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A de Finetti theorem for quantum causal structures
F. Costa1,2, J. Barrett3, and S. Shrapnel4
1Nordita, Stockholm University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12 Stockholm, 106 91, Sweden
2School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia
3Quantum Group, Department of Computer Science, University of Oxford
4ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia,
QLD 4072, Australia
What does it mean for a causal struc-
ture to be ‘unknown’? Can we even talk
about ‘repetitions’ of an experiment with-
out prior knowledge of causal relations?
And under what conditions can we say that
a set of processes with arbitrary, possi-
bly indefinite, causal structure are inde-
pendent and identically distributed? Sim-
ilar questions for classical probabilities,
quantum states, and quantum channels
are beautifully answered by so-called “de
Finetti theorems”, which connect a simple
and easy-to-justify condition—symmetry
under exchange—with a very particular
multipartite structure: a mixture of iden-
tical states/channels. Here we extend the
result to processes with arbitrary causal
structure, including indefinite causal order
and multi-time, non-Markovian processes
applicable to noisy quantum devices. The
result also implies a new class of de Finetti
theorems for quantum states subject to a
large class of linear constraints, which can
be of independent interest.
1 Introduction
The possibility to repeat an experiment is one of
the fundamental tenets of the scientific method:
after a sufficient number of repetitions, statisti-
cal analysis typically enables us to characterise
the system or process of interest, to decide be-
tween competing hypotheses, and so on. How-
ever, a potentially unsettling question underlies
this paradigm: what counts as a repetition? If
we toss a coin hundreds of thousands of times [1],
are we repeating the same experiment or are we
F. Costa: fabio.costa@su.se
rather performing a set of different experiments?
And are we allowed to combine tosses from dif-
ferent coins?
The de Finetti theorem [2–4] offers an elegant
answer: it states that a joint probability for a
set of random variables that is a)invariant under
permutations and b)the marginal of a probability
of an arbitrarily larger set, also invariant under
permutations, must take the form
P(n)(a1, . . . , an)
=ZdqP (q)q(a1)· · · q(an),(1)
for a unique normalised measure P(q)over the
space of single-variable probabilities q. The strik-
ing consequence is that a set of exchangeable vari-
ables (i.e., satisfying conditions aand babove)
can always be interpreted as independent and
identically distributed (i.i.d.), up to a global un-
certainty P(q)on the single-trial probability q.
De Finetti’s theorem is particularly satisfying
from a Bayesian point of view: one never needs
to invoke an ‘unknown’ probability—which would
make little sense if probabilities represent degrees
of belief—nor needs one introduce any ad-hoc no-
tion of repeatability. As long as they can jus-
tify exchangeability, the Bayesian can treat a set
of variables as if they were multiple trials of
the same experiment, with each trial distributed
according to the ‘unknown’ probability q, and
where P(q)represents the prior knowledge about
such a probability. Conditioning on a larger and
larger set of observed variables allows one to up-
date P(q), eventually converging to a particular
single-trial q∗,P(q)→δ(q−q∗). This justifies
the idea of ‘discovering’ the unknown probabil-
ity through repeated trials and gives a principle-
based support for Bayesian methods. Apart from
the foundational significance, classical de Finetti-
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arXiv:2403.10316v3 [quant-ph] 4 Feb 2025
type theorems have extensive applications in pure
and applied probability theory [5–7].
De Finetti’s theorem extends in a direct way
to quantum states [8,9] and channels [10]: an ex-
changeable multipartite state/channel is always a
mixture of product states/channels. Apart from
liberating the Bayesian from the uncomfortable
notion of ‘unknown state’1[11] and grounding
the use of Bayesian methods in quantum infor-
mation [12,13], these results have wide-ranging
applications in many-body physics [14,15], cryp-
tography [16–18], quantum information [19–23],
and quantum foundations [24–27].
However, some natural questions emerging in
quantum theory escape known de Finetti results.
Consider an experiment that comprises multi-
ple measurements and operations distributed in
space and time. Can the causal relations be-
tween such operations be unknown and discov-
ered through multiple trials? Does it even make
sense to repeat such an experiment, given that
each spacetime event can only happen once?
Within a broader effort to understand the role
of causal structure in quantum theory [28–42], a
framework has recently emerged in which causal
relations need not be fixed in advance [43–45]
and can be discovered through experiments. The
framework—often dubbed the “process matrix
formalism”—also includes scenarios where causal
relations are genuinely indefinite [46], with poten-
tial applications to quantum information process-
ing [47–51] and fundamental models of quantum
gravity [52–55]. Furthermore, the special case of
causally ordered, multi-time processes is emerg-
ing as a powerful tool to tackle temporal corre-
lations in non-Markovian quantum processes [56–
61], an increasingly prominent feature in complex
quantum devices [62,63].
As for ordinary quantum theory, the process
matrix formalism makes probabilistic predictions,
tacitly assuming that the processes it describes
can be repeated arbitrarily many times. Indeed,
the paradigm has been employed in several ex-
periments probing processes with definite [64–
68] and indefinite [69–76] causal structure. How-
1Regardless of one’s ontological view on pure quantum
states, it should always be possible to use mixed states to
represent incomplete subjective knowledge. In this sense,
‘unknown mixed states’ are as problematic as ‘unknown
probability distributions’, calling for a principle-based ap-
proach to repeatability and discovery.
Figure 1: Process repeatability. Repetitions of an ex-
periment are modelled as a one-shot scenario, comprising
multiple operations that are only performed once. A de
Finetti theorem for processes seeks to group these oper-
ations such that they represent independent trials under
equivalent conditions, with each trial involving multiple
operations in arbitrary, possibly unknown or indefinite,
causal relations.
ever, a foundational justification for repeatability
is missing. In fact, without additional assump-
tions, even the quantum de Finetti theorem for
states does not apply to typical ‘repeated state
preparation’ scenarios: If the repetitions are tem-
porally separated, they cannot be modelled as a
joint state, as this would constitute a “state over
time” [77]. This undermines the applicability of
standard statistical analysis and characterisation
techniques in the most common quantum exper-
iments.
The goal of this work is to extend the link be-
tween exchangeability and repeatability to quan-
tum processes with arbitrary causal structure. To
this end, multiple putative trials of an experi-
ment are modelled as a single process, where all
measurements and operations are performed only
once, Fig. 1. A global process matrix encapsu-
lates prior knowledge about the entire setup, and
the task is to establish that an assumption of ex-
changeability enables the recovery of a decompo-
sition into several repetitions of ‘the same’ pro-
cess. This requires extending the de Finetti the-
orem to quantum processes with general causal
structures. At first glance, it may seem that any
additional causal constraint (such as global causal
order or other no-signalling assumptions) necessi-
tates its own de Finetti theorem. This is because,
as made clear through the process-state duality,
each such assumption is equivalent to a differ-
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ent set of linear constraints on quantum states,
which, a priori, need not be preserved by the de
Finetti representation.
Here we solve the above hurdle by proving a
new generalised de Finetti theorem for exchange-
able states subject to a broad class of linear con-
straints. We then specialise the general theorem
to show that an exchangeable process subject to a
set of no-signalling constraints has a unique rep-
resentation as a mixture of i.i.d. processes sub-
ject to the same constraints. We further consider
possible extensions of the result and find that,
remarkably, even mild generalisations of the the-
orem’s hypothesis do not lead to corresponding
de Finetti representations.
2 The process matrix formalism
The process matrix formalism [43–45] charac-
terises the most general scenario where a quan-
tum system, or multiple quantum systems, can
be probed an arbitrary number of times, with-
out prior assumptions regarding spatiotempo-
ral or causal relations between different opera-
tions. It relates closely to other frameworks, such
as the general boundary formalism [78], higher-
order quantum transformations [79–81], multi-
time states [82–84], entangled histories [85] , and
superdensity operators [86].
We call a site the abstract location of an oper-
ation and we label sites as A, B, . . . Concretely,
these labels can be understood as spacetime co-
ordinates at which the operations take place, or
more general ways to identify the abstract loca-
tion of the site, which may be delocalised in space
and time. The most general operation performed
at a site A, yielding some measurement outcome
a, is a completely positive (CP) and trace non-
increasing map [87]Ma:L(HAI)→L(HAO),
where HAI,HAOare respectively the input and
output Hilbert spaces assigned to site Aand L(H)
denotes the set of linear operators on H. We will
always use (a version of) the Choi-Jamiołkowski
(CJ) isomorphism to represent CP maps [88,89]:
Ma7→ Ma∈L(HA),
Ma:=
X
jk
|j⟩⟨k|⊗Ma(|j⟩⟨k|)
T
,(2)
for a chosen orthonormal basis {|j⟩}jof HAI,
where Tdenotes transposition in that basis and
(a) (b)
(c)
Figure 2: Operations and processes (a) Quantum op-
erations, represented as CJ operators MA
a,MB
b, trans-
form an input to an output quantum system—here de-
picted as wires. The labels A,Bfunction as generalised
coordinates, identifying the operations without necessar-
ily referring to a background causal structure, while a,
bdenote measurement outcomes. (b) A process matrix
Wrepresents the most general way to connect opera-
tions. (c) Inserting operations into a process, with no
open wires left, returns the probability for observing out-
comes a, b through the Born rule for processes, Eq. (4).
we use the short-hand HA≡ HAI⊗ HAO. The
CP condition translates to the CJ operator being
positive semidefinite, MA
a≥0. For simplicity, we
will nominally identify CP maps with their CJ
representations.
A deterministic operation M—one with a sin-
gle measurement outcome that happens with
probability one—is represented by a CP and trace
preserving (CPTP) map, which in CJ form trans-
lates to the condition
TrAOM=1AI,(3)
where 1Xdenotes the identity operator on HX
(we may skip the superscripts if the context is
sufficiently clear). An instrument is a collection
of CP maps, {Ma}a,Ma≥0, that sums to a
CPTP map, TrAOPaMa=1AI.Note that, even
when an operation can be regarded as a trans-
formation of a single system, we treat input and
output as distinct (although possibly isomorphic)
spaces. Furthermore, we assign different Hilbert
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spaces to different sites, even though they might
represent the same system at different times. The
space of operations across all sites spans the ten-
sor product of all input and output spaces.
The probability to obtain a set of outcomes
a, b, . . . at sites A, B, . . . is given by a general-
isation of the Born rule:
P(a, b, . . .) = Tr hMA
a⊗MB
b· · · Wi,(4)
where the operator W∈L(HA⊗ HB⊗ · · · )is the
process matrix, Fig. 2, which satisfies
W≥0,(5)
Tr hMA⊗MB· · · Wi= 1 (6)
for all CPTP maps MA, M B, . . . These con-
straints can be derived on abstract grounds by
assuming that quantum theory is valid at each
site—which can be formalised, e.g., through a
non-contextuality assumption [90]—and impos-
ing positivity and normalisation of probabilities.
They hold in particular for all ordinary scenar-
ios in quantum mechanics, where all sites can be
ordered according to a background time.
The normalisation constraint (6)is equivalent
to a set of linear-affine constraints:
Tr W=dO,(7)
L(W)=0,(8)
where dO=dAOdBO. . . is the prod-
uct of all output-space dimensions and
L:L(HA⊗ HB⊗ · · · )→L(HA⊗ HB⊗ · · · )is
a linear function whose particular form is not
relevant here, see, e.g., Appendix B in Ref. [45].
Additional causal assumptions between sites—
specifically, no signalling assumptions—can be
enforced by adding linear constraints to the pro-
cess matrix. For example, imposing no signalling
between the sites A,Bof a bipartite process is
equivalent to [45]
W= (T rAOBOW)⊗1AOBO,(9)
where (here and in similar expressions) a re-
ordering of tensor factors is implied. Similarly,
one-way no signalling from Bto Acorresponds
to the linear constraints
W= (T rBOW)⊗1BO,(10)
T rBIBOW= (T rAOBIBOW)⊗1AO.(11)
Similar constraints characterise one-way no sig-
nalling for an arbitrary sequence of causally or-
dered sites A, B, C, . . . Causally ordered pro-
cesses, also known as quantum channels with
memory [91], quantum strategies [92], and quan-
tum combs [93–95], can always be realised as a
sequence of channels connecting the sites, pos-
sibly with the addition of an auxiliary system
(an environment), thus modelling the most gen-
eral non-Markovian open-system dynamics [56–
58]. More complex assumptions, such as an arbi-
trary partial-order relation between sites, can be
obtained by combining non-signalling constraints
[37,40].
In the following, we will need the reduced pro-
cess matrix obtained by removing one or more
sites from a larger set [45]. Unlike for states, the
reduced process matrix is not always unique: if a
site Bhas causal influence on (i.e., it can signal
to) a site A, then, by definition of signalling, the
reduced process for Acan depend on the opera-
tion performed at B:
WA
MB:= TrBh1A⊗MBWAB i,(12)
where MB=PbMB
bis the CPTP map repre-
senting the unconditional transformation at Bif
we ignore the outcomes of the instrument {MB
b}b.
3 Exchangeable processes
As in ordinary quantum mechanics, the process
matrix formalism makes probabilistic predictions,
through the Born rule in Eq. (4). In order for such
predictions to be experimentally meaningful, one
typically assumes that an experiment involving a
number of sites A, B, . . . can be repeated an arbi-
trary number nof times, where each trial should
be modelled by the same process W. In a consis-
tent probabilistic framework—and in particular,
in a Bayesian perspective—it should be possible
to combine all trials into a single experiment, and
then specify the assumptions that entitle us to
view the total experiment as repetitions of indi-
vidual trials under equivalent conditions.
With this in mind, we will use the la-
bels A, B, . . . to denote the sites in a sin-
gle, unspecified trial, while we will use indices
A1, . . . , An, B1, . . . , Bn, . . . to distinguish the dif-
ferent trials. A priori, an n-trial scenario with m
sites per trial is described by a single one-shot
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(a) (b)
Figure 3: Exchangeable and i.i.d. processes (a) A
priori, nrepetitions of an experiment constitute a one-
shot process W(n). (b) The process can be interpreted
as ni.i.d. trials if it is a product of identical processes,
W(n)=W⊗n. Our result guarantees that exchangeable
processes are always mixtures of i.i.d. processes.
scenario with N=n m sites, with a process ma-
trix
W(n)∈L(H⊗n
ST ),
Fig. 3, where the single-trial Hilbert space HST is
the tensor product of the input and output spaces
associated with all msingle-trial sites:
HST := HA⊗ HB⊗ · · ·
In the following, the decomposition of an indi-
vidual trial into sites A, B, . . . will mostly be ir-
relevant, so we will effectively treat a single trial
as a single site. In line with this, we introduce
the short-hand notation Mj≡MAj⊗MBj⊗ · · ·
to denote a collection of CP maps for trial j, and
simply refer to it as aCP map. With this conven-
tion, conditions (5),(6)for a valid n-trial process
matrix can be written as2
W(n)≥0,(13)
Tr hM1⊗ · · · ⊗ MnWi= 1 (14)
for all CPTP maps M1,· · · , M n.
As in other de Finetti theorems, we need to
capture the idea that different trials are equiv-
alent to each other and that, in principle, there
is no bound to the number of trials. The first
condition is simply modelled by requiring invari-
ance under relabelling of trials. More specifically,
given an n-trial process matrix W(n), let us de-
note by |µ⟩∈HST the states in a chosen orthog-
onal basis of the single-trial space. We define the
2Here and for most of this work, we depart from the
common notation in the process matrix literature and
drop the superscripts to denote tensor factors. Instead,
unless otherwise specified, we assume that tensor factors
are always in a reference order, with trials in increasing
order from left to right.
action of an n-element permutation σ∈Snas
U(σ) := X
µ1...µn
|µ1. . . µn⟩Dµσ(1) . . . µσ(n)
.(15)
This means that all single-trial sites are permuted
together: Aj7→ Aσ(j),Bj7→ Bσ(j), . . . with the
same permutation σ. The permuted process ma-
trix is then
W(n)
σ:= U(σ)W(n)U†(σ),(16)
and we require that it should be equal to the ini-
tial process matrix.
The second condition, extendibility, means that
an n-trial scenario can be obtained by ‘ignor-
ing’ the last trial in an n+ 1-trial scenario:
ρ(n)= Trn+1 ρ(n+1) for states. For process matri-
ces, there are potentially different candidate def-
initions, because of the non-uniqueness of the re-
duced process mentioned above. A simple choice,
which is sufficient to deduce a de Finetti represen-
tation, is to require that no signalling is possible
from the sites in the n+ 1 trial to all the sites
from trial 1to n. With this choice, we have the
following definition:
Definition 1. A sequence of process matrices
W(n),n≥1, is called exchangeable if it satis-
fies
1. Symmetry: W(n)is invariant under per-
mutations of trials:
W(n)
σ=W(n)∀σ∈Sn;(17)
2. Process extendibility: For every nand for
every CPTP map M,
W(n)= Trn+1 hW(n+1)(1⊗n⊗M)i,(18)
where the partial trace is over the n+1 factor
in H⊗(n+1)
ST .
We say that an n-trial process matrix is ex-
changeable if it is part of an exchangeable se-
quence.
In the definition of extendibility, no-signalling
is enforced by requiring that we obtain the same
W(n)for any CPTP map Min the last trial. To-
gether with the symmetry assumption, this im-
plies that no signalling is possible between any
(sets of) trials. We explore the consequences of
different extendibility conditions in Appendix A.
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4 Constrained states and processes
Since process matrices are positive semidefinite
and have fixed trace, they are always proportional
to density operators, up to a constant. This im-
plies that it is possible to treat processes and
states in a unified way, which also allows us to
leverage known results, such as the quantum de
Finetti theorem for states. To this end, it is useful
to work with re-normalised process matrices
ρ:= W/ Tr W=W/dO.(19)
The process matrix normalisation constraints,
Eq. (14), are strictly stronger than state normal-
isation, Tr ρ= 1. Therefore, the mapping (19)
identifies processes with states that are subject
to additional linear constraints. In view of iden-
tifying a broader class of constraints, a natural
generalisation of Eq. (14)is the following:
Definition 2. Given a set of single-trial opera-
tors R⊂L(HST), and a function r:R→C,
we say that an n-trial operator ρ(n)satisfies a
product expectation constraint if
Tr
n
O
j=1
Rj
ρ(n)
=
n
Y
j=1
r(Rj)(20)
for all choices of R1, . . . , Rn∈R.
The process normalisation constraints,
Eq. (14), correspond to
R=nR=RA⊗RB⊗ · · · ∈ L(HST)
s.t. TrXORX=dXO1XI, X =A, B, . . . o(21)
and r(R) = 1, where the conditions in Eq. (21)
are the rescaled CJ form of the trace-preserving
condition, Eq. (3). More generally, constraints of
the form (20)can be interpreted as the assump-
tion that the expectation value of tensor products
of certain observables is equal to the product of
the expectation values. See Sec. 6below for a
further discussion.
In order to simplify the proof of our main re-
sult, and make it directly applicable to a vari-
ety of scenarios, it is useful to re-write, and fur-
ther generalise, constraints of the form (20). We
present here the key steps and leave proof details
to Appendix B.
The first observation is that the affine com-
ponent of the constraints—i.e., the term on the
right-hand side of Eq. (20)—can be absorbed in
the state constraint Tr ρ(n)= 1 (which we always
assume), while maintaining the product structure
of the constraint. This can be stated as follows:
Lemma 3. A sequence of exchangeable states
ρ(n)satisfies product expectation constraints of
the form (20), for a set of operator R⊂L(HST)
and a function r:R→C, if and only
Tr[
n
O
j=1
σj
ρ(n)]=0 (22)
for all
σj:=Rj−r(Rj),(23)
Rj∈R,j= 1, . . . , n.
Next, we observe that a set of linear con-
straints, such as Eq. (22), can always be rewrit-
ten as a single linear constraint defined by an ap-
propriate vector-valued function. Explicitly, we
prove in Appendix B.2 that
Lemma 4. Given two inner product spaces V1,
V2, a measurable space Y, and a set of linear
functions Ly:V1→V2,y∈Y,
Ly(v)=0 a.e. for y∈Y(24)
for every v∈V1if and only if, for any strictly
positive measure qover Y(q(y)>0a.e.)
ZY
dy q(y)L†
y◦Ly(v)=0,(25)
where “a.e.” stands for “almost everywhere”,
meaning that the property in question may not
hold in at most a measure-zero set.
As an example of Lemma 4in action, consider a
linear constraint as in Eq. (22), for n= 1. Defin-
ing LR(ρ):= Tr σρ, where the dependence on R
is given by Eq. (23), the constraint is defined by
a (possibly infinite) set of equations: LR(ρ)=0
∀R∈R. However, thanks to Lemma 4, these
constraints are equivalent to the single (operator
valued) equation
ZR
dR L†
R◦LR(ρ)
=ZR
dR σ Tr [σρ]=0.(26)
Using the same notation, the constraint in
Eq. (22)for n > 1is given by a set of equations
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LR1⊗ · · · ⊗ LRn(ρ(n)) = 0, for all combinations
of R1, . . . , Rn∈R. In particular, this implies
the constraints (LR)⊗n(ρ(n))=0, with the same
R∈Rfor every trial. Using again Lemma 4, this
can be expressed with a single equation:
ZR
dR (L†
R◦LR)⊗n(ρ(n))=0.(27)
Even though, for generic states, this is a strictly
weaker condition than Eq. (22), it will turn out to
be sufficient to prove our constrained de Finetti
theorem. This motivates us to introduce the fol-
lowing class of de Finetti-type constraints:
Definition 5. Let H(H)denote the real vector
space of self-adjoint operators on a Hilbert space
H. Given a set of real vector spaces nVkok, a
measurable space Y, a set of linear functions Lk
y:
H(HST)→Vkdefined for all y∈Y, and a set
of strictly positive measures qkover Y, we say
that an n-trial state ρ(n)satisfies de Finetti-type
constraints if
ZY
dy qk(y)Lk
y⊗n(ρ(n))=0 ∀k. (28)
Note that, using Lemma 4, a set of constraints
of the form (28)can always be re-written as a
single constraint (eliminating the dependence on
k). However, for later purposes it is useful to refer
to the general form (28).
5 Constrained quantum de Finetti the-
orem
In order to give a unified proof of a constrained
de Finetti theorem for processes and states, let us
first note that Definition 1of process exchange-
ability implies state exchangeability:
Lemma 6. If, for n≥1,W(n)define a sequence
of exchangeable process matrices, then the nor-
malised operators ρ(n):= W(n)/Tr W(n)define a
sequence of exchangeable states, i.e., they satisfy
the conditions
1. Symmetry.ρ(n)is symmetric under per-
mutations
2. State extendibility. For every n,
ρ(n)= Trn+1 ρ(n+1).(29)
To see that this holds, it is sufficient to note
that M=1/dOis the Choi representation of
a CPTP map (the maximally depolarising chan-
nel). By substituting this map into the definition
of process extendibility, Eq. (18), we obtain state
extendibility, Eq. (29), after normalisation.
Note that the result above only works one
way: even if we assume that ρ(n)satisfy the lin-
ear constraints for processes, state extendibility
is a strictly weaker condition than process ex-
tendibility, as discussed in Appendix A.3. How-
ever, the mapping from exchangeable processes
to exchangeable states is all we need to invoke
the standard quantum de Finetti theorem:
Theorem 7 (De Finetti theorem for quantum
states [8,11]).If, for n≥1,ρ(n)is a sequence of
exchangeable states, then there is a unique prob-
ability measure Pover the space of single-trial
density operators S(ρ≥0,Tr ρ= 1) such that
ρ(n)=ZS
dρP (ρ)ρ⊗n.(30)
After restoring the normalisation factors,
Eq. (30)almost gives us a de Finetti represen-
tation for processes. The reason this is not quite
the desired result is that the integral extends over
the whole space of single-trial density operators,
which also includes operators that are not valid
processes. In order to interpret Eq. (30)as a mix-
ture of i.i.d. processes, we have to show that the
probability measure Phas support only on the
space of valid single-trial processes, that is, pro-
cesses that satisfy the constraint (6).
As discussed in the previous section, we prove
a more general result that applies to all states
subject to de Finetti-type constraints, Eq. (28).
We want to show that a set of states ρ(n)satis-
fying constraints of the form (28)can be written
as a mixture of i.i.d. states subject to the same
constraint at the single-trial level. Perhaps sur-
prisingly, a much weaker assumption leads to the
desired result: it is sufficient to assume that the
constraints hold for two trials. Our central result
can then be formulated as follows:
Theorem 8. For a set of indices k, given a set
of real vector spaces Vk, a measurable space Y,
a set linear functions Lk
y:H(HST)→Vkfor
y∈Y, and a set of strictly positive measures qk
over Y, a state ρ(n)that is
1. exchangeable,
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2. subject to a set of constraints of the form
(28)for n= 2:
ZY
dy qk(y)Lk
y⊗Lk
y(ρ(2))=0 ∀k(31)
has a unique representation of the form
ρ(n)=ZLk
y(ρ)=0 a.e.y∈Y , ∀k
dρP (ρ)ρ⊗n(32)
for some probability measure P(ρ), where the no-
tation denotes an integral ranging over single-
trial states subject to the constraints Lk
y(ρ) = 0
∀kand a.e. for y∈Y.
Interestingly, this theorem can almost be de-
rived as the asymptotic limit of a result in
Ref. [96], Theorem 2.3, which considers a finite
version of a constrained de Finetti theorem. How-
ever, the constraints considered in Ref. [96] are
stronger than in our theorem, as they only apply
to one factor: I⊗(n−1) ⊗L(W(n)) = W(n−1) ⊗v
for some vin the co-domain of Land where I
denotes the identity function. It was left as an
open question if a weaker condition, as in our
theorem, would still lead to a de Finetti theorem.
Our result answers the question in the affirmative
for the asymptotic limit of infinite exchangeable
sequences.
Let us now prove Theorem 8.
Proof. For each vector space Vk, consider an ar-
bitrary dual vector ωk, that is, an arbitrary lin-
ear function ωk:Vk→R. Applying ωk⊗ωkto
Eq. (31), we have, for every k,
ZY
dy qk(y)×
×ωk◦Lk
y⊗ωk◦Lk
y(ρ(2))=0.(33)
Since ρ(2) is exchangeable, we can expand it in
the de Finetti form, Eq. (30), which substituted
into Eq. (33) gives
ZY
dy qk(y)ZS
dρP (ρ)ωk◦Lk
y(ρ)2= 0.(34)
By assumption, P(ρ)≥0,qk(y)>0a.e., and
ωk◦Lk
j(ρ)∈R, so the expression (34) is a pos-
itive linear combination of non-negative terms
ωk◦Lk
j(ρ)2. This means that the expression
can only vanish if
P(ρ)ωk◦Lk
y(ρ)2= 0
∀kan a.e. for ρ∈S, y ∈Y.
(35)
This is only possible if, for each kand for almost
all ρ,y, at least one of the two factors vanishes,
which means that either
•P(ρ)=0or
•ωk◦Lk
y(ρ)=0.
Since this has to hold for every dual vector ωk,
we conclude that P(ρ) = 0 a.e. for all ρsuch
that Lk
y(ρ)= 0. Therefore, the integral in the
de Finetti representation for ρ(n)can be limited
to those states that satisfy Lk
y(ρ)=0∀kand
a.e. for y∈Y. This yields the desired result,
Eq. (32).
We will now discuss some consequences of this
result.
General process matrices
Because of Lemmas 3and 4, Theorem 8imme-
diately implies that a sequence of exchangeable
states subject to product expectation constraints,
Eq. (20), for some set Rof operators and some
function r, has a unique de Finetti representation
over states subject to the same constraints at the
single-trial level:
ρ(n)=ZTrRρ=r(R)∀R∈R
dρP (ρ)ρ⊗n.(36)
In particular, by choosing Ras in (21)(CPTP
maps, up to normalisation), r(R) = 1, and after
the appropriate rescaling of operators, we obtain
our original goal:
Corollary 9. An n-trial process matrix W(n)is
exchangeable if and only if it has a de Finetti
representation
W(n)=ZW
dW P (W)W⊗n,(37)
for a unique probability measure P(W), where W
denotes the set of single-trial process matrices.
Processes with no-signalling constraints
Let us now consider n-trial processes subject to
specific no-signalling constraints. These can arise
if the relative spatiotemporal location of some
sets of sites is fixed and known, or from structural
properties of the experimental protocol, such as
isolation between components of a device. For
example, if Bjis in the future of Ajin every trial
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j, we can impose that W(n)permits no signalling
from Bjto Aj. The goal is to show that we can
also restrict the single-trial probability Pto have
support on processes that satisfy the same no sig-
nalling constraint, i.e., that we can interpret our
scenario as consisting of i.i.d. processes with no
signalling from Bto A.
As shown explicitly in Ref. [97], no-signalling
from a set of parties K2to a disjoint set of par-
ties K1can be characterised through a linear con-
straint
LK1≺K2(W)=0,(38)
where we refer to Ref. [97] for the explicit form of
the linear function LK1≺K2:L(HST)→L(HST).
A given scenario may give rise to multiple no-
signalling constraints, corresponding to a set of
linear functions Lk, with each klabelling no-
signalling from a set of sites to another set of
sites.
Given an n-trial process matrix W(n), impos-
ing a set of no-signalling conditions for each trial
results in the set of constraints
Lk⊗ I · · · ⊗ I(W(n))=0 ∀k,
I ⊗ Lk⊗ · ·· ⊗ I(W(n))=0 ∀k,
. . .
These constraints can be combined to give
Lk⊗ · · · ⊗ Lk(W(n))=0 ∀k. (39)
This is now a set of de Finetti-type constraints,
Eq. (28), so we can apply the constrained de
Finetti theorem and obtain the following result:
Corollary 10. An exchangeable n-trial process
matrix W(n), subject to single-site no-signalling
constraints encoded in a set of linear functions
Lk, has a unique representation as a mixture of
i.i.d. single-site process matrices subject to the
same no-signalling constraints:
W(n)=ZW∈W
Lk(W)=0 ∀k
dW P (W)W⊗n.(40)
In particular, this result applies to a typical
experimental setup where, in each run of the
experiment, a sequence of temporally separated
measurements is performed. In such a case, we
are entitled to impose one-way no-signalling con-
straints to the global n-trial process matrix W(n).
If we can also assume exchangeability, then we
can write the n-trial process matrix as a mixture
of i.i.d. causally ordered processes, recovering a
de Finetti theorem for combs, which was proven
independently [98].
6 Frequentist interpretation of the
constraints
In line with the original de Finetti theorem, we
have presented our result within a Bayesian mind
set, where probabilities, states, and processes are
representations of an agent’s knowledge of (or ex-
pectations about) a given scenario, without any a
priori frequentist interpretation. The frequentist
interpretation emerges for the single-trial proba-
bilities appearing in the de Finetti representation.
However, it can be useful to re-interpret the
result in a frequentist approach, as this can give
some intuition about when the state constraints
we have introduced may arise. For simplicity, we
will only consider states, although the discussion
directly extends to processes.
In a frequentist view, the n-trial state ρ(n), and
any probability we associate to it, describes a sce-
nario where the entire set of trials can be repeated
an arbitrary number of times. We refer to a set
of ntrials as a supertrial. It is of course possible
to recover the frequentist interpretation of ρ(n)
by enlarging the Bayesian description to include
an unbounded number of supertrials. The fact
that each supertrial is represented by the same,
‘unknown’ ρ(n)is then recovered by assuming ex-
changeability of the sequence of supertrials. How-
ever, we will stick to a frequentist language for
describing the repetitions of supertrials.
For concreteness, we can imagine a source of
quantum systems that is turned on every day.
Each day, the source produces nsystems, which
we collectively describe by some state ρ(n), and on
which we can perform arbitrary measurements.
After repeating the same measurements for an
arbitrary number of days, the frequency of any
measurement converges to the probability cal-
culated from ρ(n). If ρ(n)is exchangeable (and
hence has a de Finetti representation), it means
that, within any given day, the source prepares
the same state ρover and over, but a different
ρis picked randomly each day according to the
probability P(ρ), so that the supertrial state is
represented by the mixture ρ(n)=RSP(ρ)ρ⊗n.
The frequentist language makes it easier to
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formulate a context in which constraints on the
expectation values of observables, such as in
Eq. (20), may arise. Imagine that, each day, we
only use the source twice, and each time we mea-
sure some chosen observable R. This means that
we are measuring the product observable R⊗R
over the two-trial state ρ(2) = Tr3...n ρ(n). Af-
ter repeating the experiment over many days, we
record the one-trial and two-trial expectation val-
ues
⟨R⟩= Tr Rρ(1),(41)
⟨R⊗R⟩= Tr R⊗Rρ(2).(42)
Note that, at this point, we do not know what ρ(n)
is—we just assume that there must be some ρ(n)
that reproduces our observed expectation values
according to expressions (41)and (42), where
ρ(1) = Tr2ρ(2).
In this scenario, we say that we have a product
expectation constraint if
⟨R⊗R⟩=⟨R⟩2.(43)
Indeed, this coincides with the form (20)where
the set of observables Rcontains a single element
R,r(R) = ⟨R⟩, and n= 2.
The constrained de Finetti theorem implies
that, having observed the relation (43)(and as-
suming exchangeability), we can conclude that,
at each trial, the source always produces a state
ρwith expectation value
Tr (Rρ) = r(R) = ⟨R⟩.(44)
Note that this is not the same as the expectation
value (41), which is obtained from one measure-
ment per day repeated over many days, for which
we use ρ(1) =RSdρP (ρ)ρ. Eq. (44)means that if,
over a single day, we measure Rarbitrarily many
times (not just two), the average of all observed
results will approach ⟨R⟩. More general product
expectation constraints arise in analogous scenar-
ios for a larger number of observables.
As a concrete example, if the single-trial system
has two levels, and we take the Pauli observable
R=Z, then the two-trial constraint ⟨Z⊗Z⟩=
⟨Z⟩2=: ¯z2on an exchangeable state ρ(n)implies3
ρ(n)=Zx2+y2≤1−¯z2dxdy P (x, y )×
×1
2n(1+xX +yY + ¯zZ)⊗n,(45)
where Xand Yare the two remaining Pauli op-
erators. This tells us that the source always pre-
pares a state in the section of the Bloch sphere
with zcoordinate equal to ¯z, but the xand yco-
ordinates are picked randomly each day according
to the distribution P(x, y).
7 (Non)-extensions of the result
It is natural to ask whether there are other types
of constraints, not covered by Theorem 8, that
still lead to a constrained de Finetti representa-
tion. It is in fact quite instructive that this does
not work for some seemingly mild generalisations
of our constraints. In rather general terms, we
can formulate the question as follows:
Given a sequence of exchangeable states ρ(n)
and a sequence of functions F(n)such that, for
some values of n,F(n)(ρ(n)) = 0, does ρ(n)have
a de Finetti representation with support on states
satisfying F(1)(ρ)=0? In other words, can we
write
ρ(n)=ZF(1)(ρ)=0
dρ P (ρ)ρ⊗n(46)
for some probability measure P(ρ)?We explore
examples of this type in Appendix C. Here we
only summarise the key findings.
•Instead of constraints that fix the expec-
tation values of some observables through
equalities, as in Eq. (20), we can consider in-
equalities instead. As it turns out, inequality
constraints such as Tr hNn
j=1 Rjρ(n)i≥
Qn
j=1 r(Rj),do not imply that the de Finetti
representation can be constrained to states
with Tr (Rjρ)≥r(Rj).
•In Theorem 8, we have seen that it is suffi-
cient to impose the linear constraints on ρ(2)
in order to derive a constrained de Finetti
representation. By contrast, imposing the
constraints only on ρ(1), or on ρ(n)for any
3A similarly constrained state was considered in
Ref. [12], although there the constraint emerged from arbi-
trarily many measurements of Zwithin a single supertrial.
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odd n, is generally not sufficient to derive a
constrained de Finetti representation.
•However, if Ris positive semidefinite, it is
sufficient to impose Tr Rρ(1) = 0 to derive
a de Finetti representation constrained on
states with Tr Rρ = 0. Since R > 0can be
interpreted as a measurement operator, the
implication is that if a measurement outcome
cannot occur in a single trial, then the ‘un-
known state’ must also give 0probability for
that outcome.
•If, on top of exchangeability, we require ρ(n)
to be invariant under the action of a group
acting on the single-trial space, we cannot
conclude that the corresponding de Finetti
representation is constrained to states invari-
ant under the group’s action.
8 Conclusions
At the technical level, we have shown that an
exchangeable state or process subject to a prod-
uct of linear constraints, or mixtures thereof,
can always be expressed as a mixture of product
states or processes, with each factor satisfying the
same constraints. In particular, our result implies
that exchangeable processes with arbitrary causal
structure are mixtures of i.i.d. processes; that ex-
changeable multi-time, causally ordered processes
are mixtures of i.i.d. multi-time, causally ordered
processes; and that exchangeable processes with
specified no-signalling constraints are mixtures of
i.i.d. processes with the same no-signalling con-
straints.
Our result shows that an exchangeability as-
sumption can ground the notion of repeatability
for experiments involving quantum causal struc-
ture. The result clarifies under what conditions
we are entitled to regard an experiment as mul-
tiple repetitions, under equal conditions, of a sin-
gle experiment involving multiple events. This
paradigm entitles us to ‘discover an unknown
causal structure’ without any ontological com-
mitment regarding quantum states, processes, or
causal structure: as long as we can justify the
equivalence of different trials under permutations,
and the possibility to repeat an experiment indef-
initely, we can treat our scenario as if it is gov-
erned by a ‘real’ underlying process matrix W,
which in turn encodes all potentially accessible
information about causal relations.
In practice, once exchangeability is established,
the distribution P(W)appearing in the de Finetti
representation, Eq. (37), effectively describes
prior knowledge of the process of interest. Fol-
lowing a similar argument as for states [12], ob-
serving a set of outcomes a in each trial leads to
an update of the prior according to Bayes rule:
Pupd(W|a) = P(W)P(a|W)
P(a),(47)
where P(a|W)is calculated using the Born rule
for processes, Eq. (4), and P(a)is the marginal
outcome probability given the prior: P(a) =
RdW P (a|W)P(W). Crucially, given prior infor-
mation about the causal relations between the
events involved, we can constrain accordingly
the prior P(W), without additional assumptions
apart from exchangeability. This opens the task
to formalise the procedure into concrete algo-
rithmic routines, extending existing methods for
quantum states and quantum channels [13].
Just like the original de Finetti theorem, our re-
sult relies on the unphysical assumption that the
experiment can be repeated an arbitrarily large
number of times. However, finite de Finetti the-
orems exist for classical probabilities [99,100],
quantum states [101,102], and quantum channels
[96]. The general idea is that a subsystem of a fi-
nite, symmetric state (or channel, process, etc.),
should approximate a state (or channel, process,
etc.) in the de Finetti form. Formulating a finite
version of a de Finetti theorem for processes could
provide a concrete starting point to treat practi-
cal scenarios where full exchangeability cannot be
guaranteed. As noted earlier, finding a finite ver-
sion of a constrained de Finetti theorem such as
Theorem 8was also posed as an open question in
Ref. [96].
Finally, it is interesting that the constraints in
our theorem do not include causal separability
[45]. This is relevant, for example, in a scenario
where all operations are well localised in time,
but there is uncertainty about their order: In
this case, one can constrain processes to prob-
abilistic mixture of causally ordered ones, which
is a special instance of causal separability [44,97].
However, working from first principles, this only
holds for the entire, one-shot process. Even as-
suming exchangeability, our current result does
not guarantee that a causally separable process
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can be written as a mixture of i.i.d. causally sep-
arable ones, and it is a compelling open question
whether such an extension holds.
Acknowledgments
We thank Eric Cavalcanti, Andrew Doherty,
Omar Fawzi, Matthew Pusey, and Frank Wilczek
for valuable comments and discussions. This
work has been supported by the Australian
Research Council (ARC) Centre of Excel-
lence for Engineered Quantum Systems (EQUS,
CE170100009) and by the John Templeton Foun-
dation through the ID # 62312 grant, as part
of the ‘The Quantum Information Structure of
Spacetime’ Project (QISS). The opinions ex-
pressed in this publication are those of the au-
thors and do not necessarily reflect the views of
the John Templeton Foundation. Nordita is sup-
ported in part by NordForsk. We acknowledge
the traditional owners of the land on which the
University of Queensland is situated, the Turrbal
and Jagera people.
References
[1] František Bartoš, Alexandra Sarafoglou,
Henrik R. Godmann, Amir Sahrani, et al.
“Fair coins tend to land on the same
side they started: Evidence from 350,757
flips” (2023). arXiv:2310.04153.
[2] Bruno De Finetti. “Funzione caratter-
istica di un fenomeno aleatorio”. In
Atti del Congresso Internazionale dei
Matematici: Bologna del 3 al 10 de set-
tembre di 1928. Pages 179–190. (1929).
url: http://www.brunodefinetti.it/
Opere/funzioneCaratteristica.pdf.
[3] Bruno de Finetti. “La prévision : ses lois
logiques, ses sources subjectives”. Annales
de l’institut Henri Poincaré 7, 1–68 (1937).
url: http://eudml.org/doc/79004.
[4] Edwin Hewitt and Leonard J. Savage.
“Symmetric measures on cartesian prod-
ucts”. Trans. Am. Math. Soc. 80, 470–
501 (1955).
[5] J. F. C. Kingman. “Uses of exchangeabil-
ity”. Ann. Probab. 6, 183–197 (1978).
[6] David J. Aldous. “Exchangeability and re-
lated topics”. In P. L. Hennequin, editor,
École d’Été de Probabilités de Saint-Flour
XIII — 1983. Pages 1–198. Springer Berlin
Heidelberg (1985).
[7] Raymond J. O’Brien. “Bayesian Inference
and Decision Techniques: Essays in Honor
of Bruno de Finetti. Studies in Bayesian
Econometrics and Statistics, Vol. 6”. The
Economic Journal 98, 883–884 (1988).
[8] Erling Størmer. “Symmetric states of in-
finite tensor products of C∗-algebras”. J.
Funct. Anal. 3, 48–68 (1969).
[9] R. L. Hudson and G. R. Moody. “Locally
normal symmetric states and an analogue
of de Finetti’s theorem”. Z. Wahrschein-
lichkeitstheorie verw Gebiete 33, 343–
351 (1976).
[10] Christopher A. Fuchs, Rüdiger Schack, and
Petra F. Scudo. “De Finetti representation
theorem for quantum-process tomography”.
Phys. Rev. A 69, 062305 (2004).
[11] Carlton M. Caves, Christopher A. Fuchs,
and Rüdiger Schack. “Unknown quantum
states: The quantum de Finetti representa-
tion”. J. Math. Phys. 43, 4537–4559 (2002).
[12] Rüdiger Schack, Todd A. Brun, and Carl-
ton M. Caves. “Quantum bayes rule”. Phys.
Rev. A 64, 014305 (2001).
[13] Christopher Granade, Christopher Ferrie,
Ian Hincks, Steven Casagrande, Thomas
Alexander, Jonathan Gross, Michal
Kononenko, and Yuval Sanders. “QInfer:
Statistical inference software for quantum
applications”. Quantum 1, 5 (2017).
[14] M. Fannes, H. Spohn, and A. Verbeure.
“Equilibrium states for mean field models”.
J. Math. Phys. 21, 355–358 (1980).
[15] Christian Krumnow, Zoltán Zimborás, and
Jens Eisert. “A fermionic de Finetti theo-
rem”. J. Math. Phys. 58, 122204 (2017).
[16] Renato Renner. “Symmetry of large physi-
cal systems implies independence of subsys-
tems”. Nature Physics 3, 645–649 (2007).
[17] Matthias Christandl, Robert König, and
Renato Renner. “Postselection technique
for quantum channels with applications to
quantum cryptography”. Phys. Rev. Lett.
102, 020504 (2009).
[18] R. Renner and J. I. Cirac. “de Finetti repre-
sentation theorem for infinite-dimensional
quantum systems and applications to quan-
tum cryptography”. Phys. Rev. Lett. 102,
110504 (2009).
Accepted in Quantum 2025-01-23, click title to verify. Published under CC-BY 4.0. 12
[19] Fernando G. S. L. Brandão and Martin B.
Plenio. “A generalization of quantum stein’s
lemma”. Commun. Math. Phys. 295, 791–
828 (2010).
[20] Miguel Navascués, Masaki Owari, and Mar-
tin B. Plenio. “Power of symmetric exten-
sions for entanglement detection”. Phys.
Rev. A 80, 052306 (2009).
[21] Fernando G. S. L. Brandão, Matthias Chri-
standl, and Jon Yard. “Faithful squashed
entanglement”. Commun. Math. Phys. 306,
805 (2011).
[22] Fernando G.S.L. Brandão, Matthias Chris-
tandl, and Jon Yard. “A quasipolynomial-
time algorithm for the quantum separabil-
ity problem”. In Proceedings of the Forty-
Third Annual ACM Symposium on The-
ory of Computing. Page 343–352. STOC
’11New York, NY, USA (2011). Association
for Computing Machinery.
[23] Fernando G. S. L. Brandão and Aram W.
Harrow. “Quantum de Finetti Theorems
Under Local Measurements with Applica-
tions”. Commun. Math. Phys. 353, 469–
506 (2017).
[24] R. L. Hudson. “Analogs of de Finetti’s the-
orem and interpretative problems of quan-
tum mechanics”. Found Phys 11, 805–
808 (1981).
[25] Jonathan Barrett and Matthew Leifer.
“The de Finetti theorem for test spaces”.
New J. Phys. 11, 033024 (2009).
[26] Matthias Christandl and Ben Toner. “Fi-
nite de Finetti theorem for conditional
probability distributions describing phys-
ical theories”. J. Math. Phys. 50,
042104 (2009).
[27] Rotem Arnon-Friedman and Renato Ren-
ner. “de Finetti reductions for correlations”.
J. Math. Phys. 56, 052203 (2015).
[28] K. B. Laskey. “Quantum Causal Net-
works” (2007). arXiv:0710.1200.
[29] Matthew S Leifer and Robert W Spekkens.
“Towards a formulation of quantum theory
as a causally neutral theory of bayesian in-
ference”. Phys. Rev. A 88, 052130 (2013).
[30] Eric G Cavalcanti and Raymond Lal. “On
modifications of reichenbach’s principle of
common cause in light of bell’s theorem.”. J.
Phys. A: Math. Theor. 47, 424018 (2014).
[31] Tobias Fritz. “Beyond bell’s theorem ii:
Scenarios with arbitrary causal structure”.
Comm. Math. Phys.Pages 1–44 (2015).
[32] Christopher J. Wood and Robert W.
Spekkens. “The lesson of causal discov-
ery algorithms for quantum correlations:
Causal explanations of Bell-inequality vio-
lations require fine-tuning”. New J. Phys.
17, 033002 (2015).
[33] Joe Henson, Raymond Lal, and Matthew F
Pusey. “Theory-independent limits on cor-
relations from generalized bayesian net-
works.”. New J. Phys. 16, 113043 (2014).
[34] Jacques Pienaar and Časlav Brukner.
“A graph-separation theorem for quan-
tum causal models.”. New J. Phys. 17,
073020 (2015).
[35] Rafael Chaves, Christian Majenz, and
David Gross. “Information–theoretic impli-
cations of quantum causal structures”. Nat.
Commun.6(2015).
[36] Katja Ried, Megan Agnew, Lydia Vermey-
den, Dominik Janzing, Robert W Spekkens,
and Kevin J Resch. “A quantum advantage
for inferring causal structure”. Nat. Phys.
11, 414–420 (2015).
[37] Fabio Costa and Sally Shrapnel. “Quan-
tum causal modelling”. New J. of Phys. 18,
063032 (2016).
[38] Sally Shrapnel and Fabio Costa. “Causation
does not explain contextuality”. Quantum
2,63(2018).
[39] John-Mark A. Allen, Jonathan Barrett, Do-
minic C. Horsman, Ciarán M. Lee, and
Robert W. Spekkens. “Quantum common
causes and quantum causal models”. Phys.
Rev. X 7, 031021 (2017).
[40] Christina Giarmatzi and Fabio Costa. “A
quantum causal discovery algorithm”. npj
Quant. Inf. 4, 17 (2018).
[41] Jonathan Barrett, Robin Lorenz, and
Ognyan Oreshkov. “Quantum causal mod-
els” (2019). arXiv:1906.10726v1.
[42] J. C. Pearl and E. G. Cavalcanti. “Classi-
cal causal models cannot faithfully explain
Bell nonlocality or Kochen-Specker contex-
tuality in arbitrary scenarios”. Quantum 5,
518 (2021).
[43] O. Oreshkov, F. Costa, and Č. Brukner.
“Quantum correlations with no causal or-
der”. Nat. Commun. 3, 1092 (2012).
[44] Ognyan Oreshkov and Christina Giarmatzi.
Accepted in Quantum 2025-01-23, click title to verify. Published under CC-BY 4.0. 13
“Causal and causally separable processes”.
New J. of Phys. 18, 093020 (2016).
[45] Mateus Araújo, Cyril Branciard, Fabio
Costa, Adrien Feix, Christina Giarmatzi,
and Časlav Brukner. “Witnessing causal
nonseparability”. New J. Phys. 17,
102001 (2015).
[46] G. Chiribella, G. M. D’Ariano, P. Perinotti,
and B. Valiron. “Quantum computations
without definite causal structure”. Phys.
Rev. A 88, 022318 (2013).
[47] M. Araújo, F. Costa, and Č. Brukner.
“Computational Advantage from
Quantum-Controlled Ordering of Gates”.
Phys. Rev. Lett. 113, 250402 (2014).
[48] Adrien Feix, Mateus Araújo, and Časlav
Brukner. “Quantum superposition of the or-
der of parties as a communication resource”.
Phys. Rev. A 92, 052326 (2015).
[49] Philippe Allard Guérin, Adrien Feix, Ma-
teus Araújo, and Časlav Brukner. “Expo-
nential communication complexity advan-
tage from quantum superposition of the di-
rection of communication”. Phys. Rev. Lett.
117, 100502 (2016).
[50] Ding Jia and Fabio Costa. “Causal order
as a resource for quantum communication”.
Phys. Rev. A100 (2019).
[51] Kaumudibikash Goswami and Fabio Costa.
“Classical communication through quan-
tum causal structures”. Phys. Rev. A 103,
042606 (2021).
[52] L. Hardy. “Towards quantum gravity: a
framework for probabilistic theories with
non-fixed causal structure”. J. Phys. A:
Math. Gen. 40, 3081–3099 (2007).
[53] Magdalena Zych, Fabio Costa, Igor
Pikovski, and Časlav Brukner. “Bell’s the-
orem for temporal order”. Nat. Commun.
10, 3772 (2019).
[54] Lucien Hardy. “Implementation of the
quantum equivalence principle”. In Fe-
lix Finster, Domenico Giulini, Johannes
Kleiner, and Jürgen Tolksdorf, editors,
Progress and Visions in Quantum Theory
in View of Gravity. Pages 189–220. Springer
International Publishing (2020).
[55] Lachlan Parker and Fabio Costa. “Back-
ground Independence and Quantum Causal
Structure”. Quantum 6, 865 (2022).
[56] Kavan Modi. “Operational approach to
open dynamics and quantifying initial cor-
relations”. Scientific Reports 2, 581 (2012).
[57] Simon Milz, Felix A. Pollock, and Ka-
van Modi. “An introduction to operational
quantum dynamics”. Open Syst. Inf. Dyn.
24, 1740016 (2017).
[58] Felix A. Pollock, César Rodríguez-Rosario,
Thomas Frauenheim, Mauro Paternostro,
and Kavan Modi. “Operational markov con-
dition for quantum processes”. Phys. Rev.
Lett. 120, 040405 (2018).
[59] Sally Shrapnel, Fabio Costa, and Gerard
Milburn. “Quantum markovianity as a su-
pervised learning task”. Int. J. Quantum
Inf. 16, 1840010 (2018).
[60] Christina Giarmatzi and Fabio Costa.
“Witnessing quantum memory in non-
Markovian processes”. Quantum 5,
440 (2021).
[61] I. A. Luchnikov, S. V. Vintskevich,
H. Ouerdane, and S. N. Filippov. “Simula-
tion complexity of open quantum dynamics:
Connection with tensor networks”. Phys.
Rev. Lett. 122, 160401 (2019).
[62] Joshua Morris, Felix A. Pollock, and Kavan
Modi. “Quantifying non-markovian mem-
ory in a superconducting quantum com-
puter”. Open Systems & Information Dy-
namics 29, 2250007 (2022).
[63] Kevin Young, Stephen Bartlett, Robin J.
Blume-Kohout, John King Gamble, Daniel
Lobser, Peter Maunz, Erik Nielsen, Tim-
othy James Proctor, Melissa Revelle,
and Kenneth Michael Rudinger. “Di-
agnosing and destroying non-markovian
noise”. Technical Report SAND-2020-
10396691214. Sandia National Lab. (SNL-
CA) (2020).
[64] G. A. L. White, C. D. Hill, F. A. Pollock,
L. C. L. Hollenberg, and K. Modi. “Demon-
stration of non-markovian process charac-
terisation and control on a quantum pro-
cessor”. Nat Commun 11, 6301 (2020).
[65] K. Goswami, C. Giarmatzi, C. Monterola,
S. Shrapnel, J. Romero, and F. Costa.
“Experimental characterization of a non-
markovian quantum process”. Phys. Rev.
A104, 022432 (2021).
[66] G.A.L. White, F.A. Pollock, L.C.L. Hol-
lenberg, K. Modi, and C.D. Hill. “Non-
Accepted in Quantum 2025-01-23, click title to verify. Published under CC-BY 4.0. 14
markovian quantum process tomography”.
PRX Quantum 3, 020344 (2022).
[67] Liang Xiang, Zhiwen Zong, Ze Zhan, Ying
Fei, Chongxin Run, Yaozu Wu, Wenyan
Jin, Zhilong Jia, Peng Duan, Jianlan Wu,
Yi Yin, and Guoping Guo. “Quantify
the non-markovian process with interven-
ing projections in a superconducting pro-
cessor” (2021). arXiv:2105.03333.
[68] Christina Giarmatzi, Tyler Jones, Alexei
Gilchrist, Prasanna Pakkiam, Arkady Fe-
dorov, and Fabio Costa. “Multi-time quan-
tum process tomography of a superconduct-
ing qubit” (2023). arXiv:2308.00750.
[69] Lorenzo M Procopio, Amir Moqanaki,
Mateus Araújo, Fabio Costa, Irati A
Calafell, Emma G Dowd, Deny R Hamel,
Lee A Rozema, Časlav Brukner, and Philip
Walther. “Experimental superposition of
orders of quantum gates”. Nat. Commun.
6, 7913 (2015).
[70] Giulia Rubino, Lee A. Rozema, Adrien
Feix, Mateus Araújo, Jonas M. Zeuner,
Lorenzo M. Procopio, Časlav Brukner, and
Philip Walther. “Experimental verification
of an indefinite causal order”. Sci. Adv. 3,
e1602589 (2017).
[71] Giulia Rubino, Lee Arthur Rozema,
Francesco Massa, Mateus Araújo, Mag-
dalena Zych, Časlav Brukner, and Philip
Walther. “Experimental entanglement of
temporal orders” (2017). arXiv:1712.06884.
[72] K. Goswami, C. Giarmatzi, M. Kewming,
F. Costa, C. Branciard, J. Romero, and
A. G. White. “Indefinite causal order in
a quantum switch”. Phys. Rev. Lett. 121,
090503 (2018).
[73] Yu Guo, Xiao-Min Hu, Zhi-Bo Hou, Huan
Cao, Jin-Ming Cui, Bi-Heng Liu, Yun-Feng
Huang, Chuan-Feng Li, Guang-Can Guo,
and Giulio Chiribella. “Experimental trans-
mission of quantum information using a su-
perposition of causal orders”. Phys. Rev.
Lett. 124, 030502 (2020).
[74] K. Goswami, Y. Cao, G. A. Paz-Silva,
J. Romero, and A. G. White. “Increas-
ing communication capacity via superpo-
sition of order”. Phys. Rev. Research 2,
033292 (2020).
[75] Kejin Wei, Nora Tischler, Si-Ran Zhao, Yu-
Huai Li, Juan Miguel Arrazola, Yang Liu,
Weijun Zhang, Hao Li, Lixing You, Zhen
Wang, Yu-Ao Chen, Barry C. Sanders,
Qiang Zhang, Geoff J. Pryde, Feihu Xu,
and Jian-Wei Pan. “Experimental quantum
switching for exponentially superior quan-
tum communication complexity”. Phys.
Rev. Lett. 122, 120504 (2019).
[76] Márcio M. Taddei, Jaime Cariñe, Daniel
Martínez, Tania García, Nayda Guerrero,
Alastair A. Abbott, Mateus Araújo, Cyril
Branciard, Esteban S. Gómez, Stephen P.
Walborn, Leandro Aolita, and Gustavo
Lima. “Computational advantage from the
quantum superposition of multiple tempo-
ral orders of photonic gates”. PRX Quan-
tum 2, 010320 (2021).
[77] Dominic Horsman, Chris Heunen,
Matthew F. Pusey, Jonathan Barrett,
and Robert W. Spekkens. “Can a quantum
state over time resemble a quantum state
at a single time?”. Proc. Math. Phys. Eng.
Sci. 473, 20170395 (2017).
[78] Robert Oeckl. “A “general boundary” for-
mulation for quantum mechanics and quan-
tum gravity”. Phys. Lett. B 575, 318–
324 (2003).
[79] G. Chiribella, G. M. D’Ariano, and
P. Perinotti. “Transforming quantum op-
erations: Quantum supermaps”. EPL (Eu-
rophysics Letters) 83, 30004 (2008).
[80] Paolo Perinotti. “Causal structures and the
classification of higher order quantum com-
putations”. Pages 103–127. Springer Inter-
national Publishing. Cham (2017).
[81] Alessandro Bisio and Paolo Perinotti. “The-
oretical framework for higher-order quan-
tum theory”. Proc. Math. Phys. Eng. Sci.
475, 20180706 (2019).
[82] Yakir Aharonov, Sandu Popescu, Jeff Tol-
laksen, and Lev Vaidman. “Multiple-time
states and multiple-time measurements in
quantum mechanics”. Phys. Rev. A 79,
052110 (2009).
[83] Ralph Silva, Yelena Guryanova, Nicolas
Brunner, Noah Linden, Anthony J. Short,
and Sandu Popescu. “Pre- and postselected
quantum states: Density matrices, tomog-
raphy, and kraus operators”. Phys. Rev. A
89, 012121 (2014).
[84] Ralph Silva, Yelena Guryanova, Anthony J.
Short, Paul Skrzypczyk, Nicolas Brunner,
Accepted in Quantum 2025-01-23, click title to verify. Published under CC-BY 4.0. 15
and Sandu Popescu. “Connecting processes
with indefinite causal order and multi-
time quantum states”. New J. Phys. 19,
103022 (2017).
[85] Jordan Cotler and Frank Wilczek. “En-
tangled histories”. Physica Scripta 2016,
014004 (2016).
[86] Jordan Cotler, Chao-Ming Jian, Xiao-
Liang Qi, and Frank Wilczek. “Super-
density operators for spacetime quantum
mechanics”. J. High Energ. Phys. 2018,
93 (2018).
[87] Teiko Heinosaari and Mário Ziman. “The
mathematical language of quantum theory:
From uncertainty to entanglement”. Cam-
bridge University Press. (2011).
[88] A. Jamiołkowski. “Linear transformations
which preserve trace and positive semidefi-
niteness of operators”. Rep. Math. Phys 3,
275–278 (1972).
[89] Man-Duen Choi. “Completely positive lin-
ear maps on complex matrices”. Linear Al-
gebra Appl. 10, 285–290 (1975).
[90] Sally Shrapnel, Fabio Costa, and Gerard
Milburn. “Updating the born rule”. New
J. Phys. 20, 053010 (2018).
[91] Dennis Kretschmann and Reinhard F.
Werner. “Quantum channels with memory”.
Phys. Rev. A 72, 062323 (2005).
[92] Gus Gutoski and John Watrous. “Toward
a general theory of quantum games”. In
Proceedings of 39th ACM STOC. Pages
565–574. (2006). arXiv:quant-ph/0611234.
[93] G. Chiribella, G. M. D’Ariano, and
P. Perinotti. “Quantum circuit architec-
ture”. Phys. Rev. Lett. 101, 060401 (2008).
[94] G. Chiribella, G. M. D’Ariano, and
P. Perinotti. “Theoretical framework for
quantum networks”. Phys. Rev. A 80,
022339 (2009).
[95] A. Bisio, G. Chiribella, G. D’Ariano, and
P. Perinotti. “Quantum networks: General
theory and applications”. Acta Phys. Slo-
vaca 61, 273–390 (2011).
[96] Mario Berta, Francesco Borderi, Omar
Fawzi, and Volkher B. Scholz. “Semidefinite
programming hierarchies for constrained bi-
linear optimization”. Mathematical Pro-
gramming 194, 781–829 (2022).
[97] Julian Wechs, Alastair A Abbott, and
Cyril Branciard. “On the definition
and characterisation of multipartite causal
(non)separability”. New J. of Phys. 21,
013027 (2019).
[98] Matthew F. Pusey. Private communica-
tion (2019).
[99] Persi Diaconis. “Finite forms of de Finetti’s
theorem on exchangeability”. Synthese 36,
271–281 (1977).
[100] P. Diaconis and D. Freedman. “Finite Ex-
changeable Sequences”. Ann. Probab. 8,
745 – 764 (1980).
[101] Robert König and Renato Renner. “A de
Finetti representation for finite symmet-
ric quantum states”. J. Math. Phys. 46,
122108 (2005).
[102] Robert König and Graeme Mitchison. “A
most compendious and facile quantum de
Finetti theorem”. J. Math. Phys. 50,
012105 (2009).
A On the definition of extendibility
The definition of process extendibility, Eq. (18),
is rather strong: we ask that we recover the same
W(n)from W(n+1) for every CPTP map Mn+1
(in particular, together with symmetry, this im-
plies no signalling across different trials). This
is a strictly stronger condition than state ex-
tendibility, for which we only ask to recover W(n)
for Mn+1 =1/dO—we show this with an explicit
example in Sec. A.3. As we have seen, state ex-
tendibility is sufficient to prove our main result,
Theorem 8, but we can ask whether other def-
initions work too. We look at two meaningful
options, one turning out to be too weak, while
the second still being sufficient to prove the de
Finetti theorem for processes.
A.1 Weak extendibility
It is meaningful to consider scenarios where we
only impose that the n-trial process is recovered
from the n+1-trial one for some particular CPTP
map.
Definition 11. A sequence of process matrices
W(n),n≥1, is weakly extendible if there is a
particular CPTP map Msuch that
W(n)= Trn+1 hW(n+1)(1A1...An⊗M)i.(48)
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We say that W(n)is weakly exchangeable if it is
symmetric and weakly extendible. The following
counterexample shows that weak exchangeability
is not sufficient to ensure a de Finetti representa-
tion for the processes W(n).
Counterexample. Consider a scenario with one
site Aper trial, with isomorphic input and output
spaces AI,AOof finite dimension d. For ntrials,
define the “idle” process
WA1→···→An:= ρ⊗[[1d]]⊗n⊗1d,(49)
where 1dis the d-dimensional identity operator
and we use the notation
[[1d]] :=
d
X
j,k=1
|j⟩⟨k|⊗|j⟩⟨k|(50)
to denote the Choi representation of the identity
channel (and, as in the rest of this work, we order
tensor factors according to A1, . . . , An). The idle
process is a causally ordered process where the
first site, A1, receives the state ρ, and each site is
linked to the next one through the identity map,
in the order A1≺ · · · ≺ An.
Now consider the symmetrised process
W(n)
sym := 1
n!X
σ∈Sn
WAσ(1)→···→Aσ(n),(51)
where the sum is over all n-element permu-
tations. By construction, W(n)
sym is symmetric
under permutations. We can also see that it
satisfies weak extendibility for the CPTP map
M= [[1d]]. Indeed, plugging the identity map
into site An+1 of the permuted idle process,
WAσ(1)→···→Aσ(n+1) , is equivalent to plugging the
identity map into site Aσ(n+1) of the unpermuted
process, Eq. (49), which, for any permutation
σ, always gives back the idle process with one
less site. Summing over all permutations then
yields Trn+1 hW(n+1)
sym (1A1...An⊗[[1d]])i=W(n)
sym,
in agreement with the definition of weak ex-
tendibility. In summary, W(n)
sym is a weakly ex-
changeable process for every n, but it is clearly
not in the de Finetti form4, so weak exchangeabil-
ity is not sufficient to deduce a de Finetti repre-
sentation.
Note that weak exchangeability for M=
1AIAO/dAOreduces to state exchangeability,
4A simple way to prove this is to note that W(n)
sym allows
some signalling between any pair of sites, which is not
possible for processes in the de Finetti form.
which we have seen does lead to a de Finetti theo-
rem. Hence, although weak exchangeability does
not work in general, it does for specific choices of
CPTP Min Eq. (48).
A.2 Channel extendibility
A process can be seen as a channel from all out-
put spaces to all input spaces, where the Choi
representation of the channel coincides with the
process matrix. Operationally, this corresponds
to restricting each party’s operations to be a mea-
surement followed by an independent state prepa-
ration. The state that the parties receive and
measure in their input spaces is given by the
channel acting on the output spaces. In formu-
las, a process matrix W∈L(HA⊗ HB⊗ · · · )
defines a channel W:L(HAO⊗ HBO⊗ · · · )→
L(HAI⊗ HBI⊗ · · · )acting as
W(ρAOBO...)
:= TrAOBO... ρAOBO...TW.(52)
In this view, it is meaningful to consider a ver-
sion of extendibility that follows the definition for
channels given by Fuchs, Schack, and Scudo [10]:
Definition 12. A sequence of process matrices
W(n),n≥1, is channel extendible if, for every
state ρ,
W(n)
= Trn+1 hW(n+1)(1A1...An⊗1An+1
I⊗ρAn+1
O)i.
(53)
We say that W(n)is channel exchangeable if it
is symmetric and channel extendible.
According to this definition, we only require
to recover W(n)when we restrict An+1 to “trace
and re-prepare” operations, where the input is
traced out and an arbitrary state is prepared.
For example, the process WA1→···→Andefined in
Eq. (49)satisfies channel extendibility (because
any state prepared at the last site simply gets
traced out), although it does not satisfy symme-
try. On the other hand, the symmetrised process
W(n)
sym, Eq. (51)is not channel extendible.
It is easy to see that channel extendibility can
replace process extendibility in the de Finetti the-
orem. Indeed, fixing the re-prepared state ρto be
the maximally mixed one, channel extendibility
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implies that the normalised states W(n)/Tr W(n)
are state extendible, which is sufficient to prove
the constrained de Finetti theorem.
A.3 State extendibility does not imply process
extendibility
We have mentioned in Sec. 5of the main text
that state extendibility does not imply process
extendibility. Let us see an explicit example.
Example. Consider a scenario with one site per
trial, with isomorphic input and output spaces
of dimension d. Consider a particular case of
the idle process, Eq. (49), with initial maximally
mixed state ρ=1d/d =: 1◦, but with reversed
causal order An≺ · · · ≺ A1:
W(n)=WAn→···→A1
1◦
:= U(rn)1◦⊗[[1d]]⊗n⊗1dU†(rn),(54)
where rnis the reverse permutation of nelements,
rn(j) := n+ 1 −j, and U(rn)its unitary repre-
sentation as defined in Eq. (15).
To see that WAn→···→A1
1◦is not process ex-
tendible (in fact, that is not even channel ex-
tendible), consider a trace-replace CPTP map
M=1d⊗ρ, where ρ=1◦is an arbitrary non-
maximally-mixed state. Plugging Minto the site
An+1 (which is the first in the causal order) of
W(n+1) gives an n-site idle process, in the order
An≺ · · · ≺ A1, starting with ρ:
Trn+1 hW(n+1)(1A1...An⊗M)i
Trn+1 hWAn→···→A1
1◦(1A1...An⊗1d⊗ρ)i
=U(rn)ρ⊗[[1d]]⊗n⊗1dU†(rn)
=: WAn→···→A1
ρ,(55)
which is not the same as W(n)as defined in
Eq. (54).
On the other hand, substituting ρ=1◦in
the above calculation gives back WAn→···→A1
1◦=
W(n). This implies that the sequence of states
ρ(n):= W(n)/dnsatisfies state extendibility.
Hence we have found a sequence of (causally or-
dered) processes that is state extendible (after
normalisation) but not process (nor channel) ex-
tendible.
B Proofs of lemmas
B.1 Proof of Lemma 3
To simplify the notation, let us write rj:= r(Rj),
so we can restate the lemma as
Lemma. A sequence of exchangeable states ρ(n)
satisfies product expectation constraints, of the
form
Tr
n
O
j=1
Rj
ρ(n)
=
n
Y
j=1
rj(56)
for a set of operator R⊂L(HST), if and only if
it satisfies
Tr
n
O
j=1
σj
ρ(n)
= 0 (57)
for all
σj:=Rj−rj,(58)
Rj∈R,j= 1, . . . , n.
Proof. Let us introduce the notation
X0
j:= Rj, X1
j:= rj(59)
(where, as usual, identity operators are implied
in rj≡rj1).
We can expand the left hand side of Eq. (57)
as
Tr
n
O
j=1
σj
ρ(n)
=X
µ1...µn=0,1
Tr
n
O
j=1
(−1)µjXµj
j
ρ(n)
.(60)
Let us now assume that Eq. (56) holds for all n.
Using the exchangeability of ρ(n), and hence its
de Finetti representation, we can see that
Tr
n
O
j=1
Xµj
jρ(n)
=
n
Y
j=1
rj(61)
for all values of µ1, . . . , µn= 0,1. So Eq. (60)
becomes
X
µ1...µn=0,1
Tr
n
O
j=1
(−1)µjXµj
j
ρ(n)
=
n
Y
j=1
X
µj=0,1
(−1)µjrj
= 0.
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This proves that condition (56) implies (57).
For the other direction, we can proceed by in-
duction. For n= 1, using Tr ρ(1) = 1 we see
directly that
Tr h(R1−r1)ρ(1)i= 0 (62)
is equivalent to
Tr hR1ρ(1)i=r1.(63)
Assume now that Eq. (57) holds for some n=
¯nand that Eq. (56) holds for all n < ¯n. This
implies
Tr
¯n
O
j=1
Xµj
jρ(¯n)
=
¯n
Y
j=1
rj(64)
for P¯n
j=1 µj>0; that is, for all choices of
µ1, . . . , µ¯nexcept µ1=· · · =µ¯n= 0, which cor-
responds to the term Tr hN¯n
j=1 Rjρ(¯n)i. Using
again the expansion (60), we have
Tr
¯n
O
j=1
σj
ρ(n)
= Tr
¯n
O
j=1
Rjρ(¯n)
−
¯n
Y
j=1
rj.(65)
Together with (57), this implies (56) for an arbi-
trary ¯n, concluding the proof.
B.2 Proof of Lemma 4
Let us re-state the lemma:
Lemma. Given two inner product spaces V1,
V2, a measurable space Y, a set of linear func-
tions Ly:V1→V2,y∈Y, and for any v∈V1
Ly(v)=0 a.e. for y∈Y(66)
if and only if, for any strictly positive measure q
over Y(q(y)>0a.e.)
ZY
dy q(y)L†
y◦Ly(v)=0,(67)
Proof. If L†
y= 0 a.e. for y∈Y, then it is clear
that ZY
dy q(y)L†
y◦Ly(v)=0 (68)
for any measure q. We need to prove that, for any
strictly positive q, Eq. (68) implies Ly(v)=0a.e.
for y∈Y. To this end, it is sufficient to take the
inner product of Eq. (68) with v:
0 = v
ZY
dy q(y)L†
y◦Ly(v)
=ZY
dy q(y)⟨Ly(v)|Ly(v)⟩.(69)
As the integrand is non-negative a.e., and q(y)is
strictly positive, Eq. (69) implies that Ly(v) = 0
a.e. in Y, which proves the lemma.
C Other classes of constraints
Here we consider some modifications to the hy-
pothesis of Theorem 8, some of which lead to a
corresponding de Finetti representation and some
of which do not. Below, it will always be assumed
that ρ(n)are exchangeable states for all n.
C.1 Inequality constraints
As in the case of product expectation constraints,
consider a set of single-trial observables Rand a
function r:R→R. However, assume we are
only given a bound on the expectation values, in
the form
tr
n
O
j=1
Rj
ρ(n)
≥
n
Y
j=1
r(Rj).(70)
for an exchangeable ρ(n). This constraint does
not imply a representation of the form
ρ(n)=ZTr(Rρ)≥r(R)∀R∈R
dρP (ρ)ρ⊗n.(71)
Counterexample As single-trial space, let us
take a two-dimensional system with basis vectors
|0⟩and |1⟩. Consider an operator Rsuch that
⟨0|R|0⟩=r0≥0,⟨1|R|1⟩=r1> r0.(72)
Now consider the exchangeable states
ρ(n):= 1
2|0⟩⟨0|⊗n+|1⟩⟨1|⊗n.(73)
These states satisfy
Tr R⊗nρ(n)=1
2(rn
0+rn
1)≥rn,(74)
r:=r0+r1
2.
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Eq. (74)is an inequality constraint of the form
(70). However, ρ(n)cannot be decomposed in the
form (71): Eq. (73)is already in the de Finetti
form, where the probability
P(ρ) = 1
2[δ(ρ− |0⟩⟨0|) + δ(ρ− |1⟩⟨1|)]
does not vanish on ρ=|0⟩⟨0|, and Tr (R|0⟩⟨0|) =
r0< r.
C.2 Single- and odd-trial constraints
C.2.1 Single- and odd-trial constraints without a
constrained de Finetti representation
Let us consider some single-trial operator Rand
a real number r. We can find exchangeable
states ρ(n)that satisfy the single-trial constraint
Tr Rρ(1) =rbut whose de Finetti representa-
tion does not have support limited to states with
Tr Rρ =r.
As a counterexample, we can take again the
state in Eq. (73),ρ(n):= 1
2|0⟩⟨0|⊗n+|1⟩⟨1|⊗n,
and as observable the Pauli operator R=Z.
Since ρ(1) =1
2, we have ⟨Z⟩= 0, even though
Tr Zρ =±1for the two states appearing in the
de Finetti representation, ρ=|0⟩⟨0|,|1⟩⟨1|. In
fact, for any odd nwe have ⟨Z⊗n⟩= 0 = ⟨Z⟩n,
so, in general, imposing a product expectation
constraint only on an odd number of trials does
not imply a constrained de Finetti representation.
Note also that, since ρ(n)is symmetric, single-
trial constraints can be expressed equivalently as
1
n⟨R⊗1· · · ⊗ 1+· · · +1· · · ⊗ 1⊗R⟩=r, namely
as a constraint on the sample mean of the oper-
ator R. The failure of the constrained de Finetti
representation tells us (the known fact) that fix-
ing the expectation value of the sample mean of
an observable does not fix the expectation value
of that observable for the ‘unknown state’.
C.2.2 Single-trial constraints with a constrained
de Finetti representation
Consider a self-adjoint operator R≥r; that is,
such that R−ris positive semidefinite for some
given real number r. In this case, given an ex-
changeable sequence of states ρ(n), it is sufficient
to impose the single-trial constraint
Tr Rρ(1) =r(75)
to deduce that the de Finetti representation can
be restricted to states ρsuch that Tr Rρ =r.
Proof. R≥rimplies that, for every state ρ,
Tr Rρ ≥r. Therefore, the de Finetti represen-
tation for a single trial gives
Tr Rρ(1) =ZS
dρP (ρ) Tr Rρ ≥r, (76)
with equality only possible if P(ρ)=0a.e. for
Tr Rρ > r. This means that we can write the
n-trial state as
ρ(n)=Zρ∈S
TrRρ≥r
dρP (ρ)ρ⊗n.(77)
C.3 States invariant under joint group action
– constraints with the wrong sign
Given a group Gand a unitary representation
Ugon the single-trial space, g∈G, consider ex-
changeable states ρ(n)that are invariant under
the joint action of G:
U⊗n
gρ(n)U†
g⊗n=ρ(n)∀g∈G. (78)
One may ask whether states of this type can be
decomposed as
ρ(n)=ZUgρU†
g=ρ
dρ P (ρ)ρ⊗n,(79)
but this is not necessarily true. A counterexample
is given by
ρ(n)=Zdψ |ψ⟩⟨ψ|⊗n,(80)
where dψ is the measure on pure states induced
by the Haar measure (i.e., the unique measure in-
variant under arbitrary unitary transformations).
ρ(n)is exchangeable and invariant under joint lo-
cal unitaries, however, the probability measure
has support over pure states, which are not in-
variant under arbitrary unitaries.
It is interesting that constraint (78)is almost
of the form (28)of a de Finetti-type constraint,
but it has the wrong sign: defining the linear op-
erators Lg
1(ρ) := UgρU†
gand Lg
2(ρ) := −ρ, we can
write the single-trial constraint as (Lg
1+Lg
2)(ρ) =
Ugρ(1)U†
g−ρ(1) = 0. However, this choice of func-
tions implies the wrong two-trial constraint: in
2
X
j=1
Lg
j⊗Lg
j(ρ(2)) = Ug⊗Ugρ(2) U†
g⊗U†
g+ρ(2) = 0
the sign in front of ρ(2) is the opposite as com-
pared to (78)for n= 2.
Accepted in Quantum 2025-01-23, click title to verify. Published under CC-BY 4.0. 20
