VII. OUTLOOK AND DISCUSSION
We have demonstrated that there is a rich resource
theory of different ways of distributing local multiparty
entanglement over hypergraphs, such as lattices, under
applying local operations. This is a resource theory of
tensor networks, in the sense that it allows us to com-
pare different entanglement structures as a resource for
creating many-body tensor network states. This resource
theory establishes a framework which accommodates the
previously proposed special instances of tensor networks
using multiparty entangled states, extending beyond the
traditional use of maximally entangled pair states. The
resource theory effectively generalizes the notion of bond
dimension, allowing for a systematic comparison of dif-
ferent entanglement structures.
Our first main result is that this resource theory goes
beyond transformations which only act on individual pla-
quette states and that certain transformations only be-
come possible on a lattice. By enabling transformations
between different entanglement structures, our approach
allows for the conversion between different tensor net-
work representations of the same quantum many-body
state. This may for example be used to relate repre-
sentations which are natural from the physics of a many-
body Hamiltonian, to representations which are practical
and efficient with respect to computational implementa-
tions. We have showcased a number of principles which
can be used to construct such transformations, laying the
groundwork for systematic development of transforma-
tions between entanglement structures. In the converse
direction, we provide techniques to prove obstructions to
the existence of transformations of entanglement struc-
tures on a lattice. For these methods we draw on pow-
erful techniques developed in order to understand the
computational complexity of matrix multiplication. We
have adapted generalized flattening bounds, the substitu-
tion method and asymptotic spectral functionals to lat-
tice problems and use these to prove no-go theorems for
transformations between entanglement structures. These
results illustrate that the profound mathematics devel-
oped in algebraic complexity theory has interesting ap-
plications in the theory of tensor networks. The resource
theory of tensors has been developed in different scientific
communities, on the one hand to study the complexity of
matrix multiplication and on the other hand to classify
multipartite entanglement with respect to SLOCC trans-
formations. Interestingly, the resource theory of tensor
networks operates at the intersection of these two view-
points. Indeed, our fundamental object of interest are
local entanglement structures on a lattice; at the same
time, tensor networks are a computational tool and we
show that the resource theory of tensor networks is di-
rectly related to the complexity of tensor network con-
tractions.
The resource theory of tensor networks opens up nu-
merous promising avenues for future research. In the al-
gorithmic direction one could further advance the usage
of entanglement structures in variational methods, for in-
stance following up on the proposal [60]. It is also clear
that the resource theory of tensor networks can be ap-
plied to infinite Projected Entangled Pair States (iPEPS)
[112] where finding good resource states could lead to
more efficient algorithms. Finally, the connection to al-
gebraic complexity theory we have emphasized may lead
to the development of improved contraction algorithms;
or in the converse direction could be used to import ten-
sor network algorithms to different problems in computer
science. Besides this it would also be interesting to ex-
plore the relevance of our resource theory to the theory of
condensed matter systems, for instance relating to (sym-
metry protected) topological phases and using symme-
tries of entanglement structures to define new canonical
forms of tensor networks, for instance using the frame-
work of [113]. Finally, we believe that our viewpoint on
tensor networks may also be of benefit outside the realm
of quantum many-body states and quantum computa-
tion, in particular in more data-driven subjects such as
machine learning.
ACKNOWLEDGEMENT
We acknowledge financial support from the European
Research Council (ERC Grant Agreement No. 818761),
VILLUM FONDEN via the QMATH Centre of Ex-
cellence (Grant No.10059) and the Villum Young In-
vestigator program (Grant No. 25452) and the Novo
Nordisk Foundation (grant NNF20OC0059939 ‘Quan-
tum for Life’). V.L. additionally acknowledges finan-
cial support from the European Union (ERC Grant No.
101040907). Views and opinions expressed are however
those of the author(s) only and do not necessarily reflect
those of the European Union or the European Research
30
Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for
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Appendix A: Proof of Theorem 14
Theorem 14 states that there is no restriction
A B
C
A′B′
C′
λ
2 8
2
3
where we have labelled the parties on the right-hand side
with a prime to indicate that they have different Hilbert
spaces. We will follow a strategy which is a generalization
of the substitution method we used for Theorem 12.
Consider the general situation where we have 3-tensors
|ϕABC ⟩and |ψA′B′C′⟩on three parties. The Hilbert
spaces are potentially different; we will assume that
dim(HA)≥dim(HA′).
For |x⟩∈HA, similar to before we let
|ϕ(x)⟩= (⟨x| ⊗
1
BC )|ϕABC ⟩
and similarly for |ψ(x)
ABC ⟩for |x⟩∈HA′. These are 2-
tensors on systems Band C(or B′and C′). As before
we define
X(k)
ϕ={|x⟩such that rk(|ϕ(x)⟩)≤k}
X(k)
ψ={|x⟩such that rk(|ψ(x)⟩)≤k}.
We now assume that |ϕ⟩≥|ψ⟩, so there exists a trans-
formation (MA⊗MB⊗MC)|ϕ⟩=|ψ⟩. If this is the
case, we can relate X(k)
ϕand X(k)
ψby the following two
lemmas.
Lemma 23. Suppose that |ϕ⟩ ≥ |ψ⟩. If for |y⟩ ∈ HA′we
have M†
A|y⟩ ∈ X(k)
ϕ, then |y⟩ ∈ X(k)
ψ.
Proof. If M†
A|y⟩ ∈ X(k)
ϕ, and we let |x⟩=M†
A|y⟩then
rk(|ψ⟩(y)) = rk((⟨y|MA⊗MB⊗MC)|ϕ⟩)
= rk((MB⊗MC)|ϕ(x)⟩)
≤rk(|ϕ(x)⟩)≤k
so |y⟩ ∈ X(k)
ψ.
Lemma 24. Suppose that |ϕ⟩ ≥ |ψ⟩and suppose that
the reduced density matrix of |ψ⟩has full rank on A′.
Let U⊆ HAbe the subspace given by M†
AHA′. Then
dim(U) = dim(HA′)and there exists an invertible map
NA:U→ HA′such that if |x⟩ ∈ X(k)
ϕ∩U, then we have
NA|x⟩ ∈ X(k)
ψ.
Proof. The fact that the reduced density matrix of |ψ⟩
has full rank on A′means that MAmust be surjective,
and hence M†
Ais injective. Therefore, restricting it to
its image U, it is invertible and we let NAbe its inverse.
If |x⟩ ∈ X(k)
ϕ∩U, then by Lemma 23 we find NA|x⟩ ∈
X(k)
ψ.
34
Now we specialize to the situation in Eq. (17) we are
interested in. We assign subsystems as follows:
≱
A2
A1
B2
B1
C1C2
A′B′
1
B′
2
C′
1C′
2
λ
2 8
2
3
|ϕ⟩ |ψ⟩
Proof of Theorem 14.For our purposes we take k= 8
and write X(8)
ϕ=Xϕand X(8)
ψ=Xψ. We will identify
the sets Xϕand Xψand show that no subspace as in
Lemma 24 can exist, from which we conclude that there
can be no restriction.
First of all, similar to the way we reasoned in the proof
of Theorem 12
rk(|ϕ(x)⟩) = 8 rk(|x⟩)
for any |x⟩ ∈ HA1⊗ HA2=C2⊗C2. Thus, if |x⟩ ∈ Xϕ
it must have rk(|x⟩)≤1 and
Xϕ={|x1⟩ |x2⟩,|xi⟩ ∈ C2}.
Next, for Xψ, note that
rk|ψ(x)⟩= rk((⟨x| ⊗
1
B′C′)|ψA′B′C′⟩)
= rk(⟨x| ⊗
1
B′
1C′
1)|λA′B′
1C′
1⟩rk(EPR3)
= 3 rk(⟨x| ⊗
1
B′
1C′
1)|λA′B′
1C′
1⟩.
Therefore, |x⟩ ∈ Xψif and only if
rk(⟨x| ⊗
1
B′
1C′
1)|λA′B′
1C′
1⟩≤2.
Suppose
|x⟩=x0|0⟩+x1|1⟩+x2|2⟩
then writing λ(x)= (⟨x| ⊗
1
B′C′
1)|λA′B′C′
1⟩as a 3 ×3
matrix
Λ(x)=
0x2−x1
−x20x0
x1−x0x2
.
We have rk(λ(x))≤2 if and only if this matrix has deter-
minant zero. This is the case if and only if x2= 0 since
det(Λ(x))=(x2)3. We conclude that
Xψ= span{|0⟩,|1⟩}.
Now we assume that a restriction exists, and therefore
we find a subspace U⊂ HA=C2⊗C2of dimension
dim(HA′) = 3 from Lemma 24 (note that the tensors
are concise, so the full rank condition is satisfied). Since
dim(HA)−dim(U) = 1, the subspace Umust be de-
scribed a by a single linear equation. We may assume
without loss of generality (after a choice of basis on HA2)
that Uis given by
U={|x⟩=x00 |00⟩+x01 |01⟩+x10 |10⟩
+ (ax00 +bx01 +cx10)|11⟩, xij ∈C}.
We claim that span(U∩Xϕ) = U. Recall that Xϕis the
set of product states. The following vectors are elements
of U∩Xϕ:
|x1⟩=|1⟩(|0⟩+c|1⟩)
|x2⟩= (|0⟩+b|1⟩)|1⟩
|x3⟩=|0⟩(b|0⟩ − a|1⟩)
|x4⟩= (c|0⟩ − a|1⟩)|0⟩.
E.g., for |x3⟩, we have x00 = b,x01 =−a,x10 = 0, and
the coefficient of |11⟩is given by ax00 +bx01 +cx10 = 0
so |x3⟩ ∈ U. For any choice of a, b, c for which we do not
have b=c= 0, these vectors span a three-dimensional
space, so span(U∩Xϕ) = U. In the case where b=c= 0,
the following three vectors in U∩Xϕ
|x1⟩=|1⟩ |0⟩
|x2⟩=|0⟩ |1⟩
|x3⟩= (|0⟩+|1⟩)(|0⟩+a|1⟩)
are linearly independent, so span(U∩Xϕ) = Uin this
case as well. However, this leads to a contradiction: we
saw that dim(Xψ) = 2, but by Lemma 24 we have a
linear invertible map NAmapping U∩Xϕ(and hence
this maps span(U∩Xϕ)) into Xψbut this is not possible
if dim(span(U∩Xϕ)) = 3.
As a consequence of Theorem 14, there is no bond
dimension D= 2 representation for the λtensor on a
kagome lattice (Corollary 15). However, we proved this
with a folding using open boundary conditions. We con-
jecture that the same is true with periodic boundary con-
ditions. One way to prove this is to use the same folding
as in Corollary 13. In this case, it would suffice to show
≱λ
8 8
2
33
We have obtained numerical evidence that there is indeed
no such restriction, but were unable to give a complete
proof, so we leave this as a conjecture.
Appendix B: Tensor rank of two copies of the W
state
The goal of this section is to prove Theorem 22, which
determines the tensor rank of two copies of the Wstate
|W⟩=|001⟩+|010⟩+|100⟩ ∈ C2⊗3
on the hypergraph G4.
Proof of Theorem 12.Our proof strategy is inspired by
[80]. We consider |W⟩G4with subsystems A, B, C, D
35
A D
B
C
Suppose that |W⟩G4has rank 7 or less, then there must
exist a decomposition
|W⟩G4=
7
X
i=1
|ai⟩ |bi⟩ |ci⟩ |di⟩
with |ai⟩,|di⟩ ∈ C2and |bi⟩,|ci⟩ ∈ C2⊗C2. Since
(
1
ACD ⊗ ⟨11|B)|W⟩G4= 0 we may assume without loss
of generality that ⟨11|b7⟩= 1. So, we expand
|b7⟩=x00 |00⟩+x01 |01⟩+x10 |10⟩+|11⟩
for some xij ∈C. We will now use a symmetry of a single
copy of the Wstate. If we define
h(p) : C2→C2
|0⟩ 7→ |0⟩,|1⟩ 7→ |1⟩+p|0⟩
then
(h(p)⊗h(q)⊗h(r)) |W⟩=|W⟩
if p+q+r= 0. We now apply the map
1
A⊗(h(−x01)⊗h(−x10 ))B⊗(h(x01)⊗h(x10 ))C⊗
1
D
to |W⟩G4. It leaves the state invariant, but we get a new
decomposition
|W⟩G4=
7
X
i=1
|a′
i⟩ |b′
i⟩ |c′
i⟩ |d′
i⟩
which now has
|b′
7⟩=x′
00 |00⟩+|11⟩
for some x′
00 ∈C. Next, we use the symmetry given by
1
A⊗g(−q)B⊗g(q)C⊗
1
D
for g(q) with any q∈Cas in Eq. (18). This allows us,
using q=x′
00, to transform to a decomposition
|W⟩G4=
7
X
i=1
|a′′
i⟩ |b′′
i⟩ |c′′
i⟩ |d′′
i⟩
with
|b′′
7⟩=|11⟩.
Next, we observe that
(
1
ACD ⊗(⟨0| ⊗
1
)B)|W⟩G4=|ϕ⟩ |W⟩
where |ϕ⟩=|01⟩+|10⟩.
|ϕ⟩ |W⟩ |W⟩ |ϕ⟩
AD
B
C
A D
B
C
Similarly,
(
1
ACD ⊗(
1
⊗ ⟨0|)B)|W⟩G4=|W⟩ |ϕ⟩.
By applying the same map to our rank 7 decomposi-
tion, we see that these tensors have rank at most 6 (since
|b′′
7⟩=|11⟩):
|ϕ⟩ |W⟩=
6
X
i=1
|a′′
i⟩(⟨0| ⊗
1
)|b′′
i⟩ |c′′
i⟩ |d′′
i⟩
|W⟩ |ϕ⟩=
6
X
i=1
|a′′
i⟩(
1
⊗ ⟨0|)|b′′
i⟩ |c′′
i⟩ |d′′
i⟩.
Now, grouping Dtogether with Bturns |W⟩ |ϕ⟩into a
3-tensor
A
BD
C
of rank 6 (this can be shown using the substitution
method). In particular, the vectors |αi⟩=|a′′
i⟩⊗|c′′
i⟩
for i= 1,...,6 are linearly independent. We claim that
the vectors |βi⟩= (⟨0| ⊗
1
)|b′′
i⟩⊗|d′′
i⟩for i= 1,...,6
must span at least a dimension 3 subspace. To see this,
from the decomposition of |ϕ⟩ |W⟩and letting
|˜αi⟩= (⟨ϕ| ⊗
1
)|αi⟩
on C(since (⟨ϕ| ⊗
1
)(|ϕ⟩ |W⟩) = |W⟩) we obtain a de-
composition
|W⟩=
6
X
i=1
|˜αi⟩ |βi⟩
where the |βi⟩are product states on BD. This implies
that if there were at most two linearly independent ele-
ments amongst the |βi⟩,|W⟩would have tensor rank at
most two, but we know that R(|W⟩) = 3. Now we con-
sider |ϕ⟩ |W⟩as a bipartite tensor between AC and BD.
It is clear that it has rank 2:
A D
B
C
However, we can write
|ϕ⟩ |W⟩=
6
X
i=1
|αi⟩AC ⊗ |βi⟩BD .
The existence of three independent vectors amongst the
|βi⟩, and the linear independence of the |αi⟩implies that
this has rank at least 3 as a 2-tensor, which is a contrac-
tion with the fact that it has rank 2.
36
Appendix C: Hardness of tensor network contraction
We will now discuss a hardness result for the com-
plexity of tensor network contraction in the arithmetic
circuit model. Recall that in this model of computation,
the goal is to compute some polynomial through an arith-
metic circuit. We will first define the complexity classes
V P and V N P . A sequence of functions fn,n= 1,2, . . .
is in V P if fnare polynomials in a number of variables
and with a degree that is polynomial in n, and moreover
there exists a family of arithmetic circuits of polynomial
size in ncomputing fn. A representative example is the
n×ndeterminant polynomial. Next we define the class
of V N P , which is the analog of N P . A sequence of func-
tions fn,n= 1,2, . . . is in V NP if fnhas a number of
variables v(n) and degree which are polynomial in n, and
moreover there exists a number m(n) and a sequence of
polynomials gnwith v(n) + m(n) variables such that gn
is in V P and fn(x1, . . . , xv(n)) can be computed as
X
y∈{0,1}m(n)
gn(x1, . . . , xv(n), y1, . . . , ym(n)).
The paradigmatic example is the n×npermanent poly-
nomial. It is conjectured that V P =V N P . There are
connections to the problem of Pversus N P ; for instance
it is known that V P =V NP over a field of characteristic
zero, together with the generalized Riemann hypothesis,
would imply P/ poly = N P/ poly, see Corollary 4.6 in
[114]. We now formally state our result on the hardness
of tensor network contraction, showing that tensor net-
work contraction is V N P -complete.
Theorem 25. Let f|ϕ⟩Gbe defined as in Eq. (9)for an
arbitrary hypergraph Gand entanglement structure |ϕ⟩G.
We denote by
fn=f|ϕ⟩Gn
the family of polynomials where we take Gnto be the n×n
square lattice graph (so each edge is a 2-edge), and the
entanglement structure |ϕ⟩Gnis given by placing level-2
EPR pairs on each edge.
1. The problem of computing tensor network contrac-
tion coefficients, given by the polynomial f|ϕ⟩Gon
hypergraphs with nedges and constant degree, and
local Hilbert spaces of polynomial size in nwith an
arbitrary entanglement structure |ϕ⟩G, is in V N P .
2. The problem of computing tensor network contrac-
tion coefficients with bond dimension D= 2 on a
square lattice, given by the polynomial fn, is V N P -
hard.
Proof. Note that in Eq. (9), the argument of f|ϕ⟩Gncon-
sists of (Tv)v∈V, and each Tvconsists of dim(Hv) vari-
ables, so if all local Hilbert spaces have polynomial di-
mension and the vertices have constant degree, then
f|ϕ⟩Gnhas as polynomial number of variables and poly-
nomial degree (it is linear in each of the variables). Given
a hypergraph Gwith nedges, and plaquette states |ϕe⟩,
for edges e∈Eon Hilbert spaces of polynomial dimen-
sion, we can find a restriction |ψ⟩G≥ |ϕ⟩Gwith plaquette
states |ψe⟩on edge Hilbert spaces of polynomial dimen-
sion and which consist of a collection of level-2 EPR pairs.
By the argument in Theorem 1this implies that it suf-
fices to prove the result for graphs Gwith only 2-edges,
with edge states |ϕe⟩which are level-2 EPR pairs. For
each edge e∈Ewe may define
|ϕe(ye)⟩= ((1 −ye)|0⟩+ye|1⟩)⊗2
where yeis a variable on the edge e. We let |ϕ(y)⟩Gnbe
the entanglement structure where we place these states
on Gn, which is a state depending on the variables y=
(ye)e∈E. Then we define the following polynomial:
g|ϕ⟩G((Tv)v∈V,(ye)e∈E) = O
v∈V
Tv!|ϕ(y)⟩Gn.
The g|ϕ⟩Gare in V P , since they can be computed as
Y
v∈V TvO
e,v∈e
((1 −ye)|0⟩+ye|1⟩)!
i.e. since the edge states are product states, the compu-
tation factorizes over the vertices. Moreover, the tensor
network contraction f|ϕ⟩G((Tv)v∈V) is computed by
X
(ye∈{0,1})e∈E
g|ϕ⟩G((Tv)v∈V,(ye)e∈E)
so we conclude f|ϕ⟩Gis in V N P , proving 1.
We proceed to prove 2. We will do so by reducing the
computation of a partition function of matchings on a
square lattice graph to the tensor network contraction fn.
We will then use that computing the partition function
of weighted (non-perfect) matchings on a square lattice
graph is V N P -hard. That is, one considers the graph
Gn. A matching is a subset of edges M⊆Esuch that
the edges in Mhave no common vertices. We let
x={x(e)}e∈E
be set of variables assigned to the edges e∈E. We then
define a partition function as
Zn(x) = X
MY
e∈M
x(e)
where the sum is over all matchings Mof Gn. By Theo-
rem 3.7 in [114], based on [115], computing this polyno-
mial is V N P -hard. We now argue that it can be com-
puted as a tensor network contraction. To this end, we
need to assign the edge variables to vertices. We will
do so by assigning them to the vertex to the right or
37
to the vertex below. We can then enforce the match-
ing constraint locally at the vertices. Consider a vertex
vwhich has edges e1,e2,e3,e4(starting from the left
edge in clockwise order). We then define the tensor Tv
as follows:
Tv(xe1, xe2) = xe1⟨1|e1⟨0|e2⟨0|e3⟨0|e4
+xe2⟨0|e1⟨1|e2⟨0|e3⟨0|e4
+⟨0|e1⟨0|e2⟨1|e3⟨0|e4
+⟨0|e1⟨0|e2⟨0|e3⟨1|e4
+⟨0|e1⟨0|e2⟨0|e3⟨0|e4
or in diagrammatic notation
xe1+xe2
+ + +
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
We find that in the resulting tensor network contraction,
we sum over all assignments of 0,1 to the edges where
there is at most one 1 neighboring each vertex, and we
assign variable xeto each edge which is labelled by 1:
P
matchings
xe1xe2
xe3xe4
xe5
xe6
The resulting polynomial over the variables xeis precisely
Zn(x).
