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Genuine Network Multipartite Entanglement
Miguel Navascués,
1
Elie Wolfe,
2
Denis Rosset,
2
and Alejandro Pozas-Kerstjens
3,4
1
Institute for Quantum Optics and Quantum Information (IQOQI) Vienna,
Austrian Academy of Sciences, Boltzmanngasse 3,1090 Vienna, Austria
2
Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, Canada, N2L2Y5
3
Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain
4
ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
The standard definition of genuine multipartite entanglement stems from the need to assess the
quantum control over an ever-growing number of quantum systems. We argue that this notion is easy
to hack: in fact, a source capable of distributing bipartite entanglement can, by itself, generate genuine
k
-partite entangled states for any
k
. We propose an alternative definition for genuine multipartite
entanglement, whereby a quantum state is genuinely network
k
-entangled if it cannot be produced by
applying local trace-preserving maps over several (
k
-1)-partite states distributed among the parties,
even with the aid of global shared randomness. We provide analytic and numerical witnesses of genuine
network entanglement, and we reinterpret many past quantum experiments as demonstrations of this
feature.
The existence of multipartite quantum states thatcannot
be prepared locally is at the heart of many communication
protocols in quantum information science, such as quantum
teleportation [
1
], dense coding [
2
], entanglement-based
quantum key distribution [
3
] and the violation of Bell
inequalities [
4
,
5
]. Most importantly, for the last two
decades, the ability to entangle an ever-growing number of
photons or atoms has been regarded as a benchmark for
the experimental quantum control of optical systems [
6
–
9
].
Since any multipartite quantum state where two parts
share a singlet can be regarded as “entangled,” another,
more demanding notion of entanglement was required to as-
sess the progress of quantum technologies. The accepted an-
swer was genuine multipartite entanglement [
10
–
12
]. Gen-
uine multipartite entanglement has since become a stan-
dard for quantum many-body experiments [
6
–
9
,
13
]. But,
is it a universal measure?
In this paper, we argue the opposite and present an al-
ternative and stronger definition, genuine network multi-
partite entanglement, which we formulate in terms of quan-
tum networks [
14
]. First, we define and compare the two
approaches. Next, we present general criteria to detect
genuine network entanglement and discuss the tightness
of the bounds so obtained. Finally, we single out past ex-
periments in quantum optics that can be reinterpreted as
stronger demonstrations of genuine network entanglement.
Multipartite entanglement. A
n
-partite quantum state
can be identified with a bounded Hermitian positive
semidefinite operator
ρ
acting on a composite Hilbert space
H1⊗···⊗Hn
such that
tr(ρ)=1
. Each factor
Hi
with
i=1,...,n
represents the local Hilbert space of the
ith
party.
For a subset
S⊆ {Hi}i
, we denote by
ρ(S)= trS(ρ)
the
density matrix of the reduced state on the subsystems
S
,
where
S
is the complement of
S
. We say that an
n
-partite
state is fully separable if it can be written as a convex mix-
ture of product states as follows:
ρ=X
j
wjρj
1⊗···⊗ρj
n,X
j
wj=1,(1)
where the
{ρj
i}
are normalized density matrices and the
weights
wj
are nonnegative. If
ρ
does not admit a decom-
position of the form
(1)
, we say that it is entangled. The
problem with the definition of full separability is that any
technology capable of entangling, say, the first two parti-
cles could claim the generation of “entangled states” com-
posed of arbitrarily many particles. Indeed, the reader can
check that any state ˆρof the form
ˆρ=|φ+ihφ+|⊗ρ(H3,...,Hn),|φ+i=|00i+|11i
√2(2)
does not admit a decomposition of the form of Eq. (1).
In order to address this issue, an extended definition of
multipartite separability was proposed [
10
–
12
]. Intuitively,
a state is
k
-partite entangled if, in order to produce it, one
must create
k
-partite entangled states and distribute them
among the
n
parties in such a way that no party receives
more than one subsystem. More formally, we say that
an
n
-partite state is separable with respect to a partition
S1|...|Ssof {H1,...,Hn}if it can be expressed as
ρ=X
j
wjρj
(S1)⊗···⊗ρj
(Ss).(3)
An
n
-partite state is genuinely
k
-partite entangled (or has
entanglement depth
k
) if it cannot be expressed as a convex
combination of quantum states, each of which is separable
with respect to at least one partition
S1|S2|...
of
{1,...,n}
with
|S`|≤k91
, for all
`
. Using this definition, the state
ˆρ
in Eq.
(2)
is certainly genuinely
2
-entangled. However,
ˆρ
is
not genuinely
3
-entangled so long as its marginal
ˆρ(H3,...,Hn)
is fully separable.
This notion of multipartite entanglement is easy to cheat,
as we show next. For simplicity, we consider a tripartite sce-
nario (
n=3
) and rename the Hilbert spaces
A
,
B
and
C
; we
split
A
into three local subsystems
A0
,
A00
and
A000
, the same
for
B
and
C
. Now, let
ρA0B0C0=|φ+ihφ+|A0B0⊗|0ih0|C0
,
and similarly
ρA00B00 C00 =|φ+ihφ+|B00C00 ⊗ |0ih0|A00
while
ρA000B000 C000 =|φ+ihφ+|C000A000 ⊗|0ih0|B000
. Following the
arXiv:2002.02773v3 [quant-ph] 18 Dec 2020
2
same discussion as for
ˆρ
, each of these three states in-
dividually is genuinely
2
-entangled but not genuinely
3
-
entangled. However, if we consider those three states col-
lectively (i.e., distributed at the same time), then the re-
sulting state
ρABC =ρA0B0C0⊗ρA00B00 C00 ⊗ρA000 B000C000
is gen-
uinely
3
-entangled when considering the partition
A|B|C
.
Accordingly, the established definition of genuine
k
-partite
entanglement is unstable under parallel composition (i.e.,
under simultaneous distribution of states).
In fact, enough copies of the state
ρABC
enable the distri-
bution of any tripartite state using the standard quantum
teleportation protocol [
1
]. Any definition of genuine tri-
partite entanglement that regarded states like
ρABC
as not
genuinely tripartite entangled and, at the same time, were
stable under composition and local operations and classical
communication (LOCC), would thus be necessarily void.
Namely, it would not apply to any physical tripartite state.
In this paper, we introduce the concept of genuine net-
work
k
-entanglement, an alternative operational definition
of multipartite entanglement that is stable under composi-
tion and where
ρABC
is not genuinely tripartite entangled.
The drawback, as it will be evident from the definition,
is that non-genuine network entanglement is not closed
under LOCC, but under the subset of LOCC transforma-
tions known as Local Operations and Shared Randomness
(LOSR) [
15
,
16
]. This set of operations has been argued to
be more relevant than LOCC for the study of Bell nonlocal-
ity [
17
,
18
]. Note that LOSR is a natural set of operations
when the parties being distributed the states are separated
in space and do hot hold a quantum memory.
Genuine network entanglement. We explain our defini-
tion using an adversarial approach. Eve is a vendor selling
a source of tripartite quantum states to three honest scien-
tists Alice, Bob and Charlie. Eve pretends that her device
produces a valuable entangled tripartite state
ρABC
. Unbe-
known to the scientists, the source sold to them is actually
composed of cheaper components: quantum sources that
produce the bipartite entangled states
σA0B00 ,σC0A00 ,σB0C00
,
see Figure 1. Alice receives the
A0,A00
subsystems of the
states
σA0B00 ,σC0A00
. Those can in principle interact within
Alice’s experimental setup, giving rise to a new quantum
system
A
: that is what Alice eventually probes. Similarly,
Bob (resp. Charlie) will have access to system
B
(resp.
C
),
whose state is the result of a deterministic interaction be-
tween systems
B0,B00
(resp.
C0,C00
). In addition, we provide
Eve with unlimited shared randomness
Λ
to jointly influ-
ence the local operations acting on systems
A0A00
,
B0B00
and
C0C00
. It is worth remarking that we do not make any
assumption on the dimensionality of the “hidden” states
σA0B00 ,σC0A00 ,σB0C00
: even if the systems
A,B,C
accessible to
Alice, Bob and Charlie are a qubit each, the Hilbert space
dimension of the hidden systems might well be infinite.
By performing local tomography on the state
ρABC
, can
Alice, Bob and Charlie certify that the state produced by
Eve’s network is indeed a valuable tripartite quantum state?
The family of states that they try to rule out can be
FIG. 1. Network producing a nongenuine network
3
-entangled
state; quantum resources and spaces are denoted using dotted
lines, while classical variables are drawn using solid lines.
defined formally. Let
Λ
be a classical random variable
with distribution
PΛ(λ)
sent to the three labs (for example
through radio broadcast). Denoting by
B(H)
the set of
bounded operators on the Hilbert space
H
, we describe
the deterministic operation at Alice’s by a family of linear
maps Ωλ
Aλ, where each Ωλ
Ahas type
Ωλ
A:B(A0⊗A00)→B(A)
and each
Ωλ
A
is completely positive and trace preserv-
ing. For completeness, the other maps correspond to
Ωλ
B:B(B0⊗B00)→B(B)
and
Ωλ
C:B(C0⊗C00)→B(C)
, so that
the state ρABC is
ρABC =XPΛ(λ)Ωλ
A⊗Ωλ
B⊗Ωλ
C(σ),(4)
where σ=σA0B00 ⊗σB0C00 ⊗σC0A00 .
The valuable states, those genuinely network 3-entangled,
are those that cannot be written the way described by
Eq.
(4)
. It is easy to see that the set of states of the form of
Eq.
(4)
is closed under tensor products and LOSR transfor-
mations. That is, the set of network
2
-entangled states is
a self-contained class within the resource theory of LOSR
entanglement [15,16]. This property has obvious implica-
tions for the monotonicity of any network
3
-entanglement
measure. Think for instance of the robustness of entangle-
ment [
19
]. We could define its network
3
-entanglement gen-
eralization as the minimum amount of network
2
-entangled
noise
R(ρABC)
which one must add to a tripartite quan-
tum state
ρABC
to make it network
2
-entangled. Closure
under LOSR implies that
R(ρABC)
is monotonically de-
creasing under LOSR operations. From our motivating dis-
cussion, though, it follows that
R(ρABC)
can be arbitrarily
increased by means of LOCC protocols.
Note that, in the considered adversarial scenario, rather
than the state
σA0B00 ⊗σC0A00 ⊗σB0C00
, Eve could also dis-
tribute Alice, Bob and Charlie arbitrary convex combi-
nations of states of the form
σ(i)
A0B00 ⊗σ(i)
C0A00 ⊗σ(i)
B0C00
, for
some values of
i
. Since the dimensionality of the primed
spaces is unbounded, though, this strategy can be sim-
ulated with the operations allowed by Eq.
(4)
. Indeed,
3
it suffices to distribute the tensor product of the states
σ(i)
A0B00 ⊗σ(i)
C0A00 ⊗σ(i)
B0C00
and embed the index
i
within the
hidden variable
Λ
(whose dimension is also unbounded).
The index
i
would then signal in which pair of Hilbert
spaces at party Z’s the map Ωλ
Zis to be applied.
The definition of genuine network entanglement can be
straightforwardly extended to the n-partite case.
Definition 1.
A multipartite quantum state is genuinely
network
k
-entangled if it cannot be generated by distributing
entangled states among subsets of maximum
k91
parties,
and letting the parties apply local trace-preserving maps,
those maps being possibly correlated through global shared
randomness.
Witnesses of genuine network entanglement. The cer-
tification of
ρABC
being genuinely network
3-entangled
is complicated, as the dimensions of the Hilbert spaces
A0,...,C00
are in principle unbounded. To classify the degree
of a state’s network multipartiteness, we must somehow de-
termine if the state can come about from a particular quan-
tum causal process. The study of quantum causal processes
has experienced great progress [
14
,
20
–
23
], and many tech-
niques have recently been developed [
22
,
24
,
25
]. Herein,
we adapt the inflation technique forcausal inference [
22
,
26
]
in order derive witnesses for genuine network entanglement.
As a starter, we consider a three qudit state
ρABC
, and
quantify its proximity to the Greenberger-Horne-Zeilinger
(GHZ) state [27] via the fidelity
FGHZd≡hGHZd|ρABC|GHZdi,(5)
where |GHZdi=
d
X
i=1
|iiii
√d.
If
ρABC
is of the form of Eq.
(4)
, then there exists a ran-
dom variable
Λ
, quantum states
σA0B00
,
σB0C00
and
σC0A00
and families of completely positive and trace-preserving
(CPTP) maps
Ωλ
Aλ
,
Ωλ
Bλ
and
Ωλ
Cλ
that generate
ρABC
. To derive bounds on the maximum fidelity achiev-
able by network
2
-entangled states, we next imagine what
states one could prepare by combining multiple realiza-
tions of the above state and channel resources. As we will
see, some of the reduced density matrices of the resulting
many-body inflated states are fully determined by the orig-
inal tripartite state
ρABC
. The property of
ρABC
admitting
a decomposition of the form of Eq.
(4)
will then be relaxed
to that of admitting positive semidefinite inflated states
satisfying said linear constraints. In the language of [
26
],
we will be defining a nonfanout inflation of the causal sce-
nario depicted in Figure 1.
In this regard, consider the ring inflation scenario de-
picted in Figure 2. If one acts on two copies of the states
σA0B00
,
σB0C00
and
σC0A00
with the maps
Ωλ
Aλ
,
Ωλ
Bλ
and
Ωλ
Cλ
in the ways indicated in the figure, one ob-
tains the six-partite density matrices
τA1B1C1A2B2C2
and
γA3B3C3A4B4C4
. Those are essentially unknown to us, as we
FIG. 2. Ring inflation of the triangle scenario in Figure 1,
containing copies of the state processing devices
Ωλ
A,B,C
; we label
such copies according to their output Hilbert space
Ai,Bj,Ck
,
where
i,j,k
is the index of the copy. These devices process copies
of the quantum resources
σA0B00
,
σB0C00
and
σC0A00
. To simplify
the drawing, we omitted the indices of these copies and only
indicate their original type. Note that, despite the fact that the
wirings between states and CPTP maps are different than in
the original scenario, every copy of a CPTP map acts on copies
of the states determined by the original scenario.
do not know how Eve’s devices act when they are wired
differently.
However, the states
τ
and
γ
are subject to several con-
sistency constraints. To begin, with,
τ
is symmetric under
the exchange of systems
A1B1C1
by systems
A2B2C2
, and
so is
γ
under the exchange of
A3B3C3
by
A4B4C4
. In addi-
tion, we observe that
τ(A1B1C1)=τ(A2B2C2)=ρABC.(6)
Still, we cannot say that
τA1B1C1A2B2C2=ρABC ⊗ρABC
as
the production of the two triangles could be classically corre-
lated through the shared randomness
Λ
. However, the state
τ
is separable across the
A1B1C1/A2B2C2
partition. Both
γand τare related to each other through the constraints
γ(A3B3A4B4)=τ(A1B1A2B2)(7)
and
γ(B3C3B4C4)=τ(B1C1B2C2)
and
γ(C3A4C4A3)=
τ(C1A1C2A2)
. Furthermore,
τ
and
γ
have trace one and are
semidefinite positive. Finally, the reduced state
γ(A3B3C3B4)
is separable across the
A3B3C3/B4
partition; and addi-
tional constraints of that type follow from cyclic symmetry.
Let us now provide some intuition as to why any state
ρABC
admitting such extensions
τ,γ
cannot be arbitrarily
close to the GHZ state. Suppose, indeed, that
FGHZd= 1
,
i.e.,
ρABC =|GHZdihGHZd|
and that there exist extensions
γ,τ
satisfying the constraints above. A measurement in
the computational basis of the sites
A3,B3,C3
of
γ
will
generate the random variables
a3,b3,c3
. Since
γ(A3B3)=
ρ(AB)=1
dPd
i=1|i,iihi,i|
, it must be the case that
a3,b3
are
perfectly correlated. The same considerations hold for
b3
4
and
c3
. Since
a3,b3
and
b3,c3
are pair-wise perfectly corre-
lated, so are
a3,c3
. Now, from the condition
γ(C3A4C4A3)=
τ(C2A2C1A1)
, we have thatthe distribution of
c3
and
a3
must
be the same as that of
c2
and
a1
. Hence,
c2
and
a1
must
be perfectly correlated. However,
τ(A1B1C1)
is a pure state,
since
τ(A1B1C1)=ρABC =|GHZdihGHZd|
, and hence it must
be in a product state with respect to any other system, such
as
C2
. It follows that a measurement in the computational
basis of the sites
A1
and
C2
will produce two uncorrelated
random variables a1,c2. We thus reach a contradiction.
The previous argument just invalidates the case
FGHZd=
1
. A more elaborate argument (see Appendix A for a proof)
shows that, if
a,b,c
are the random variables resulting from
measuring
ρABC
locally, then any network 2-entangled
state ρABC must satisfy
H(a:b)+H(b:c)−H(b)≤
S(ρ(A))+S(ρABC)−S(ρ(BC)).(8)
Here
H(x)
,
H(x:y)
and
S(ρ)
respectively denote the
Shannon entropy of variable
x
, the mutual information
between the random variables x,y, and the von Neumann
entropy of state
ρ
. Condition (8) is clearly violated if
ρABC ≈|GHZdihGHZd|
and the measurements are carried
in the computational basis.
Another constraint satisfied by states satisfying Eq.
(4)
,
expressed in terms of the GHZ fidelity, is
FGHZd≤2d(3d+√2d−1)
1−2d+9d2.(9)
Remarkably, in order to derive Eqs.
(8)
and
(9)
, it is not
necessary to invoke the existence of the six-partite states
τ,γ
, but that of their reduced density matrices
τ(A1B1C1C2)
,
γ(A3B3C3)
. As shown in Appendix B, both expressions,
Eqs.
(8)
and
(9)
, can be generalized to detect genuine
network k-entanglement.
For
d=2
, Eq.
(9)
establishes that any tripartite state
with
FGHZ2>4
33 6+√3≈0.9372
is genuinely network
3
-
entangled. As it turns out, this inequality is not tight: it
can be improved to
FGHZ2>1+√3
4≈0.6803
by means of
semidefinite programming applied to the ring inflation.
The variables in the corresponding program are trace-
one positive semidefinite matrices
τA1B1C1A2B2C2
and
γA3B3C3A4B4C4
of size
64×64
, subject to linear constraints
of the form of Eqs.
(6)
and
(7)
, as well as to the permu-
tational symmetry
1↔2
,
3↔4
. For all states
µXY
sepa-
rable across a
X/Y
partition, we add a Positivity under
Partial Transposition (PPT) constraint
(µXY)>Y
PSD
0
[
28
].
This applies to
τ
across the
A1B1C1/A2B2C2
partition,
and to reduced states of
γ
for the partitions
A3B3C3/B4
,
B3C3A4/C4,C3A4B4/A3.
The bound
FGHZ2>1+√3
4
is obtained by maximizing
hGHZ2|ρABC|GHZ2i
subject to the constraints above—a
typical instance of a semidefinite program—using the opti-
mization toolbox CVX [29] and the solver M o s e k [30].
We also employed the semidefinite optimization
procedure using as reference the Wstate [
31
],
|Wi≡ |001i+|010i+|100i
√3
, concluding that any
3
-qubit
state
ρABC
with
hW|ρABC|Wi>0.7602
is genuinely net-
work 3-entangled.
Armed with these witnesses, we find that several past ex-
periments in quantum optics can be interpreted as demon-
strations of genuine network tripartite entanglement [
32
–
35
]. Indeed, in all those experiments, the fidelity of the
prepared states with respect to GHZ or Wstates is greater
than the bounds derived above for network-bipartite states.
The prepared states are thus certified to contain genuine
network tripartite entanglement.
Robustness to detection inefficiency. In many experi-
mental setups, due to low detector efficiencies, the carri-
ers transmitting the quantum information are often un-
observed. The standard prescription in such a predica-
ment consists in discarding the experimental data gath-
ered when not all detectors click. Coming back to our ad-
versarial setup, this postselection of measurement results
opens a loophole that Eve can in principle exploit to fool
Alice, Bob, and Charlie. It is possible to contemplate this
contingency in the calculations above, and thus bound the
detection efficiency needed for certifying genuine network
entanglement under post-selection.
Let
p
indicate the fraction of experimental data preserved
by postselection, i.e., the probability that all three detectors
click. If
ρp
ABC
is the state reconstructed after postselection,
then all that can be said about the true tripartite quantum
state ρABC before the postselection took place is that
ρABC −p×ρp
ABC 0.(10)
As before, linear optimizations over the set of postselected
states
ρp
ABC
can be conducted via semidefinite program-
ming. In such instances one continues to relate the inflated
states
τ
and
γ
to the true (albeit unknown) tripartite state
ρABC
, and Eq.
(10)
is merely added as an extra constraint.
We find critical postselection probabilities beyond which
one can still certify genuine network tripartite entangle-
ment via GHZ fidelity (
pc≈0.685
) or Wfidelity (
pc≈0.765
).
Conclusions. In this paper we have argued that the
standard definition of genuine multipartite entanglement
is not appropriate to assess the quantum control over an
ever-growing number of quantum systems. We proposed
an alternative definition, genuine network multipartite en-
tanglement, that captures the potential of a source to dis-
tribute entanglement over a number of spatially separated
parties. We provided analytic and numerical tools to de-
tect genuine network tripartite entanglement, and also in-
dicated how the definition can be adapted to situations
where there may be local postselections on each party’s lab.
Furthermore, the construction can be adapted to detect
genuine network n-partite entanglement for any n.
While quite general, our numerical methods to detect
genuine network entanglement demand considerable mem-
5
ory resources to the point that we were not able to derive
new entanglement witnesses for tripartite qutrit states in a
normal computer. In addition, there exist significant gaps
between the bounds we derived on GHZ and Wstate fideli-
ties via SDP relaxations and the lower bounds obtained
using standard variational techniques [
36
,
37
]. Using such
algorithms, we were not able to give lower bounds to the
GHZ and Wfidelities larger than
0.5170
and
2/3
, respec-
tively. A topic for future research is thus to develop bet-
ter techniques for the characterization of genuine network
multipartite entanglement.
Note added. After completing this manuscript, we be-
came aware of the work of [
38
,
39
], whose authors consider
a scenario very similar to that depicted in Figure 1. Cru-
cially, they restrict the maps
Ωλ
A,B,C
to be unitary trans-
formations, acting on convex combinations of bipartite
states. The restriction to unitary maps not only allows
upper-bounding the dimensionality of the source states
σA0B00 ,σC0A00 ,σB0C00
, but it also severely constrains the re-
sulting set of states
∆C
: as shown in [
38
], tripartite qubit
states in
∆C
cannot be genuinely tripartite entangled. This
contrasts with the GHZ fidelity greater than
1/2
reported
above, achievable by states of the form of Eq. (4).
Acknowledgments. We thank Antonio Acín, Jean-
Daniel Bancal, T.C. Fraser, Yeong-Cherng Liang, David
Schmid, and Robert W. Spekkens for useful discussions.
M.N. was supported by the Austrian Science fund
(FWF) stand-alone project P 30947. The work of A.P.-
K. was supported by Fundació Obra Social “la Caixa”
(LCF/BQ/ES15/10360001), the ERC (CoG QITBOX and
the European Union’s Horizon 2020 research and innova-
tion programme - grant agreement No 648913), the Spanish
MINECO (FIS2016-80773-P and Severo Ochoa SEV-2015-
0522), Fundació Cellex, and Generalitat de Catalunya
(SGR 1381 and CERCA Programme). This research was
supported by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innova-
tion, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Economic
Development, Job Creation and Trade. This publication
was made possible through the support of a grant from the
John Templeton Foundation. The opinions expressed in
this publication are those of the authors and do not neces-
sarily reflect the views of the John Templeton Foundation.
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Appendix A: analytic witnesses for
genuine network tripartite entanglement
The goal of this appendix is to prove the following result:
Theorem 1.
Let
ρABC ∈B(Cd⊗Cd⊗Cd)
be a tripartite
quantum state, and let
a,b,c
be the outcomes which result
when we locally probe subsystems
A,B,C
. If
ρABC
is not
genuinely network
3
-entangled, then it must satisfy the
relations:
hGHZd|ρ|GHZdi≤ 2d(3d+√2d−1)
1−2d+9d2,(11)
H(a:b)+H(b:c)−H(b)≤S(A)+S(A|BC),(12)
where
S(A)
,
S(A|BC)
respectively denote the von Neumann
entropy of
ρ(A)
and the conditional entropy of system
A
with respect to BC, i.e., S(ρABC )−S(ρ(BC)).
The intuition behind the proofs of both inequalities is the
same. First, we assume that the state
ρABC
admits the
six-partite extensions
τA1B1C1A2B2C2
and
γA3B3C3A4B4C4
described in the main text. Then we prove that a strong
correlation between the random variables
a3,b3
and
b3,c3
implies a strong correlation between the variables
a3,c3
,
and hence a strong correlation between the variables
a1,c2
.
Next, we show that the correlation between the variables
a1,c2
is upper bounded in some way by the purity of the
original state
ρABC
. To obtain one bound or another we
rely on different measures of correlation and purity.
The following lemma will establish the transitivity of
strongly correlated variables.
Lemma 1.
Let
x,y,z
be jointly distributed random vari-
ables. Then, the fol lowing inequalities hold:
P(x=z)≥P(x=y)+P(y=z)−1,(13)
H(x:z)≥H(x:y)+H(y:z)−H(y).(14)
Proof.
Let
P(x,y,z)
be the joint probability distribution
of the three variables. Then we have that
P(x=y)+P(y=z)−P(x=z) = X
i
P(i,i,i)+
X
j6=i
P(i,i,j)+P(j,i,i)−P(i,j,i)
≤X
k
P(i,i,i)+X
j6=i
P(i,i,j)+P(j,i,i).(15)
The right-hand side of the above equation contains the
probabilities of a set of incompatible events. Its sum is
thus bounded by 1, hence proving inequality (13).
To prove Eq.
(14)
, we invoke strong subadditivity.
Namely, for any three random variables
x,y,z
, it holds that
H(x,y,z)≤H(x,y)+H(y,z)−H(y).(16)
The left-hand side of the equation above can be lower
bounded by
H(x,z)
. It follows that
H(x:z) = H(x) +
H(z)−H(x,z)
is lower bounded by
H(x)+H(z)−H(x,y)−
H(y,z)+H(y)
. This, in turn, equals the right-hand side of
Eq. (14).
The next lemma will relate the purity of a tripartite state
with the correlations it can establish with other systems.
7
Lemma 2.
Consider a four-partite quantum state
σABCY
,
with
F=hGHZd|σ(ABC)|GHZdi
, and suppose that
a,y
are
the result of measuring systems A,Yin the computational
basis, then the inequality
P(a=y)≤1+1
d−1F+2rF(1−F)
d(17)
holds. Moreover, independently of the nature of the mea-
surements, the relation
H(a:y)≤S(A)+S(A|BC)(18)
is satisfied.
Proof.
Suppose that we measure systems
A
and
Y
in the
computational basis, and define the operator
E≡
d
X
i=1|iihi|⊗I⊗2⊗|iihi|.(19)
Then,
P(a=y)= tr[Eσ]
. Furthermore, one can verify that
(P0⊗I)E(P0⊗I)= 1
d|GHZdihGHZd|⊗I,(20)
where
P0, P1
are the projectors defined by
P0=
|GHZdihGHZd|,P1=I−P0.
We have that
tr[Eσ]= X
i,j=0,1
ωij ,(21)
where ωis the 2×2matrix defined by
ωij =trσ(Pi⊗I)E(Pj⊗I).(22)
ω
is positive semidefinite. Indeed, take an arbitrary vector
|ci. Then,
hc|ω|ci=tr
σ X
i
c∗
iPi⊗I!E
X
j
cjPj⊗I
≥0,
(23)
where the last inequality stems from the fact that both
σand Eare positive semidefinite.
From the positive-semidefiniteness of
ω
it follows that
|ω01|≤√ω00ω11
. On the other hand, by (20) we have that
ω00 =1
dtr[σ(|GHZdihGHZd|⊗Id)] = F
d.(24)
In addition,
ω11 =tr[˜σE]
, where
˜σ
is the positive semidefi-
nite operator defined by
˜σ≡(P1⊗I)σ(P1⊗I).(25)
Note that
tr[˜σ] = trσ(ABC)P1= 1−F
. Since the operator
E
has norm
1
, it follows that
ω11 =tr[˜σE]≤1−F
. Putting
all together, we have that
P(a=y)≤F
d+1−F+2rF(1−F)
d.(26)
This proves Eq. (17).
Let
α, β
be two quantum systems. By
S(α|β)= S(αβ)−S(β)
we denote the conditional quantum
information; by
S(α:β)= S(α)+S(β)−S(αβ)
, the quan-
tum mutual information. To prove Eq.
(18)
, we invoke
weak monotonicity, namely, the fact that for any three
quantum subsystems
α,β,γ
,
S(α|β)+S(α|γ)≥0
. Taking
α=A
,
β=BC
,
γ=Y
, we have that
−S(A|Y)≤S(A|BC)
.
By the data processing inequality it thus follows that
H(a:y)≤S(A:Y)= S(A)−S(A|Y)(27)
≤S(A)+S(A|BC).
Having reached this point, we are ready to prove part
of Theorem 1. Choose Positive Operator Valued Measures
(POVMs)
MA
,
MB
,
MC
and use them to probe the type-
A
,
type-
B
and type-
C
subsystems of
γ
and
τ
, thus obtaining
the random variables
a1,a2,a3,a4
,
b1,b2,b3,b4
,
c1,c2,c3,c4
.
From the constraints
ρ(AB)=γ(A3B3)
,
ρ(BC)=γ(B3C3)
and
Lemma 1, we arrive at the relations
P(a=b)+P(b=c)−1 =
P(a3=b3)+P(b3=c3)−1≤P(a3=c3),(28)
and
H(a:b)+H(b:c)−H(b) =
H(a3:b3)+H(b3:c3)−H(b3)≤H(a3:c3),(29)
where
a,b,c
is the result of locally measuring
ρABC
according
to the POVMs MA,MB, and MC.
On the other hand,
γ(A3C3)=τ(A1C2)
, and hence
H(a3:c3)= H(a1:c2)
and
P(a3=c3)= P(a1=c2)
. Invok-
ing Lemma 2, the right hand side of Eq.
(29)
is upper
bounded by the right-hand side of Eq.
(18)
. This proves
Eq. (12).
If
MA
,
MB
,
MC
moreovercorrespond to measurements in
the computational basis, then we can invoke again Lemma 2
to bound the right-hand side of Eq.
(28)
with the right-
hand side of Eq. (17). This gives:
P(a=b)+P(b=c)−1≤
1+1
d−1F+2rF(1−F)
d.(30)
To prove Eq.
(11)
, we need to lower bound
p(a=b)
and
p(b=c)
in terms of the GHZ fidelity. The necessary bound
is provided by the next lemma.
Lemma 3.
Let
ρABC ∈B(Cd⊗Cd⊗Cd)
be a tripartite
quantum state, and let
F=hGHZd|ρ|GHZdi
be its GHZ
fidelity. If we measure any two systems
A,B,C
in the
computational basis, the corresponding random variables
a,b will satisfy:
P(a=b)≥F. (31)
8
Proof.
Without loss of generality, suppose that we measure
systems
A
and
B
, obtaining the random variables
a
and
b
,
respectively. Notice that
P(a=b)= tr[Sρ]
, with
S
defined
by
S≡
d
X
i=1|iihi|⊗2⊗Id.(32)
Then one can verify that
P0SP0=P0=|GHZdihGHZd|,
P1SP0=P0SP1= 0.(33)
It follows that
P(a=b)= tr[Sρ]= X
i,j=0,1
tr[ρPiSPj] = (34)
tr|GHZdihGHZd|ρ+tr[P1SP1ρ]≥F.
Now, use the previous lemma to lower bound the left-
hand side of Eq.
(30)
by
2F−1
. Solving the inequality for
F, we arrive at Eq. (11).
Appendix B: analytic witnesses for
genuine network k-entanglement
The arguments leading to Theorem 1can be ex-
tended to detect network
k
-entanglement. Consider
this time a
k
-partite quantum state
ρX0...Xk91
, and sup-
pose that it can be generated by applying correlated lo-
cal maps to
(k91)
-partite quantum states of the form
{σ0⊕j,...,(k92)⊕j:j=0,...,k91}
, where
⊕
indicates addition
modulo
k
. As in the tripartite case, we consider a dou-
ble network with nodes
X0
1,X1
1,...,Xk91
1,X0
2,X1
2,...,Xk91
2
.
For
j=0,...,k91
, we distribute two copies of the
states
σ0⊕j,...,(k92)⊕j
to systems
Xj
1,X1⊕j
1,...,X(k91)⊕j
1
,
Xj
2,X1⊕j
2,...,X(k91)⊕j
2
, respectively. By applying the maps
Ωλ
Xi
over systems
Xi
1
,
Xi
2
and averaging over
λ
, we ob-
tain a
2k
-partite quantum state
τX0
1...Xk91
1X0
2...Xk91
2
with
τ(X0
1...Xk91
1)=τ(X0
2...Xk91
2)=ρX0...Xk91.
On the contrary, consider the systems
X0
3,X1
3,...,Xk91
3,X0
4,X2
4,...,Xk91
4
, order them as
0,1,...,2k91
,
and distribute each state
σ0⊕j,...,(k92)⊕j
to the systems
0+j(mod 2k),...,k−1+j(mod 2k)
, for
j=0,...,2k91
. Ap-
plying the maps
Ωλ
Xi
and averaging over
λ
, we end up with
a2k-partite state γ0,...,2k91with the following properties:
1.γ(i,i+1) =ρ(XiXi⊕1)
. This is so because, in the previ-
ous construction, each node
i
shares with node
i+1
the
k−2
states
{σ...,i,i+1,...,σi,i+1,...}
. These are all
the states which those two nodes would have shared
had they been part of the network that built
ρX0...Xk91
.
Hence, their joint state must correspond to to the
latter’s reduced state ρ(XiXi⊕1).
2.γX0
3Xk91
3=τX0
1Xk91
2
. This follows from the fact that
the states used to generate
γ
were distributed in such
a way that no states are shared by systems
i
and
i+k−1(mod 2k).
We are ready to derive new witnesses. Say that the
original state
ρX0...Xk91
has a high fidelity with the
k
-partite
GHZ state, i.e.,
FGHZk
d≡hGHZk
d|ρ|GHZk
di,(35)
where |GHZk
di=
d
X
j=1
|ji⊗k
√d.
We locally measure
ρ
in some basis, obtaining the ran-
dom variables
x0,...,xk91
. If
FGHZk
d
is high enough; and
the measurement basis is close to the computational one,
then one should expect to find a high correlation between
xi
,
xj
, for
i,j =0,...,k91
. This implies that a measure-
ment of
γ
in the computational basis will produce random
variables
x0
3,...,xk91
4
with very high coincidence probabil-
ity
P(xi
3=xi+1
3)
and mutual information
H(xi
3:xi+1
3)
be-
tween neighboring sites. Applying Lemma 1recursively,
and taking into account that the distributions of
xi
3
,
xi+1
3
and xi,xi+1 are the same, we have that
k92
X
i=0
H(xi:xi+1)−
k92
X
i=1
H(xi)≤H(x0
3:xk91
3)(36)
and
k92
X
i=0
P(xi=xi+1)−k+2 ≤P(x0
3=xk91).(37)
In turn, the right-hand sides of the above equations respec-
tively equal H(x0
1:xk91
2)and P(x0
1=xk91
2).
The proofs of Lemmas 2and 3easily generalize to the
case of
k
parties: it amounts to replacing expressions such
as
|iihi|⊗I⊗2⊗|iihi|
by
|iihi|⊗I⊗k91⊗|iihi|
, and redefining
systems
A,B,C,Y
as
A=X0
1
,
B=X1
1
,
C=X2
1... Xk91
1
,
Y=Xk91
2
. Putting all together, we arrive at the entropic
inequality
k92
X
i=0
H(xi:xi+1)−
k92
X
i=1
H(xi)≤
S(X0)+S(X0|X1...Xk91),
(38)
valid for arbitrary physical measurements of subsystems
X0,...,Xk91, and at
(k−1)F−k+2 ≤1+1
d−1F+2rF(1−F)
d,(39)
where Fis shorthand for FGHZk
d.
Equivalently, one can solve for
FGHZk
d
and write the last
witness in linear form as
FGHZk
d≤
d3−k(d+1)+k2d+2p2+k(d−1)−d
1+4d−2dk+k2d2,(40)
where inequalities
(38)
and
(40)
are satisfied by all states
not genuinely network k-entangled.
