The Multi-round Process Matrix
Timothée Hoffreumon and Ognyan Oreshkov
Centre for Quantum Information and Communication (QuIC), École polytechnique de Bruxelles, CP 165, Université libre de
Bruxelles, 1050 Brussels, Belgium.
We develop an extension of the pro-
cess matrix (PM) framework for correla-
tions between quantum operations with no
causal order that allows multiple rounds
of information exchange for each party
compatibly with the assumption of well-
defined causal order of events locally. We
characterise the higher-order process de-
scribing such correlations, which we name
the multi-round process matrix (MPM),
and formulate a notion of causal nonsep-
arability for it that extends the one for
standard PMs. We show that in the multi-
round case there are novel manifestations
of causal nonseparability that are not cap-
tured by a naive application of the stan-
dard PM formalism: we exhibit an in-
stance of an operator that is both a valid
PM and a valid MPM, but is causally sepa-
rable in the first case and can violate causal
inequalities in the second case due to the
possibility of using a side channel.
1 Introduction
There has recently been significant interest in
quantum processes in which the operations per-
formed by separate parties exhibit ‘indefinite
causal order’ [2, 3, 5, 6, 9, 11, 12, 17, 20–
23, 25, 31, 33, 37–39, 42–47, 49–52]. A formal
definition of this feature, termed causal nonsepa-
rability [6, 37, 38, 50], has been given in the pro-
cess matrix (PM) framework [38], which describes
correlations between elementary quantum exper-
iments, each defined by a pair of input and an
output system, also referred to as a quantum node
[3, 7, 8], over which an agent could apply differ-
ent operations, without presuming the existence
of global causal order among the separate oper-
ations but only the validity of causal quantum
Timothée Hoffreumon: hoffreumon.timothee@ulb.ac.be
Ognyan Oreshkov: oreshkov@ulb.ac.be
theory [16] for their local description. Causally
nonseparable processes have been shown to ac-
complish informational tasks that are not possible
via processes in which the operations are used in a
defined order [5, 14, 20, 23, 44]. They have been
conjectured to be potentially relevant in quan-
tum gravity scenarios [17, 25, 38, 52] where the
causal structure of spacetime may be subject to
quantum indefiniteness, as well as in the presence
of closed timelike curves [4, 10, 12, 17, 38, 49].
But some of these processes, such as the quan-
tum SWITCH [17] for which most of the exam-
ples of advantage have been found, also admit
realisations within standard quantum mechan-
ics on time-delocalized systems [35] via coher-
ent control of the order of operations. This has
been demonstrated in several experimental setups
[22, 33, 43, 45, 46, 51], offering blueprints for pos-
sible applications.
In view of developing potential applications of
indefinite causal order, the standard PM frame-
work appears limited by the fact that each party
is assumed restricted to a single round of informa-
tion exchange, where from the local causal per-
spective of that party [24], information is received
once via the input system and subsequently sent
out once via the output system. Practical com-
munication protocols and distributed computing
algorithms generally involve multiple rounds of
information exchange between separate parties
that can use local memory and condition the op-
erations applied at a given time on information
obtained at other times. It is therefore natural
to ask whether the PM framework can be ex-
tended to allow for multiple rounds of communi-
cation and whether such an extension would con-
tain any new possibilities that are not captured
by the standard PM framework.
In this paper, we formulate a multi-round ex-
tension of the PM framework in which each party
can perform operations over an ordered sequence
of quantum nodes, assuming standard causality
locally but not globally. The formalism is analo-
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arXiv:2005.04204v2 [quant-ph] 19 Jan 2021
gous to the PM formalism, except that the local
operations are now generalised to quantum net-
works [15]—the most general causal operations
that can be applied over a given sequence of in-
put and output systems, and which are described
by the theory of quantum combs [15]—while the
process matrix is replaced by an operator that
we call the multi-round process matrix (MPM).
The MPM is a specific higher-order process in
the hierarchy of higher order processes classified
by Perinotti and Bisio [11, 39]. We derive handy
necessary and sufficient conditions for an opera-
tor to be a valid MPM, which are expressed via a
generalisation of the projector techniques intro-
duced for PMs in Ref. [6]. Given a set of par-
ties and an ordered set of nodes for each party,
a valid MPM is in particular a valid PM on all
nodes, if each node is regarded as belonging to a
separate party. However, it respects additional
constraints that ensure the possibility of using
side channels for causal communication between
the nodes within the laboratory of each actual
party. As we show, these constraints amount to
the condition that the operator of a valid MPM
is an affine combination of deterministic quantum
combs that are compatible with the local orders
of the nodes assumed for each different party.
We further formulate a notion of causal separa-
bility in the MPM framework, building upon the
one defined for PMs [37, 50]. The key insight be-
hind this definition is that since every quantum
comb can be implemented as a sequence of inde-
pendent operations connected via side channels,
by viewing these side channels as ‘external’ to the
parties’ operations, we can reduce the problem
of defining causal separability to that for stan-
dard PMs. Remarkably, we show that the possi-
bility of using side channels can make a radical
difference on the causal (non)separability of the
process: we describe an example of a bipartite
MPM with two nodes for Alice and one node for
Bob, which is such that, when viewed as a stan-
dard PM on three nodes it is causally separable
(and admitting a simple physical realisation), but
when the possibility of using a side channel in the
presumed direction between the nodes of Alice is
considered, this opens up the possibility of vio-
lating causal inequalities. This shows that the
standard PM framework, applied naively on the
nodes over which the parties can operate, does
not suffice to capture the causal nonseparability
of the process on those nodes: the in-principle
possibility of using side channels, which under-
lies the MPM concept, can have nontrivial conse-
quences.
2 The framework
Consider a set of parties, N ≡ {A, B, C, . . .}, of
finite cardinality |N|. In the original process ma-
trix (PM) framework [38], each party X is as-
sumed associated with a pair of finite-dimensional
input and output quantum systems, X
in
and
X
out
with respective Hilbert spaces H
X
in
and
H
X
out
, hereby referred to as a (quantum) node
[3, 7, 8]. The party X can perform an arbitrary
causal quantum operation (also called quantum
instrument) from the input to the output. This
is described by a collection of completely pos-
itive (CP) maps [34] corresponding to the dif-
ferent possible outcomes i
X
of the operation,
n
M
X
in
→X
out
i
X
o
, M
X
i
X
: L(H
X
in
) → L(H
X
out
),
where L(H) denotes the space of linear operators
over H. These CP maps obey the constraint that
M
X
in
→X
out
≡
P
i
X
M
X
in
→X
out
i
X
is a completely
positive and trace-preserving (CPTP) map.
In the PM formalism, CP maps are repre-
sented via positive semidefinite operators us-
ing the Choi-Jamiołkowski (CJ) isomorphism
[18, 29]. Here, we take a basis-independent
version of the isomorphism following [3, 7, 8]
(mudulo an overall transposition) for which a
CP map M
X
in
→X
out
is represented by a CJ op-
erator M
X
in
X
∗
out
∈ L
H
X
in
⊗ (H
X
out
)
∗
, where
(H
X
out
)
∗
is a copy of the dual of H
X
out
1
, via
M
X
in
X
∗
out
≡
X
i,j
M
X
in
→X
out
(|iihj|
X
in
) ⊗ |iihj|
X
∗
in
T
,
(1)
where
n
|ii
X
in
o
is an orthonormal basis of H
X
in
,
n
|ii
X
∗
in
o
the corresponding dual basis, and we
1
Note that this copy of the dual of the output Hilbert
space can be seen related to the time-reversed version of
the output Hilbert space [36] via a concrete linear map,
which was incorporated in the time-neutral generalisation
of the formalism developed in Ref. [36]. We do not invoke
this here for simplicity.
Accepted in Quantum 2021-01-03, click title to verify. Published under CC-BY 4.0. 2
identify (H
∗
)
∗
with H via the canonical iso-
morphism. For a CPTP map M
X
in
→X
out
,
Tr
X
∗
out
[M
X
in
X
∗
out
] = 1
X
in
.
Given a choice of instrument at each node, the
joint probabilities for the outcomes of all parties
are then given by the ‘generalised Born rule’
p(i
A
, i
B
, . . . |{M
A
in
→A
out
i
A
}, {M
B
in
→B
out
i
B
}, . . .) =
Tr
h
W
A
in
A
∗
out
B
in
B
∗
out
...
M
A
in
A
∗
out
i
A
⊗ M
B
in
B
∗
out
i
B
. . .
i
,
(2)
where W
A
in
A
∗
out
B
in
B
∗
out
...
∈ L(H
A
in
⊗ (H
A
out
)
∗
⊗
H
B
in
⊗ (H
B
out
)
∗
. . .). This follows simply from
the assumption that, as in standard quantum
theory, the probabilities are linear functions of
the CP maps corresponding to the outcomes.
The operator W
A
in
A
∗
out
B
in
B
∗
out
...
is called the
process matrix (PM) (also process operator
[7, 8]). The only conditions it has to satisfy
are W
A
in
A
∗
out
B
in
B
∗
out
...
≥ 0 (coming from the
requirement of non-negativity of the probabil-
ities, assuming the parties’ operations can be
extended to act on local input ancillas pre-
pared in arbitrary joint quantum states), and
Tr
h
W
A
in
A
∗
out
B
in
B
∗
out
...
M
A
in
A
∗
out
⊗ M
B
in
B
∗
out
. . .
i
=
1 on all CPTP maps M
A
in
A
∗
out
, M
B
in
B
∗
out
, . . .
(coming from the requirement that probabilities
sum up to 1). Practical necessary and sufficient
conditions for an operator to be a valid PM have
been formulated based on the types of nonzero
terms appearing in the expansion of the operator
in a Hilbert-Schmidt basis [37, 38], as well as
based on a superoperator projector [6]. In the
next section, we will use the latter, of which we
will derive a generalisation. A review of it can
be found in Appendix A.1.
Formula (2) can be interpreted as the result
of composing in a loop the local operations of
the parties with a channel
˜
W from the output
systems of all nodes to the inputs systems of all
nodes: the PM is the transpose of the CJ operator
of
˜
W, where the transposition simply reflects the
link product [15] for composing channels in the
CJ representation.
Here, we generalise the framework by relaxing
the assumption that each party is restricted to a
single round of receiving a system in and send-
ing a system out. We now assume that each
party is associated with an ordered sequence of
quantum nodes over which they can operate in
a causal fashion. Let n
X
denote the number of
Figure 1: Graphical representation of a quantum network
with 3 nodes (plain circles), and its associated quan-
tum 3-instrument obtained through CJ isomorphism C
applied on the composition of the operations at each
node.
nodes for party X. There are then 2n
X
systems
associated with X. We will label them by X
j
,
j = 0, . . . 2n
X
− 1, with X
0
and X
1
being re-
spectively the input system and the dual of the
output system of the first node, X
2
and X
3
the
input system and the dual of the output system
of the second node, and so on, with even (re-
spectively, odd) numbering referring to an input
(dual of output) system. The i-th node of X will
be compactly denoted by X
(i)
. In other words,
for i = 1, . . . , n
X
, with X
i,in
(respectively, X
i,out
)
referring to the input (output) of the i-th node
we define
H
X
2i−2
≡ H
X
i,in
, (3a)
H
X
2i−1
≡
H
X
i,out
∗
, (3b)
H
X
(i)
≡ H
X
i,in
⊗
H
X
i,out
∗
. (3c)
The most general causal quantum operation on
a sequence of n nodes is a non-deterministic quan-
tum network [15], which can be implemented as
a sequence of n
X
instruments acting on the given
nodes plus ancillary systems connected via side
channels, as illustrated on the left-hand side of
Fig. 1 for the case of 3 nodes. Such a network
amounts to implementing a quantum operation
from the joint input system of all nodes to the
joint output system of all nodes, called an n-
instrument [15]. In the CJ representation, an n
X
-
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instrument is described by a collection of positive
semidefinite (PSD) operators {M
X
0
X
1
···X
2n
X
−1
i
},
defined on L(H
X
0
⊗H
X
1
. . . ⊗H
X
2n
X
−1
). We will
shorten the notation using X
0
X
1
···X
2n
X
−1
≡ X
to lessen clutter when there is no ambiguity, writ-
ing
n
M
X
i
o
∈ L
H
X
. Each M
X
i
is labelled
by the outcome index i, which in this case is
a poly-index corresponding to the collection of
outcomes at the different steps in the network,
i ≡ (i
X
(1)
, . . . , i
X
(n
X
)
). The operators M
X
i
are
the CJ operators of CP maps from the joint in-
put system of all nodes to the joint output system
of all nodes, called probabilistic quantum (n
X
-
)combs [15]. These operators must satisfy the
condition that M
X
≡
P
i
M
X
i
is a deterministic
quantum (n
X
-)comb [15]. The latter is the CJ op-
erator of a deterministic n
X
-instrument, i.e. the
most general CPTP map from the joint input sys-
tem of the nodes to the joint output system of the
nodes that can be implemented via a network of
channels as in the left-hand side of Fig. 1. An op-
erator M
X
is a deterministic quantum n
X
-comb
if and only if it obeys the following constraints
[15]:
M
X
0
···X
2n
X
−1
≥ 0 , (4a)
∀j : 2 ≤ j ≤ n
X
,
Tr
X
2j−1
h
M
X
0
···X
2j−1
i
= 1
X
2j−2
⊗ M
X
0
···X
2j−3
,
Tr
X
1
h
M
X
0
X
1
i
= 1
X
0
,
(4b)
where 1
Y
denotes the unit matrix on subsystem
Y .
Analogously to the PM framework, we assume
that the joint probabilities for the outcomes of
the generalised instruments performed by the sep-
arate parties in this multi-round setting are lin-
ear functions of the probabilistic quantum combs
associated with the outcomes, as would be the
case in standard quantum mechanics if the nodes
are associated with quantum systems at definite
times. This means that we can write these prob-
abilities in the ‘generalised Born rule’ form
p(i
A
, i
B
, . . . |{M
A
i
A
}, {M
B
i
B
}, . . .)
= Tr
h
W
AB...
·
M
A
i
A
⊗ M
B
i
B
. . .
i
, (5)
where for the superscripts that label the systems
over which the operators are defined we have used
the short-hand notation X ≡ X
0
X
1
. . . X
2n
X
−1
,
Figure 2: Graphical representation of an MPM for 2
parties A and B. A has 3 rounds and B has 2. Here,
the name of the subsystems associated with each wire
is apparent. By convention, the ordering of the indices
goes from bottom to top
X ∈ N. We call the operator W
AB...
the multi-
round process matrix (MPM) (see Fig. 2) .
As in the PM framework, the only property
we demand from W
AB...
is that it yields valid
probabilities through Eq. (5) for all generalised
instruments that can be applied by the parties,
including when the instruments are extended to
act on ancillary input systems in arbitrary quan-
tum states. It is straightforward to see in analogy
to the argument in Ref. [38] that this is equiva-
lent to the following constraints:
W
AB...
≥ 0 , (6)
and
Tr
h
W
AB...
·
M
A
⊗ M
B
. . .
i
= 1 (7)
for all deterministic quantum combs
M
A
, M
B
, . . ..
Remark 1. In the case when the number of
nodes per party is 1, W
AB...
reduces to a stan-
dard process matrix, whereas when the number
of parties is 1 and that party has n nodes, W
A
re-
duces to (the transpose of) a deterministic quan-
tum (n+1)-comb with a trivial first input system
and a trivial last output system [15]. See Fig. 3.
Remark 2. Note that if an operator W
AB...
is
a valid MPM, it is in particular a valid PM if
each different node is interpreted as belonging to
a separate party. This is because the probabili-
ties must be equal to 1 when the parties perform
independent CPTP maps in their different nodes,
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Figure 3: Graphical representations of an MPM W for:
(left) 3 parties with a single node each [W is a PM
in this case]; (center) a party A having a single round
and B having two rounds; (right) a single party A with
3 rounds [W is a deterministic quantum comb in this
case].
which is a special case of a deterministic network.
The converse, however, is obviously not true since
a PM is not normalised on deterministic quantum
combs on arbitrary ordered subsets of its nodes
(e.g. if the PM is equivalent to a deterministic
quantum comb for a specific order of the nodes,
it would generally not admit compositions with
deterministic quantum combs in the opposite or-
der).
3 Characterisation of the set of valid
MPMs
We now derive necessary and sufficient conditions
for an operator to be a valid MPM that are easier
to handle. They can be seen as a generalisation of
the projective characterisation of the set of PMs
obtained in Ref. [6], where the authors showed
that the set of valid PMs belongs to a subspace
of the space of operators, the projector on which
can be deduced from the one on valid quantum
channels.
Our approach goes as follows. Given the pro-
jective characterisation of the space of valid quan-
tum channels in the CJ picture [6], we can itera-
tively infer a similar projective characterisation of
the space of deterministic quantum combs. This
is done in Appendix A.2, and we find that all
the deterministic quantum combs M
X
, no mat-
ter their number of nodes (or ‘teeth’), have the
same algebraic structure:
M
X
≥ 0 , (8a)
P
X
n
X
h
M
X
i
= M
X
, (8b)
Tr
h
M
X
i
=
n−1
Y
j=0
d
X
2j
≡ d
X
in
. (8c)
That is, they are elements of a positive and trace-
normalised subset within a subspace of the space
of operators on the full system. The subspace is
defined through a projector P
X
n
X
, whose super-
script refers to the system it acts upon (we will
refine the signification of the superscript in the
next section) and whose subscript indicates the
total number of nodes. As a function of X, the
exact form of the projector is given by the recur-
sive relation
P
X
n
X
= I
X
−
X
2n
X
−1
(·) +
X
2n
X
−2
X
2n
X
−1
P
X
0
n
X
−1
,
(9)
expressed in terms of the mapping
X
i
(·) ≡
1
X
i
d
X
i
⊗
Tr
X
i
[·] introduced in Ref. [6] (see Appendix A.1)
and the identity projector I
X
[O] = O , ∀ O ∈
L
H
X
. P
X
0
n
X
−1
is the (n
X
− 1)-comb projector
that acts on subsystems X
0
≡ X
0
. . . X
2n
X
−3
(im-
plicitly understood extended by a tensor product
with the identity on X
2n
X
−2
X
2n
X
−1
). The re-
cursion starts with the 0-comb projector being 1,
P
0
≡ 1, because the space of 0-combs is the real
numbers.
Here, we are interested in characterising the set
{W } of valid MPMs as defined in Eqs. (6) and
(7). Note that condition (7), which states that
an MPM is normalised on all tensor products of
deterministic quantum combs, is equivalent to the
condition that an MPM is normalised on all affine
combinations of tensor product of such combs:
Tr
"
W
AB...
·
X
i
q
i
M
A
⊗ M
B
. . .
i
#
= 1 , (10)
with
P
i
q
i
= 1 while
M
A
⊗ M
B
. . .
i
is a ten-
sor product of deterministic quantum combs that
may be different for each index i. The fact that
(10) is necessary for (7) follows from the linearity
of the trace, while its sufficiency follows from the
fact that (7) is a special case of (10).
In Appendix B, we show that an operator M
is an affine combination of tensor product of |N|
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deterministic combs if and only if it respects the
conditions
P [M ] ≡
O
X∈N
P
X
n
X
[M] = M , (11a)
Tr [M ] =
Y
X∈N
d
X
in
≡ d
in
, (11b)
where we have defined the notations P for the
overall projector and d
in
for the input dimen-
sion of the full Hilbert space. In the following,
Span {
N
M} will be used to refer quickly to the
subspace of operators spanned by a tensor prod-
uct of deterministic combs, i.e. operators satis-
fying (11a).
With these considerations, the rationale in Ref.
[6] of inferring PM validity conditions from the
projector on the space of quantum channels and
the normalisation can be generalised to MPMs
using the projective characterisation (11) of the
affine combination of tensor products of deter-
ministic combs. This result is systematised as a
theorem:
Theorem 1. Let L(H) ≡ L
H
in
⊗
H
out
∗
be
a linear space of operators defined on a Hilbert
space of dimension d ≡ d
in
d
out
. This Hilbert
space admits a tensor factorisation into 2|N| sub-
systems associated with the inputs and outputs of
|N| parties. Let {W } ⊂ L(H) be the set of valid
MPMs shared between the parties. Let P
X
n
X
be
the projector characterising the validity subspace
of the deterministic combs that can be applied by
party X. Let P ≡
N
X∈N
P
X
n
X
be the tensor prod-
uct of the different such projectors of all parties.
Then, an operator W belongs to the set {W } if
and only if it satisfies
W ≥ 0 , (12a)
Q
P[W ] ≡ (I − P + D) [W ] = W , (12b)
Tr [W ] = d
out
, (12c)
where I is the identity projector, and D is the
projector on the span of the unit matrix, D[O] =
1
d
⊗ Tr [O].
The projector
Q
P defined in Eq. (12b) will
be referred to as the quasiorthogonal projector to
P. The proof of this theorem is presented in Ap-
pendix B. It can be understood in a more mathe-
matical terminology as “the CJ representation of
the generalised instruments is a subset of a unital
subalgebra which is the quasiorthogonal comple-
ment to the subalgebra in which lies the set of
valid processes {W }” [27, 40]. Equivalently, it
says that the traceless part of an MPM is orthog-
onal (with respect to the Hilbert-Schmidt inner
product) to the traceless part of the CPTP maps
that are plugged into it [11].
We note that an equivalent characterisation
has been independently explored in [11, 39], but
without using projective methods. In particular,
Theorem 1 corresponds to the Lemma 4 of [11].
Indeed, in Perionotti and Bisio’s classification of
higher-order quantum processes, the MPM is de-
scribed by a type ((n
A
⊗ n
B
⊗ ...) → 1). That is,
a structure taking in a tensor product of quantum
combs of types n
A
, n
B
, . . . and sending them onto
the trivial type 1. Theorem 1 is therefore a di-
rect way of finding the constraints to apply on an
operator so that it is the representation of such a
type. In a subsequent work, we will present a sys-
tematic way of associating a projector to a type
structure.
4 Link between comb and MPM sub-
spaces
Using the above projective characterisation, we
are able to highlight the link between the valid-
ity subspace of deterministic combs with trivial
first input and last output and that of MPMs.
We will need to introduce the symbol ≺ to mark
the causal relation “is before”, indicating that a
node is in the causal past of another. The pro-
jector on the subspace of valid 3-combs with teeth
named A, B, C so that A ≺ B ≺ C will be de-
noted P
A≺B≺C
3
, with the total number of teeth
indicated as a subscript and the causal order-
ing as a superscript. Permutations of the order-
ing will be denoted π
i
(A, B, C), with the sub-
script referring to a particular permutation, e.g.
π
0
(A, B, C) = A ≺ B ≺ C , π
1
(A, B, C) = B ≺
A ≺ C, . . . . When the letters refer to an or-
dered set of teeth, for example A =
n
A
(1)
o
and
B =
n
B
(1)
≺ B
(2)
o
, the permutations are limited
to those that are compatible with the the order-
ing between the elements within each set. So for
this particular example, compatibility means that
permutations containing a causal relation where
B
(2)
appears before B
(1)
are forbidden, giving
{π
i
(A, B)} = {A
(1)
≺ B
(1)
≺ B
(2)
, B
(1)
≺ A
(1)
≺
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B
(2)
, B
(1)
≺ B
(2)
≺ A
(1)
}.
Observe that Theorem 1 states that the projec-
tor on the subspace of valid MPMs is quasiorthog-
onal to the projector on Span {
N
M}, where de-
terministic combs in the tensor product are as-
sociated with the different parties. For a single
party A, it is expected that the MPM is a deter-
ministic comb [15] (see Fig. 3, rightmost case).
Indeed, it can be verified using Eq. (9) that the
quasiorthogonal projector to a single comb pro-
jector is a comb projector itself; as expected, it
sends to the validity subspace of deterministic
(n
A
+ 1)-combs with trivial first input and last
output, and with node ordering compatible with
the ordering of the teeth of A. By “compatible”,
we mean here that the nodes in the (n
A
+1)-comb,
which are now the gaps between its teeth, are
properly ordered, i.e. A
(1)
≺ A
(2)
≺ . . . . Since
we are working with orthogonal projectors, equiv-
alence between projectors is sufficient to prove
the equivalence between subspaces.
We can actually use the algebra of these projec-
tors described in Appendix A.1 to generalise this
observation to any number of parties. For |N|
parties sharing an MPM with n = n
A
+ n
B
+ . . .
nodes, what we find is that the MPM validity sub-
space is the span of all the deterministic (n + 1)-
combs with trivial first input and last output such
that their node ordering is compatible with the
local ordering of each of the |N| combs that are
to be plugged into it.
For example, if Alice has one operation and
Bob two, there are three possible such 4-combs
compatible with B
(1)
≺ B
(2)
, having pro-
jectors
Q
P
A
(1)
≺B
(1)
≺B
(2)
3
,
Q
P
B
(1)
≺A
(1)
≺B
(2)
3
, and
Q
P
B
(1)
≺B
(2)
≺A
(1)
3
sending to three different sub-
spaces. The validity subspace of the MPM is then
spanned by the union of these three subspaces.
To make this claim explicit, we need to intro-
duce the notion of union of projectors [41]. For
two arbitrary linear projectors P and P
0
acting
on the same space, the projector on the span of
the union of their subspaces is the union of the
projectors given by
P ∪ P
0
= P + P
0
− PP
0
, (13)
provided they commute. Thus, for M, N ⊂
L(H) such that M = {M|P [M] = M}
and N =
n
N|P
0
[N] = N
o
, we have O ∈
Span {M ∪ N} ⇐⇒
P ∪ P
0
[O] = O.
This allow us to formally state the result as a
theorem:
Theorem 2. For a set of parties {A, B, . . .} with
each having possibly more than one node, let {π
i
}
denote the set of valid permutations between the
nodes as defined above. The projector on the lin-
ear subspace of valid MPMs shared between the
parties is then equivalent to the union of all the
quasiorthogonal projectors to the projectors on
the linear subspaces of the deterministic quantum
combs that respect the partial ordering of the teeth
for each party:
Q
P
A
n
A
⊗ P
B
n
B
⊗ . . .
=
[
π
i
Q
P
π
i
(A,B,...)
n
A
+n
B
+...
. (14)
A proof of this theorem is given in Appendix
C. It shows that any MPM, and as a special case
any PM, is a linear combination of deterministic
quantum combs that respect the local ordering of
each party. The trace condition (12c) further con-
strains the coefficients in the combination to sum
up to one, hence this is an affine combination.
5 (Non)causal correlations and causal
(non)separability for MPMs
Now that the set of valid MPMs has been
characterised, a natural ensuing question is
which ones describe causal correlations be-
tween the nodes. On a purely theory-
independent account, let the set of nodes
A
(1)
, . . . , A
(n
A
)
, B
(1)
, . . . , B
(n
B
)
, C
(1)
, . . . receive
settings ~s ≡
s
A
(1)
, s
A
(2)
, . . .
and produce out-
comes ~o ≡
o
A
(1)
, o
A
(2)
, . . .
after performing
their operations. We can formulate a notion
of the correlations p(~o|~s) being causal, building
upon the notion of causal correlations introduced
for the original process framework in Ref. [37].
The idea of that definition is that the corre-
lations are deemed casual if they are compati-
ble with a stochastic unravelling of the events at
the nodes in a causal sequence, which can be de-
scribed as follows: first, according to some prob-
ability distribution, one of all nodes is selected to
come first; then, depending on the setting cho-
sen at that node, a specific outcome occurs at
it with a specific probability; after that, one of
the remaining nodes is selected to come second
according to a probability distribution that may
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generally depend on the setting and outcome that
have occurred at the first node; this continues un-
til all nodes have been used, with the probability
for each subsequent event generally depending on
all events that have occurred in the past. In the
multiround case, the same idea can be formalised
as follows:
Definition 1 (Causal multi-round correlations).
We call the conditional probability distribution
p(~o|~s) causal if and only if it can be decomposed
as
p(~o|~s)
=
X
X∈N
q
X
(1)
p(o
X
(1)
|s
X
(1)
)p(~o
\X
(1)
|o
X
(1)
, ~s) ,
(15)
with q
X
(1)
∈ [0; 1],
P
X∈N
q
X
(1)
= 1, where
p(o
X
(1)
|s
X
(1)
) is a single-node distribution and
p(~o
\X
(1)
|o
X
(1)
, ~s) is itself a conditional causal
probability distribution on the remaining nodes
after the relabelling X
(i)
→ X
(i−1)
for i =
2, . . . , n
X
and n
X
→ n
X
− 1. We have used the
shorthand notation ~o
\X
(1)
≡ ~o \
n
o
X
(1)
o
for the
vector of remaining outcomes.
This definition is similar to the one for the
single-round case [1, 37], with the difference that
only unravellings compatible with the local causal
orders of the nodes of the parties are permitted.
This extra condition is a nontrivial addition: the
fact that the overall correlations must be com-
patible with no signalling from future nodes to
past nodes does not guarantee that if the cor-
relations admit a causal unravelling in principle,
that unravelling would respect the local orders.
Consider for example the correlations between a
party A having two nodes and a party B having
one node. Let there be no signalling from A
(2)
to A
(1)
. Hence, the correlations are in principle
compatible with the assumed local causal order-
ing. However, with this assumption alone, there
can be perfect signalling from A
(2)
to B and from
B to A
(1)
, as long as B does not forward any in-
formation obtained from A
(2)
to A
(1)
. In such
a scenario, no unravelling is compatible with the
supplementary constraint that A
(1)
is before A
(2)
.
The new definition allows us to speak about
causal inequalities for the MPM, which are
bounds on the set of joint probability distribu-
tions whose violation by a specific distribution
implies that the distribution is not causal [38].
Investigating the subject of causal inequalities for
MPMs is left for future work.
In the process formalism, a given PM is
causally separable if and only if it produces causal
correlations in which the conditional probability
distributions appearing in the causal unravelling
can themselves be seen as arising from a quan-
tum process [37] (see precise definition later in the
case of the MPM). Note that while causal separa-
bility implies causal correlations, the converse is
not true. In accordance with this definition, we
want to define causal separability for MPMs as
the ability to unravel the causal ordering of the
operations. This unravelling should be compati-
ble with the partial ordering existing between the
nodes associated to each party, and it should also
take into account the effects of the new means
of communication provided by the side channels
between those nodes. As it turns out, this re-
quirement motivates non-trivial modifications to
the current definition for PMs [37, 50]. Noticing
that every generalised instrument performed by a
given party can be thought of as consisting of in-
dependent single-node operations connected via
side channels (see below), a concise way of taking
this into account is to apply the PM definition
of causal separability on an MPM extended by
identity side channels.
Definition 2. An MPM W is causally separable
if and only if the operator obtained by extending
the nodes of each party with identity side chan-
nels between consecutive nodes is causally sep-
arable in the sense of a PM. In equation, let
there be an extension of all the parties’ subsys-
tems in an MPM W such that
N
2n
X
−1
i=0
H
X
i
→
N
2n
X
−1
i=0
H
X
i
⊗
N
2n
X
−2
j=1
H
X
0
j
, where d
X
0
2k−1
=
d
X
0
2k
, k = 1, . . . , n
X
− 1, so that a node X
(i)
,
1 < i < n
X
− 1, is now defined on H
X
2i−2
⊗
H
X
2i−1
⊗ H
X
0
2i−2
⊗ H
X
0
2i−1
, X
(1)
is defined on
H
X
0
⊗ H
X
1
⊗ H
X
0
1
, and X
(n
X
−1)
is defined on
H
X
2n
X
−2
⊗H
X
2n
X
−1
⊗H
X
0
2n
X
−2
. We define a PM
W
0
on the extended nodes as
W
0
= W ⊗
O
X∈N
n
X
−1
O
i=1
Id
X
0
2i−1
X
0
2i
, (16)
where Id
X
0
2i−1
X
0
2i
is the transpose of the CJ oper-
ator of an identity side channel from
H
X
0
2i−1
∗
to H
X
0
2i
. Then, W is a causally separable MPM
if and only if W
0
is a causally separable PM for
all dimensions of the identity side channels.
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Figure 4: Graphical interpretation of Eq.(21): an MPM
(left) taking in a 2-comb is interpreted as a PM (right)
extended by a side-channel (dashed part) taking in two
1-combs.
For completeness, we now show explicitly that
the correlations obtained through the ‘Born rule’
(5) for MPMs can be transformed into correla-
tions obtained with a PM extended by identity
side channels. We will make use of a version of
the link product [15], which allows to merge combs
together. In the context of our basis-independent
convention for the CJ isomorphism, we define the
link product between two operators M
AB
∗
and
N
BC
∗
, where A and C
∗
are separate systems or
the trivial system, and B and B
∗
are mutually
dual (i.e. H
B
∗
and H
B
are Hilbert spaces dual to
each other, assuming the canonical isomorphism
(H
∗
)
∗
∼
=
H in the finite-dimensional case), as
L
AC
∗
= M
AB
∗
∗ N
BC
∗
:=
Tr
BB
∗
h
M
AB
∗
⊗ N
BC
∗
1
A
⊗ Id
BB
∗
⊗ 1
C
∗
i
,
(17)
where Id
BB
∗
≡
P
d
B
ij
|iiihjj|
BB
∗
.
Consider a quantum 2-comb M
A
≡
M
A
0
A
1
A
2
A
3
. The CP map it describes can
be seen as resulting from the composition of
two CP maps with an intermediate identity side
channel. In the CJ representation, this is given
by the link product between the corresponding
1-combs,
M
A
= M
A
3
A
2
L
2
∗ Id
L
∗
2
L
∗
1
∗ M
L
1
A
1
A
0
, (18)
where L
∗
1
is the ancillary output system of the
first CP map (with CJ operator M
L
1
A
1
A
0
), L
2
is the ancillary input system of the second CP
map (with CJ operator M
A
3
A
2
L
2
), and Id
L
∗
2
L
∗
1
is the CJ operator of the identity side channel
connecting these systems. This is explicitly given
by
M
A
=
Tr
L
2
L
∗
2
L
1
L
∗
1
h
M
A
3
A
2
L
2
⊗ Id
L
∗
2
L
∗
1
⊗ M
L
1
A
1
A
0
×
1
A
3
A
2
⊗ Id
L
2
L
∗
2
⊗ Id
L
∗
1
L
1
⊗ 1
A
1
A
0
i
.
(19)
Computing the trace over L
∗
2
L
∗
1
yields
M
A
=
Tr
L
2
L
1
h
M
A
3
A
2
L
2
⊗ M
L
1
A
1
A
0
1
A
⊗ Id
L
2
L
1
i
.
(20)
In this last expression, Id
L
2
L
1
≡
P
d
2
L
ij
|iiihjj|
L
2
L
1
is the transpose of the CJ operator of an identity
channel from H
L
∗
1
to H
L
2
(just like a PM is the
transpose of the CJ operator of a channel from
the outputs of nodes to the inputs of nodes).
Plugging equation (20) into the ‘Born rule’ be-
tween the 2-comb M
A
and a 2-node MPM W ,
one has
Tr
h
M
A
W
i
=
Tr
h
M
A
3
A
2
L
2
⊗ M
L
1
A
1
A
0
W ⊗ Id
L
2
L
1
i
,
(21)
where M
A
3
A
2
L
2
, M
L
1
A
1
A
0
are quantum 1-combs,
and W ⊗ Id
L
2
L
1
can be shown to be a valid
(M)PM, hinting the formula (16) for the gen-
eral case. Indeed, formula (18) can be applied
repetitively in the case of a party with more
than 2 nodes: M
A
= M
A
2n
A
−2
A
2n
A
−1
L
2n
A
−2
∗
. . . ∗ M
L
3
A
3
A
2
L
2
∗ Id
L
∗
2
L
∗
1
∗ M
L
1
A
1
A
0
, and
then the identity Tr
Y
h
A
Y
· Tr
X
h
B
XY
ii
=
Tr
XY
h
1
X
⊗ A
Y
B
XY
i
can be used to rewrite
the whole expression as one overall partial trace.
The tensor product being a special case of link
product over the trivial system [15], this argu-
ment also applies to the multipartite case.
Consequently, the correlations achievable with
an MPM are equivalent to those in a PM ex-
tended by side channels and we can appeal to the
concept of causal separability for PMs in order to
define causal separability for MPMs as it is done
in Definition 2.
As shown in Ref. [37], for the PM definition
of causal separability, the possibility of extending
the local operations of the parties to act on local
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ancillary input systems that may be entangled
with the ancillary input systems of other parties
also needs to be considered. Neglecting to do
so may mistakenly lead to the conclusion that a
PM is causally separable, although it is able to
violate a causal inequality when using such an
extension. Likewise, we will show in the next
section that neglecting the side channels leads to
the same kind of problem. We end this section
by reformulating Definition 2 into one closer to
Ref. [37, 50], so that the consequences of local
causal ordering are made explicit and extension
by arbitrary ancillas and side channels is explicit.
We only need to rewrite (21) as
Tr
h
M
A
W
i
= Tr
A
3
A
2
L
2
h
M
A
3
A
2
L
2
W
|M
L
1
A
1
A
0
i
,
(22)
where we have defined the conditional process ma-
trix
W
|M
L
1
A
1
A
0
=
c Tr
L
1
A
1
A
0
h
1 ⊗ M
L
1
A
1
A
0
(W ⊗ Id
L
2
L
1
)
i
.
(23)
This can be shown to be a valid process matrix
on the remaining node between A
2
L
2
and A
3
for
all the possible choices of M
L
1
A
1
A
0
, as long as the
normalisation factor is chosen such that Eq. (12c)
is satisfied. On the contrary, the analogously de-
fined W
|M
A
3
A
2
L
2
will not always be a valid PM
for all M
A
3
A
2
L
2
as it can lead to post-selection.
This implies that when we reformulate the ‘Born
rule’ so as to make the unravelling from an n-
node MPM to an (n − 1)-node MPM apparent,
starting the unravelling with a node that is not
the first of some party is automatically forbidden.
The generalisation of this unravelling procedure
yields the following definition:
Definition 3 (Causal separability of the MPM).
Consider an MPM W shared by |N| parties. For
|N| = 1 this MPM is a deterministic quantum
comb and thus causally separable. For |N| > 1,
the MPM is causally separable if and only if, for
any state ρ ∈ L
N
X∈N
N
n
X
−1
i=0
H
X
0
2i
defined
on an extension of the parties’ input subsystems
H
X
2i
→ H
˜
X
2i
= H
X
2i
⊗H
X
0
2i
, the extended MPM
can be decomposed as
W ⊗ ρ =
X
X∈N
q
X
(1)
W
ρ
X
(1)
, (24)
with q
X
(1)
≥ 0,
P
X∈N
q
X
(1)
= 1, and where
W
ρ
X
(1)
is an MPM compatible with party X’s first
operation being first in the causal unravelling (i.e.
there can be no signalling from the rest of the
nodes to X
(1)
[37]), so that the conditional MPM
after the first operation of X has been carried out,
W
ρ
X
(1)
M
X
0
X
0
0
X
1
L
1
= c Tr
X
0
X
0
0
X
1
L
1
h
M
X
0
X
0
0
X
1
L
1
⊗ 1
W
X
⊗ ρ
X
0
⊗ Id
L
2
L
1
i
, (25)
is itself causally separable for all possi-
ble CP maps between L
H
X
0
⊗ H
X
0
0
and
L
H
X
1
∗
⊗
H
L
1
∗
represented by the CJ
operator M
X
0
X
0
0
X
1
L
1
. Here L
2
is an extension of
the input system of X
(2)
and c is a normalisation
constant.
In the two equivalent definitions of causal sepa-
rability above, we have left the dimensions of the
side channels unbounded. Yet, it is natural to ask
whether a bounded dimension could suffice. It is
clear that no bounded dimension could reproduce
all correlations achievable with an MPM because,
for example, a given party could follow a proto-
col where she applies a different instrument at her
second node depending on the outcome of the in-
strument applied at her first node. By taking the
number of outcomes at the first node sufficiently
large, the side channel required to realise this sit-
uation could be made larger than any assumed
bound. However, it may still be the case in prin-
ciple that some bounded dimension for the side
channels (generally dependent on the dimensions
of the nodes of the MPM) is sufficient for the defi-
nition of causal separability because it implies the
analogous property for all dimensions. The ques-
tion of whether such a bound exists is left open
for future work.
6 Activation of causal nonseparability
by a side channel
Since a PM is a special case of an MPM with
no partial ordering at all assumed for its nodes,
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one may think that applying the PM definition to
an MPM would be sufficient to establish causal
(non)separability. Remarkably, this is not true,
as we will now demonstrate by an example.
Consider an MPM for two parties, A and B,
where A has two nodes and B a single node. Let
the dimensions of all input and output systems of
all nodes be equal to 2: d
X
i
= 2 ∀X ∈ N , 0 ≤
i < 2n
X
− 1. We take the MPM to have the fol-
lowing expression, here written in the Pauli basis:
W
AB
=
1
8
1 +
1
√
2
h
σ
A
0
x
σ
A
2
z
σ
A
3
z
σ
B
0
z
+ σ
A
0
z
σ
A
2
z
σ
B
1
z
i
.
(26)
To keep the above expression concise, the ten-
sor product between matrices acting on separate
systems has been omitted, and unit matrices on
subsystems are implied. σ
A
0
z
σ
A
2
z
σ
B
1
z
is then the
short form of σ
A
0
z
⊗1
A
1
⊗σ
A
2
z
⊗1
A
3
⊗1
B
0
⊗σ
B
1
z
in our notation.
One can demonstrate that (26) is a valid PM as
well as a valid MPM by using Theorem 1, see Ap-
pendix D. This is actually almost the same matrix
as the one given in the activation example in Ref.
[37], but the isolated node A
(1)
between subsys-
tems A
0
and A
1
is now getting a non-trivial input
instead of output. The matrix (26) actually cor-
responds to a deterministic quantum comb with
the fixed causal order A
(2)
≺ B ≺ A
(1)
as one can
verify using Eqs. (8). This proves that W
AB
is a
causally separable PM, which admits a physical
realisation [15]. However, in this causal realisa-
tion node A
(2)
is before node A
(1)
.
Yet, if we treat W as an MPM with node A
(1)
before node A
(2)
, Alice can use a side channel to
pass on her first input to act on it during her sec-
ond operation. The extension of W by an iden-
tity side channel Id =
P
i,j
|iiihjj|
L
1
L
2
between
the operations of A,
W
0ALB
≡ W
AB
⊗
X
i,j
|iiihjj|
L
1
L
2
, (27)
is effectively allowed in the MPM formalism; the
situation is represented in yellow in Fig. 5.
When Alice’s first action (in red) is to simply
forward her input through the channel, M
A
(1)
=
P
k,l
|kkihll|
A
0
L
1
⊗
1
2
1
A
1
, the conditional 2-partite
Figure 5: Activation of causal nonseparability by a side
channel: process matrix W is causally separable in the
PM sense. When extended by a side channel (in yellow),
there exists an operation M
A
(1)
(in red) that makes it
lose this property between the 2 remaining nodes. By
measuring A
2
and changing the basis of L
2
accordingly
(in green), A obtains the OCB PM between the remain-
ing nodes.
(M)PM on the remaining nodes is
W
A
2
A
3
L
2
B
0
B
1
= Tr
A
0
A
1
L
1
h
M
A
(1)
· W
0ALB
i
=
1
8
1 +
1
√
2
h
σ
A
2
z
σ
A
3
z
σ
L
2
x
σ
B
0
z
+ σ
A
2
z
σ
L
2
z
σ
B
1
z
i
.
(28)
If A performs a measurement in the σ
z
basis on
A
2
and obtains the outcome “+”, A and B are
left with the OCB process matrix W
A
3
L
2
B
0
B
1
=
1
4
1 +
1
√
2
h
σ
A
3
z
σ
L
2
x
σ
B
0
z
+ σ
L
2
z
σ
B
1
z
i
on the re-
maining systems, which can be used to vio-
late a causal inequality as described in Ref.
[38]. If she obtains the outcome “−”, A
and B are left with the matrix W
A
3
L
2
B
0
B
1
=
1
4
1 −
1
√
2
h
σ
A
3
z
σ
L
2
x
σ
B
0
z
+ σ
L
2
z
σ
B
1
z
i
, which differs
from the OCB matrix merely by a change of basis.
Hence the same noncausal correlations as with
the OCB PM can be obtained if A’s operations
are modified to account for this change of basis.
She can achieve this by applying a σ
y
transfor-
mation on L
2
controlled by the output of the σ
x
measurement of A
2
(in green). Consequently, the
ability for the parties to use side communication
with the MPM exhibits non-trivial differences in
terms of achievable correlations compared to the
PM.
Accepted in Quantum 2021-01-03, click title to verify. Published under CC-BY 4.0. 11
7 Conclusion
To summarise, we defined an extension of the PM
formalism that allows multiple rounds of informa-
tion exchange for each party, which we named the
“multi-round process matrix” (MPM). We pro-
vided a complete characterisation of the set of
valid MPMs in Theorem 1 using the projective
formulation of the validity constraints on deter-
ministic quantum combs. This highlighted a con-
nection between the set of MPMs and determin-
istic quantum combs: Theorem 2 demonstrates
that MPMs are affine combination of all the quan-
tum combs compatible with the local ordering of
each party. Finally, we motivated a new notion of
causal separability specific to MPMs, Definition
2, that takes into account the possibility of using
side channels. A non-trivial consequence of this
possibility was that the notion of causal nonsep-
arability for PMs and MPMs are not equivalent
when applied to the same operator – we showed
an example of an operator that is causally sep-
arable when considered as a PM, and causally
nonseparable when considered as an MPM.
Several paths can be considered for the contin-
uation of this work. First, the question of finding
whether there exists a bound on the dimensions
of the side channels needed to certify causal non-
separability in the MPM sense has been left open
in this work.
Then, a natural path is to analyse the connec-
tion of this work with the axiomatic theory of
higher-order quantum computation by Bisio and
Perinotti [11, 39], which captures the MPM as a
specific type of process within the infinite hier-
archy of higher-order processes; this will be ex-
plored in an upcoming article. Another related
work is Jia’s correlational approach to quantum
theory [30]. Several results derived here were par-
tially found or hinted in these works.
Finding new protocols that provide an ad-
vantage over regular quantum communications
[19, 20] constitutes another research direction.
We expect that the tool provided by the multi-
round process matrix would be useful to this pur-
pose, especially to formulate communication pro-
tocols using indefinite causal order as a resource
[32, 48].
Finally, one may expect that the projective
methods used here to characterise MPMs could
prove useful in the context of various other prob-
lems in the field of (M)PMs, as well as in the
broader field of higher-order processes, such as
for deriving convenient characterisations of more
general processes in the hierarchy. Understand-
ing how the notion of causal nonseparability gen-
eralises to higher orders is another question of
great interest, which could potentially unveil new
phenomena specific to these orders.
Acknowledgments
This work is an adaptation of T. H.’s master’s
thesis at the École polytechnique de Bruxelles
(ULB) that can be found on QuIC website [28].
Illustrations were drawn using draw.io. This pub-
lication was made possible through the support of
the ID# 61466 grant from the John Templeton
Foundation, as part of the “The Quantum Infor-
mation Structure of Spacetime (QISS)” Project
(qiss.fr). The opinions expressed in this publi-
cation are those of the authors and do not nec-
essarily reflect the views of the John Templeton
Foundation. This work was supported by the
Program of Concerted Research Actions (ARC)
of the Université Libre de Bruxelles. O. O. is a
Research Associate of the Fonds de la Recherche
Scientifique (F.R.S.–FNRS).
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A Projective formulation
A.1 The projective superoperator
This section aims to introduce the reader to the
mapping
X
: L(H) → L(H), defined as
X
(·) =
1
X
d
X
⊗ Tr
X
[·] , (29)
where X is a tensor factor of H, and d
X
its di-
mension. It was introduced in Ref. [6] alongside
its shorthand prescript notation. To illustrate its
properties, we consider an operator M defined
on some space L
H
A
⊗ H
B
⊗ H
C
of dimension
d
2
= (d
A
d
B
d
C
)
2
.
We define a multiplication within the prescript
that corresponds formally to a composition of
several maps as
1
A
d
A
⊗ Tr
A
"
1
B
d
B
⊗ Tr
B
[M]
#
≡
AB
M . (30)
This “multiplication” inherits the properties of
partial tracing: it possesses an identity element
(see below), it is associative as well as commu-
tative as partial tracings on different subsystems
are associative and commute, but it does not pos-
sess an inverse. Next, note that this mapping is
CPTP, as well as idempotent:
AA
M =
A
2
M =
A
M . (31)
Using the Hilbert-Schmidt product as the inner
product on the space of linear operators,
h· , ·i : L(H) × L(H) → C ,
M, N 7→ hM , N i ≡ Tr
h
M
†
· N
i
, (32)
one can verify that the mapping is self-adjoint
with respect to it. Let N ∈ L
H
A
⊗ H
B
⊗ H
C
be another arbitrary operator defined on the same
space as M, then
hM ,
A
N i = Tr
ABC
"
M
†
·
1
A
d
A
⊗ Tr
A
[N]
!#
= Tr
BC
Tr
A
h
M
†
i
·
1
d
A
Tr
A
[N]
= Tr
ABC
"
1
A
d
A
⊗ Tr
A
h
M
†
i
!
· N
#
,
(33)
where the identity
Tr
ABC
h
U
ABC
·
1
A
⊗ V
BC
i
=
Accepted in Quantum 2021-01-03, click title to verify. Published under CC-BY 4.0. 15
Tr
BC
h
Tr
A
h
U
ABC
i
· V
BC
i
have been used
to go to the second and third lines. Thus,
hM ,
A
N i = h
A
M , N i . (34)
This means that the mapping is idempotent and
self-adjoint, making it an orthogonal projector
onto a subspace of L
H
A
⊗ H
B
⊗ H
C
.
We define a linear addition operation within
the prescript as
A
M +
B
M ≡
A+B
M , (35)
and for each element
A
(·) we define the in-
verse element
−A
(·) ≡ −
1
A
d
A
⊗ Tr
A
[·], such that
A+(−A)
M ≡
0
M = 0, where
0
is the additive
identity element, which corresponds to the zero
mapping. We will introduce the minus sign as
a shorthand notation
A+(−A)
M ≡
A−A
M for ad-
dition of inverse elements. With this addition
and multiplication, the prescripts have the alge-
braic structure of a ring [26]. One could also
promote the structure to an algebra by defin-
ing a scalar multiplication the same way the in-
verse additive element have been defined, i.e.
λA
(·) ≡ λ
1
A
d
A
⊗ Tr
A
[·], for all λ ∈ C. How-
ever, note that the idempotency property is not
in general conserved under scalar multiplication
nor under addition of prescripts. Hence, for the
purposes of this article, we restrict the algebra
to a ring and we only consider combinations un-
der addition that result in valid projectors, i.e.
idempotent elements.
Finally, we define 2 particular elements of the
ring. One is the identity mapping I, such that
I[M ] = M for all operators, which corresponds
to the multiplicative identity element in the ring.
In prescript notation it is then denoted as a 1:
I[M ] ≡
1
M . (36)
The other is the multiplicative absorbing element
D : DP = PD = D, for every P in the ring,
which is no other than the projector to the sub-
space spanned by the unit matrix
1
d
1, or equiv-
alently the mapping applied on the full system
D(M) =
1
d
⊗ Tr [M ]. In our example, D is writ-
ten in prescript notation as
D[M] ≡
ABC
M . (37)
We will also use superscripts when those projec-
tors are applied on parts of composite subsys-
tems, e.g.
I
A
⊗ D
BC
[M] ≡
1BC
M =
BC
M.
A.2 Comb conditions in projective formulation
In this section, we derive an alternative formu-
lation of the set of necessary and sufficient con-
straints an operator has to satisfy to be a valid
deterministic quantum comb (Eqs. (4), see [15,
Theorems 3 and 5]). This formulation is in the
spirit of what was done in Ref. [6] for the PM va-
lidity conditions. We will find a recursive way
of building the projector onto the subspace of
valid quantum combs for an increasing number of
teeth. To do so, we use the shorthand notation
M
(j)
≡ M
X
0
X
1
···X
2j−1
, so that M
X
≡ M
(n
X
)
,
to refer to a deterministic quantum comb with j
teeth.
The set of conditions to be proven equivalent
to (4) are equations (8) and (9), rewritten here
for convenience:
M
X
≥ 0 , (38a)
P
X
n
X
h
M
X
i
= M
X
, (38b)
Tr
h
M
X
i
=
n
X
−1
Y
j=0
d
X
2j
, (38c)
where P
X
n
X
≡ P
X
(1)
≺...≺X
(n
X
)
n
X
is the projector to
the validity subspace of n
X
-combs, which is given
by the recursive relation (9)
P
0
=
1
(·) ,
P
X
(1)
≺...≺X
(n
X
−1)
≺X
(n
X
)
n
X
=
(
1−X
2n
X
−1
)
(·) +
X
2n
X
−2
X
2n
X
−1
P
X
(1)
≺...≺X
(n
X
−1)
n
X
−1
.
(39)
Keep in mind that we are using the shorthand
notation P
X
(1)
≺...≺X
(n
X
−1)
n
X
−1
≡ P
X
0
...X
2n
X
−3
n
X
−1
⊗
I
X
2n
X
−2
X
2n
X
−1
to indicate that we are project-
ing the subsystems X
0
. . . X
2n
X
−3
on a subspace
of quantum (n
X
-1)-combs with causal ordering
X
(1)
≺ . . . ≺ X
(n
X
−1)
.
Note that condition (38a) is just condition
(4a)—the condition that a quantum comb is PSD,
which reflects the fact that it is the CJ operator
of a CP map. Therefore, equivalence between
the set of conditions (4) and the set of conditions
(38) with (39) would follow if we show that (4b)
is equivalent to (38b) and (38c), with the projec-
tor in (38b) given by (39). This is what we show
next.
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The normalisation condition (38c) for de-
terministic combs is obtained directly by tak-
ing the total trace of the comb. We can
indeed use the identity Tr
AB
h
M
A
⊗ N
B
i
=
Tr
A
h
M
A
i
Tr
B
h
N
B
i
, which allows to nest the set
of linear constraints (4b) within one another. As
it yields Tr
X
2i−1
X
2i−2
h
M
(i)
i
= d
X
2i−2
M
(i−1)
,
one can see that the total trace of a valid comb
is equal to the product of its input dimensions
(reflecting the fact that a deterministic quantum
comb correspond to a channel from all inputs to
all outputs). Hence, Eq. (38c) is a necessary con-
dition.
We next show that for a 1-comb, Eq. (38b)
with the projector P
X
1
defined as in (39) is neces-
sary. Indeed, the constraint for a 1-comb is equiv-
alent to
Tr
X
1
h
M
(1)
i
= 1
X
0
⇐⇒
X
1
M
(1)
=
1
X
0
X
1
d
X
0
d
X
1
d
X
0
(38c)
=⇒
X
1
M
(1)
=
X
0
X
1
M
(1)
, (40)
where we multiplied by
1
X
1
d
X
1
⊗ on both sides be-
tween the first and second lines, and we used the
definition (29) on the left-hand side, then we used
Eq. (38c), i.e. Tr
h
M
(1)
i
= d
X
0
, together with
(29) on the right-hand side to go to from the sec-
ond line to the last. Hence, (40) is necessary.
The relation (40) is rephrased as a projector as
M
(1)
−
X
1
M
(1)
+
X
0
X
1
M
(1)
= M
(1)
[13], or
P
X
1
h
M
(1)
i
≡
1−X
1
+X
0
X
1
M
(1)
= M
(1)
, (41)
where we have defined the 1-comb projector
P
X
1
≡ P
X
(1)
1
from the space of operators
L
H
X
0
⊗
H
X
1
∗
to the subspace of (nonnor-
malised) deterministic quantum 1-combs. It is
a projector as the relation (1 − X
1
+ X
0
X
1
)
2
=
1 − X
1
+ X
0
X
1
follows from the algebraic rules
of the ring defined in Sec. A.1. This last equa-
tion, (41), is exactly (38b) with (39) in the case
of 1-combs, proving their necessity in this case.
To prove that Eqs. (38b) and (38c) with (39)
are sufficient to enforce Eqs. (4b) in the 1-comb
case, we start from the projective condition,
X
1
M
(1)
−
X
0
X
1
M
(1)
= 0
⇐⇒
1
X
1
d
X
1
⊗
Tr
X
1
h
M
(1)
i
− Tr
X
1
h
X
0
M
(1)
i
= 0
⇐⇒
Tr
X
1
h
M
(1)
i
−
1
X
0
d
X
0
Tr
X
0
X
1
h
M
(1)
i
= 0
(38c)
=⇒
Tr
X
1
h
M
(1)
i
− 1
X
0
= 0 , (42)
where we successively used the fact that pre-
scripts commute and the distributive property of
the tensor product to go to the second line, and
(29) to make the full trace appear in the third
line. Injecting (38c) we reach (4b) in the last
line. Therefore, (38b) and (38c) with (39) imply
(4b) for the case of 1-combs. This completes the
proof of equivalence in this case.
Equivalence in the general case as well as a
rule for building the n
X
-comb projector can be
proven by induction. Suppose that the reformu-
lation holds up to n
X
− 1 teeth such that, for
2 ≤ j ≤ n
X
− 1,
Tr
X
2j−1
h
M
(j)
i
= 1
X
2j−2
⊗ M
(j−1)
,
Tr
X
1
h
M
(1)
i
= 1
X
0
,
⇐⇒
P
X
n
X
−1
h
M
(n
X
−1)
i
= M
(n
X
−1)
,
Tr
h
M
(n
X
−1)
i
=
n
X
−2
Y
j=0
d
X
2j
,
(43)
where P
X
n
X
−1
≡ P
X
(1)
≺X
(2)
≺...≺X
(n
X
−1)
n
X
is the
(n
X
− 1)-comb projector. To prove that the
n
X
case of (38) follows from (4), we start with
Tr
X
2n
X
−1
h
M
(n
X
)
i
= 1
X
2n
X
−2
⊗M
(n
X
−1)
. We al-
ready know that the trace condition will be satis-
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fied, so to find the projective condition we write
Tr
X
2n
X
−1
h
M
(n
X
)
i
= 1
X
2n
X
−2
⊗ M
(n
X
−1)
,
X
2n
X
−1
M
(n
X
)
=
1
X
2n
X
−2
X
2n
X
−1
d
X
2n
X
−1
⊗ M
(n
X
−1)
,
X
2n
X
−1
X
2n
X
−2
M
(n
X
)
=
1
X
2n
X
−2
X
2n
X
−1
d
X
2n
X
−1
⊗M
(n
X
−1)
,
X
2n
X
−1
X
2n
X
−2
P
X
n
X
−1
h
M
(n
X
)
i
=
1
X
2n
X
−2
X
2n
X
−1
d
X
2n
X
−1
⊗ M
(n
X
−1)
, (44)
where we applied successively
1
X
2n
X
−1
d
X
2n
X
−1
⊗,
X
2n
X
−2
(·), and P
X
n
X
−1
[·] on both sides to show
that equivalence in the n
X
− 1 case implies the
following relation in the n
X
case:
Tr
X
2n
X
−1
h
M
(n
X
)
i
= 1
X
2n
X
−2
⊗ M
(n
X
−1)
=⇒
X
2n
X
−1
M
(n
X
)
=
X
2n
X
−1
X
2n
X
−2
P
X
n
X
−1
h
M
(n
X
)
i
.
(45)
This last relation is actually the recursive defini-
tion (39) of the projector as it can be rephrased
as
M
(n
X
)
−
X
2n
X
−1
M
(n
X
)
+
X
2n
X
−1
X
2n
X
−2
P
X
n
X
−1
h
M
(n
X
)
i
= M
(n
X
)
,
≡ P
X
(1)
≺...≺X
(n
X
−1)
≺X
(n
X
)
n
X
h
M
(n
X
)
i
= M
(n
X
)
.
(46)
This yields the definition (39).
To prove that this relation indeed defines a pro-
jector, notice the pattern in the prescripts of the
recursive relation: let P
X
(1)
≺...≺X
(n
X
−1)
≺X
(n
X
)
n
X
=
1 − B + ABC, where A ≡
X
2n
X
−2
and B ≡
X
2n
X
−1
are idempotent elements, while C ≡
P
X
(1)
≺...≺X
(n
X
−1)
n
X
−1
. In the case n
X
= 1,
C
= 1,
which is idempotent, in the case n
X
= 2,
C
=
P
X
(1)
1
, which has been proven idempotent as well.
If we assume C to be idempotent up to n
X
− 1,
then for n
X
:
(1 − B + ABC)
2
= (1 − B)
2
+ 2 (1 − B) ABC
+ (ABC)
2
=1 − B + 0 + ABC ,
(47)
where we have used the distributive property of
multiplication as well as idempotency of all ele-
ments to show that 1−B+ABC is idempotent as
well. This proves that the objects built in equa-
tion (39) are idempotent for all n
X
, thus they
define projectors.
Hence, by induction we have that conditions
(38) with (39) are necessary for (4). Sufficiency
is also proven by induction. Suppose (43) holds.
For the n
X
case we have the implication
P
X
(1)
≺...≺X
(n
X
−1)
≺X
(n
X
)
n
X
h
M
(n
X
)
i
= M
(n
X
)
⇐⇒
X
2n
X
−1
M
(n
X
)
−
X
2n
X
−1
X
2n
X
−2
P
X
n
X
−1
h
M
(n
X
)
i
= 0
=⇒
Tr
X
2n
X
−1
M
(n
X
)
−
X
2n
X
−2
P
X
n
X
−1
h
M
(n
X
)
i
= 0,
(48)
yielding
Tr
X
2n
X
−1
h
M
(n
X
)
i
=
1
X
2n
X
−2
d
X
2n
X
−2
⊗Tr
X
2n
X
−1
X
2n
X
−2
h
P
X
n
X
−1
h
M
(n
X
)
ii
.
(49)
Because of equations (43), we must have that
Tr
X
2n
X
−1
X
2n
X
−2
h
P
X
n
X
−1
h
M
(n
X
)
ii
∝ M
(n
X
−1)
.
(50)
Tracing out both sides, we find that the propor-
tionality constant has to be equal to d
X
2n
X
−2
in order for condition (38c) to hold. Thus,
Tr
X
2n
X
−1
h
M
(n
X
)
i
= 1
X
2n
X
−2
⊗ M
(n
X
−1)
follows
from (38). Adding this condition to the n
X
−
1 other conditions Tr
X
2j−1
h
M
(j)
i
= 1
X
2j−2
⊗
M
(j−1)
, 2 ≤ j ≤ n
X
− 1 and Tr
X
1
h
M
(1)
i
= 1
X
0
,
that were already assumed in Eqs. (43), gives the
n
X
conditions (4b) for an deterministic quantum
n
X
-comb. Hence (38) with (39) implies (4) by
induction, proving sufficiency.
Note that Eq. (39) admits an “unravelled” for-
mulation,
P
X
n
X
=
1−X
2n
X
−1
1−X
2n
X
−2
...
1−X
1
1−X
0
...
(·) ,
(51)
shortened in
P
X
n
X
= I−
X
2n
X
−1
I −
X
2n
X
−2
P
X
n
X
−1
. (52)
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This will be useful in the proof of Theorem 1 pre-
sented in the next section.
B Proof of Theorem 1
As (6) is the same condition as (12a), the proof
consists in showing that condition (7) is equiva-
lent to conditions (12b) and (12c).
The first step of the proof is to notice that since
N
X∈N
1
X
Q
X∈N
Q
n
X
−1
i=0
d
X
2i+1
≡
1
d
out
is a valid tensor prod-
uct of deterministic quantum combs, the normal-
isation (7) implies that the set {W } have fixed
trace norm,
Tr
W ·
1
d
out
= 1 ⇐⇒ Tr [W ] = d
out
. (53)
This is condition (12c), which is thus necessary.
It also implies that W =
N
X∈N
1
X
Q
X∈N
Q
n
X
−1
i=0
d
X
2i
≡
1
d
in
is an element of the set of valid MPMs since
Tr
h
1
d
in
·
N
X∈N
M
X
i
= 1 for all M
X
. This ac-
tually corresponds to enforcing condition (8c) on
each comb in the tensor product.
For proving the necessity of (12b) (and sub-
sequently its sufficiency together with (12c)), we
will work in a convenient Hilbert-Schmidt (HS)
basis {σ
i
} for an operator space of dimension d
2
,
whose elements satisfy
σ
†
i
= σ
i
, ∀i ,
σ
0
= 1 ,
Tr [σ
i
] = 0 , ∀i 6= 0 ,
Tr [σ
i
· σ
j
] = d δ
i,j
, ∀i, j ,
(54)
where δ
i,j
is the Kronecker delta.
An arbitrary operator O ∈
L
H
X
0
⊗ H
X
1
⊗ . . .
can always be expressed
in a product basis of the kind described above:
O =
P
i
0
,i
1
...
o
i
0
,i
1
...
σ
X
0
i
0
⊗σ
X
1
i
1
⊗···, where σ
Y
j
is
the j-th element of a basis of subsystem Y that
respects (54), while the o
i
0
i
1
...
∈ C , ∀i
0
, ∀i
1
, . . .
are the associated coefficients of the basis
expansion, and the sum runs over all in-
dices:
P
i
0
,i
1
...
≡
P
d
2
X
0
−1
i
0
=0
P
d
2
X
1
−1
i
1
=0
···. Let
i = (i
0
, i
1
, . . .) be a multi-index, ranging from 0
to d
2
X
−1, with first element 0 ≡ (0, 0, . . .). Each
sub-index corresponds to a subsystem of X, i.e.
i
0
refers to the index of the X
0
component, i
1
to the one of X
1
, etc... We use it to shorten the
basis expansion:
O =
X
i
0
i
1
...
o
i
0
i
1
...
σ
X
0
i
0
⊗ σ
X
1
i
1
⊗ . . . ≡
X
i
o
i
σ
X
i
.
(55)
The action of the mapping (29) defined in sec-
tion A.1 on an operator is to remove certain
terms in the basis expansion, as it is projecting
on a subset of the basis elements of this type
of basis. One can see it as if the superoper-
ator was setting its corresponding sub-index in
the HS basis to 0:
X
0
O =
P
i=(i
0
=0,i
1
,i
2
,...)
o
i
σ
X
i
;
the notation under the sum is to be understood
as “summing only over all i for which i
0
=
0”. For example,
X
0
P
i
0
o
i
0
σ
X
0
i
0
= o
0
σ
X
0
0
,
X
0
P
i
0
i
1
o
i
0
i
1
σ
X
0
i
0
⊗ σ
X
1
i
1
=
P
i
1
o
0i
1
σ
X
0
0
⊗ σ
X
1
i
1
,
etc...
Actually, the projector on the comb validity
subspace also shares this property, and so does
any tensor product of several projectors of this
kind (i.e. the projector on Span {
N
M}). This
is the content of the following lemma.
Lemma 1. The (tensor product of) mapping(s)
P
X
n
X
defined as in Eq. (39) is a superoperator
projector whose action is to remove certain basis
elements in the Hilbert-Schmidt expansion of an
arbitrary CJ operator.
Proof. The proof is based on the following two
observations: (a) Let P be a projector whose ac-
tion in the HS basis expansion amounts to re-
moving certain types of terms while leaving the
others intact. Then, I − P is a projector that
also has this property. Indeed, it will leave the
terms that are removed by P and will remove
those that are left by P. (b) Let P and P
0
be
two projectors whose action in the basis expan-
sion amounts to removing certain types of terms.
Then the product P
0
P is also a projector of this
kind. Indeed, P would first remove the terms it
removes, and then P
0
would remove those that
it removes from what remains after the action of
P. Note that this observation implies in particu-
lar that if the claimed property holds for P
X
and
P
0
Y
that act on separate systems, it also holds
for their tensor product
P
X
⊗ I
Y
I
X
⊗ P
0
Y
,
since
P
X
⊗ I
Y
and
I
X
⊗ P
0
Y
are obviously
also projectors with this property.
Consider the “unravelled” version of Eq. (39),
Eq. (51). We have seen that
X
0
is such
a projector, hence
1−X
0
is also a projector of
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this kind by (a), so is
X
1
(1−X
0
)
by (b), and
so is
1−X
1
(1−X
0
)
by (a) again. This proves
the lemma in the case n
X
= 1 (that is, for
Eq. (41)). The general case follows by induc-
tion: if it is true for P
X
n
X
−1
, then it is true
for
X
2n
X
−2
P
X
n
X
−1
by (b), which is true for
I−
X
2n
X
−2
P
X
n
X
−1
by (a), which is again true for
X
2n
X
−1
I −
X
2n
X
−2
P
X
n
X
−1
by (b), and which
is true for I −
X
2n
X
−1
I −
X
2n
X
−2
P
X
n
X
−1
by
(a). This is Eq. (51) for n
X
, hereby proving the
induction.
For example, in the case of a single party with
one operation O =
P
i
0
i
1
o
i
0
i
1
σ
X
0
i
0
⊗ σ
X
1
i
1
, one
has P
X
1
[O] =
1−X
1
+X
0
X
1
O = o
00
σ
X
0
0
⊗ σ
X
1
0
+
P
i=(i
0
,i
1
>0)
o
i
0
i
1
σ
X
0
i
0
⊗ σ
X
1
i
1
; the removed terms
have indices i = (i
0
> 0, i
1
= 0).
In the general case for a single party, we will
write the action of a projector on the basis ex-
pansion as P
X
n
X
h
P
i
o
i
σ
X
i
i
=
P
i∈{{M
X
}}
o
i
σ
X
i
,
where
P
i∈{{M
X
}}
is to be understood as “sum-
ming only over the basis elements that belong
to the subspace in which the set of n
X
-combs
{M
X
} is defined”. In the 1-node example of
above, i ∈ {{M
X
}} is thus equivalent to i ∈
{(0, 0)} ∪ {(i
0
, i
1
)}
d
2
X
0
−1,d
2
X
1
−1
i
0
=0,i
1
>0
.
We next observe that the subspace spanned by
the operators that are a tensor product of valid
deterministic combs for the separate parties is the
subspace on which the projector P ≡
N
X∈N
P
X
n
X
projects. Indeed, the action of P on an operator
is to remove all terms in its Hilbert-Schmidt ex-
pansion that are not tensor products of terms al-
lowed (left intact) by the local projectors P
X
n
X
. In
other words, P projects on the subspace spanned
by the tensor products of all locally allowed basis
elements. Since each locally allowed basis ele-
ment can be expressed as a linear combination
of local deterministic combs, every operator in
the subspace on which P projects is of the form
M =
P
k
q
k
N
X
M
X
k
, where M
X
is a valid
deterministic comb associated with party X, and
q
k
are real coefficients. Conversely, every opera-
tor of this form is obviously left invariant by P.
The trace of this last expression gives Tr [M ] =
P
k
q
k
(
Q
X∈N
d
X
in
)
k
, where for each k the terms
in parenthesis are exactly the input dimension of
the Hilbert space, so they can be factored out of
the sum: Tr [M ] = d
in
P
k
q
k
. Hence, requiring
that the trace is equal to d
in
≡
Q
X∈N
d
X
in
is
equivalent to requiring that
P
k
q
k
= 1. There-
fore, conditions (11) are equivalent to requiring
that M is an affine sum of tensor products of de-
terministic quantum combs.
We will now prove the necessity of (12b) by us-
ing this observation to compute the quasiorthogo-
nal projector explicitly. Let M be an operator on
L(H) that satisfies conditions (11). It is written
as
M =
O
X∈N
X
i
X
∈{{M
X
}}
m
i
X
σ
X
i
X
≡
X
i∈{{M}}
m
i
σ
i
,
(56)
where we have introduced a shortened formula-
tion by making the tensor product implicit. Let
there be an arbitrary operator W =
P
j
w
j
σ
j
,
where j is a multi-index defined analogously to i.
Applying the normalisation condition (7) on W is
equivalent to normalising it on all affine sums of
tensor product of deterministic combs (see main
text), hence Eq. (10) gives
Tr [W · M] =
X
j
X
i∈{{M}}
w
j
m
i
d δ
j,i
= 1 . (57)
Since Tr [W ] = w
0
d and Tr [M] = m
0
d, the val-
ues of these two coefficients are fixed using Eqs.
(38c) and (53) obtained above. This allows us to
write
X
j
X
i∈{{M}}
w
j
m
i
dδ
j,i
=
X
i∈{{M}}
w
i
m
i
d
= w
0
m
0
d +
X
i6=0∈{{M}}
w
i
m
i
d
= 1 +
X
i6=0∈{{M}}
w
i
m
i
d ,
(58)
turning the normalisation condition into
X
i6=0∈{{M}}
w
i
m
i
d = 0 . (59)
As w
i
, m
i
∈ R ∀i (since W and M are Her-
mitian), as d is a non-negative integer, and as
in general m
i
6= 0 for at least one given value
of i 6= 0 (otherwise we are back to the case
M =
1
Q
d
X
out
), we are left with two possibilities:
either
P
i6=0∈{{M}}
w
i
m
i
= 0 for some nonvanish-
ing w
i
, or w
i
= 0 , ∀i 6= 0 ∈ {{M}}. We will show
that only the second possibility is viable.
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Following Ref. [37], one can always construct
a valid deterministic quantum comb of the form
M
a
= m
0
σ
0
+ m
a
σ
a
for a 6= 0 ∈ {{M}} with a
small enough m
a
coefficient so that Eqs. (8) are
satisfied (it is sufficient to set m
0
to (
Q
d
X
out
)
−1
and m
2
a
≤ m
2
0
). Then, equation (59) yields
w
a
= 0. Doing it again with a different valid
comb M
b
= m
0
σ
0
+ m
b
σ
b
such that b 6= a
yields a second condition, w
b
= 0. Repeating
this argument for all possible choice of basis el-
ement, {M
k
= m
0
σ
0
+ m
k
σ
k
}
k6=0∈{{M}}
, proves
the latter possibility: w
i
= 0 , ∀ i 6= 0 ∈ {{M}}.
Note that the last condition defines a subspace
in the space of operators to which W must belong.
This subspace is quasiorthogonal to the subspace
Span {
N
M}, meaning that the intersection of
the two subspaces is the span of the unit ma-
trix. Noticing that w
0
σ
0
=
X
W ≡ D[W ], one ex-
presses this last statement in a basis-independent
formulation:
{W } :
X
i∈{{M}}
w
i
σ
i
= w
0
σ
0
⇐⇒ (P − D) [W ] = 0 . (60)
One can check that (P −D)
2
= P−D, so P−D is
a projector. Taking the orthogonal complement,
one gets
W = (I − P + D) [W ] , (61)
which is the sought formula.
The fact that Eqs. (12b) and (12c) are suffi-
cient for (10) (and therefore (7)) is straightfor-
ward to check. Indeed, plugging (60), which is
equivalent to (12b), together with (12c) in Eq.
(57), one verifies that m
0
d
d
in
= 1 because Eq.
(11b) implies that m
0
= 1/d
out
. This completes
the proof of the theorem.
C Proof of Theorem 2
The proof will be split among several lem-
mas. Given a finite set of orthogonal projectors
P
1
, P
2
, . . ., we call their intersection the orthog-
onal projector P
P
1
∩P
2
∩...
that projects on the in-
tersection of the subspaces on which P
1
, P
2
, . . .
project. As shown in [41], if the projectors com-
mute, this intersection is given by the “product”
of the projectors (formally, their composition):
P
P
1
∩P
2
∩...
≡ P
1
∩ P
2
∩ . . . = P
1
P
2
. . . . (62)
Figure 6: Examples of ways to append n
A
= 1 opera-
tions of Alice with n
B
= 2 operations of Bob into valid
3-combs, with the associated projectors on a particu-
lar subspace of valid 3-comb written below its graphical
representation.
Let P
A≺B
n
A
+n
B
be the projector on the space of valid
(n
A
+ n
B
)-combs formed by composing n
A
op-
erations (or “teeth”) of a party A with the n
B
operations of another party B into an overall de-
terministic (n
A
+ n
B
)-comb where the operations
of Alice are all before those of Bob (see the sec-
ond case from the left in Fig. 6 for a graphical
example). Let P
B≺A
n
A
+n
B
be the same kind of comb
projector but where the operations of B are put
before those of A (rightmost case in Fig. 6). Then
the intersection of these two projectors is
P
A≺B
n
A
+n
B
∩ P
B≺A
n
A
+n
B
= P
A≺B
n
A
+n
B
P
B≺A
n
A
+n
B
. (63)
Any comb in the intersection should therefore
be compatible with either all the operations of Al-
ice being first or those of Bob. Intuitively, one can
conceive that the only kind of combs valid within
this requirement are those where the part of Al-
ice is forbidden to communicate with the part of
Bob and vice versa. This is the content of the
first lemma.
Lemma 2. Consider a quantum comb in which
some of the teeth are associated with a party A
and the others with B. Then, the intersection
of the subspace of deterministic quantum combs
in which the teeth of A are all before those of
B (e.g. Fig. 6, second case from the left) with
the one in which the teeth of B are all before
those of A (Fig. 6, rightmost case) is equiva-
lent to Span
n
M
A
⊗ M
B
o
, which is the subspace
spanned by the tensor product of a smaller de-
terministic comb acting on the nodes of A only
Accepted in Quantum 2021-01-03, click title to verify. Published under CC-BY 4.0. 21
together with a smaller deterministic comb acting
on the nodes of B only (Fig. 6, leftmost case).
This is because their projectors are equivalent:
P
A≺B
n
A
+n
B
P
B≺A
n
A
+n
B
= P
A
n
A
⊗ P
B
n
B
. (64)
Proof. The proof relies upon the recursive formu-
lation of a quantum comb projector (39). When
inspecting formula (51) for P
A≺B
n
A
+n
B
, we see that
it is possible to express the projector as:
P
A≺B
n
A
+n
B
= I
A
⊗P
B
n
B
−
I
A
− P
A
n
A
⊗D
B
. (65)
Indeed, it holds for the case n
B
= 1:
P
A≺B
n
A
+1
=
1−B
1
(·) +
B
0
B
1
P
A
n
A
=
1−B
1
+B
0
B
1
(·) −
B
0
B
1
1 − P
A
n
A
= I
A
⊗ P
B
1
−
B
0
B
1
I
A
− P
A
n
A
= I
A
⊗ P
B
1
−
I
A
− P
A
n
A
⊗ D
B
.
(66)
Suppose the decomposition (65) is true for n
A
+
n
B
− 1 operations. We define the projector for
such a case as P
A≺B
n
A
+n
B
−1
≡ P
A≺B
(1)
≺...≺B
(n
B
−1)
n
A
+n
B
−1
.
Then, for n
A
+ n
B
:
P
A≺B
n
A
+n
B
=
1−B
2n
B
−1
(·)+
B
2n
B
−2
B
2n
B
−1
P
A≺B
n
A
+n
B
−1
=
1−B
2n
B
−1
(·) +
B
2n
B
−2
B
2n
B
−1
I
A
⊗ P
B
n
B
−1
−
B
2n
B
−2
B
2n
B
−1
I
A
− P
A
n
A
⊗ D
B
n
B
−1
= I
A
⊗ P
B
n
B
−
I
A
− P
A
n
A
⊗ D
B
n
B
, (67)
where we injected P
A≺B
n
A
+n
B
−1
= I
A
⊗ P
B
n
B
−1
−
I
A
− P
A
n
A
⊗ D
B
n
B
−1
to go from the first equal-
ity to the second, with D
B
n
B
−1
≡
B
0
...B
2n
B
−3
.
Next, we used Eq. (39) together with
B
2n
B
−2
B
2n
B
−1
D
B
n
B
−1
= D
B
to go to the last
equality. This proves decomposition (65) by in-
duction.
Now, applying an analogous decomposition to
P
B≺A
n
A
+n
B
, the left-hand side of Eq. (64) becomes
P
A≺B
n
A
+n
B
P
B≺A
n
A
+n
B
=
I
A
⊗ P
B
n
B
−
I
A
− P
A
n
A
⊗ D
B
×
P
A
n
A
⊗ I
B
− D
A
⊗
I
B
− P
B
n
B
= P
A
n
A
⊗ P
B
n
B
. (68)
To go to the last line, we used the fact that
D is the absorbing (i.e. “zero”) element of the
multiplication in the ring (see Sec. A.1), mak-
ing all the terms where it appears vanish, since
I
X
− P
X
D
X
=
D
X
− D
X
= 0.
The projector of this first lemma is simply
the projector on a subspace containing all the
(n
A
+ n
B
)−combs obtained by taking the ten-
sor product of an n
A
-comb with an n
B
-comb (e.g.
leftmost case in Fig. 6). It should also be compat-
ible with permutations of Bob’s and Alice’s teeth
in the full comb that respect the local causal or-
dering for each party. We will need the π
i
(A, B)
notation introduced in Sec. 4 to refer to the i-th
valid permutation in the causal ordering of the
teeth of parties A and B in order to express this
as a corollary.
Corollary 1. The subspace made by the intersec-
tion of the (n
A
+n
B
)-combs such that all the teeth
of A are before those of B with the (n
A
+ n
B
)-
combs such that all the teeth of B are before those
of A is inside the subspace of all (n
A
+ n
B
)-
combs whose teeth orderings are permutations of
the teeth of A and B that respect the local order
assumed for A and B:
P
A≺B
n
A
+n
B
P
B≺A
n
A
+n
B
P
π
i
(A,B)
n
A
+n
B
= P
A≺B
n
A
+n
B
P
B≺A
n
A
+n
B
∀i .
(69)
Proof. The subset of deterministic quantum
combs on L
H
A
⊗ H
B
formed by the tensor
product of combs on A and B is automatically
a valid comb, no matter how one defines the
relative ordering between the teeth of A and B
[15].
In other words, this corollary says that when
an operator is a tensor product of a deterministic
n
A
-comb with a deterministic n
B
-comb, then it
is a valid deterministic (n
A
+ n
B
)-comb for any
ordering of the teeth compatible with the partial
ordering of the individual combs.
The second lemma needed is just a simplifica-
tion of the formula for the union of projectors
when applied to the case of quasiorthogonal pro-
jectors for different teeth orderings.
Lemma 3. The union of two arbitrary qua-
siorthogonal projectors
Q
P
π
i
(A,B)
n
A
+n
B
and
Q
P
π
j
(A,B)
n
A
+n
B
is given by
Q
P
π
i
(A,B)
n
A
+n
B
∪
Q
P
π
j
(A,B)
n
A
+n
B
= I−P
π
i
(A,B)
n
A
+n
B
P
π
j
(A,B)
n
A
+n
B
+D.
(70)
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Proof. It is proven by doing the computation ex-
plicitly, then simplifying by using again the fact
that the projector D on the span of unity is the
zero element. Hence,
Q
P
π
i
Q
P
π
j
= (I − P
π
i
+ D) (I − P
π
j
+ D)
= I − P
π
i
− P
π
j
+ P
π
i
P
π
j
+ D,
(71)
where we have dropped the n
A
+ n
B
subscript and
simplified the superscript (i.e. π
i
(A, B) ≡ π
i
) in
order to shorten the notation. This gives, when
plugged into (13),
Q
P
π
i
∪
Q
P
π
j
= I − P
π
i
+ D + I − P
π
j
+ D
− (I − P
π
i
− P
π
j
+ P
π
i
P
π
j
+ D)
= I − P
π
i
P
π
j
+ D.
(72)
All the elements needed for the main proof are
in place. Note that the lemmas have been de-
rived assuming only 2 parties. Actually, these
hold analogously for any number of parties, since
every formula that was used is associative. In
the case where |N| > 2, one just has to group
the parties together to get back to the 2-partite
scenario.
For the main proof, we start from the right-
hand side of equation (14) to which we apply the
result of Lemma 3, equation (70):
[
π
i
Q
P
π
i
(A,B,...)
n
A
+n
B
+...
= I −
Y
π
i
P
π
i
(A,B,...)
n
A
+n
B
+...
+ D. (73)
We then apply Corollary 1 so the permutations
π
i
of all the nodes are restricted to permutations
π
0
i
of whole parties only:
I −
Y
π
0
i
P
π
0
i
(A,B,...)
n
A
+n
B
+...
+ D =
I−P
A≺B≺...
n
A
+n
B
+...
P
B≺A≺...
n
A
+n
B
+...
... + D.
(74)
Remember that here a superscript X is referring
to n
X
nodes like X ≡ X
(1)
≺ X
(2)
≺ . . . ≺
X
(n
X
)
. The final step is to apply Lemma 2, equa-
tion (64), yielding
I − P
A≺B≺...
n
A
+n
B
+...
P
B≺A≺...
n
A
+n
B
+...
. . . + D =
I −
P
A
n
A
⊗ P
B
n
B
⊗ . . .
+ D. (75)
We can now invoke the definition of the qua-
siorthogonal projector (12b) on the right-hand
side, and we have proven that
[
π
i
Q
P
π
i
(A,B,...)
n
A
+n
B
+...
=
Q
P
A
n
A
⊗ P
B
n
B
⊗ . . .
. (76)
D Details for the activation example
As an illustration of Theorem 1, we will now
sketch the proof that the operator (26), rewrit-
ten here for convenience
W
AB
=
1
8
1 +
1
√
2
h
σ
A
0
x
σ
A
2
z
σ
A
3
z
σ
B
0
z
+ σ
A
0
z
σ
A
2
z
σ
B
1
z
i
,
(77)
is a valid MPM with 3 nodes, where A
(1)
≺ A
(2)
.
This is a PSD matrix with trace Tr
h
W
AB
i
=
8 = d
A
1
d
A
3
d
B
1
= d
out
, so conditions (12a) and
(12c) are directly verified. The projective condi-
tion (12b) is given by
W
AB
= I
AB
h
W
AB
i
−
(1−A
3
+A
2
A
3
−A
1
A
2
A
3
+A
0
A
1
A
2
A
3
)(1−B
1
+B
0
B
1
)
W
AB
+ D
AB
h
W
AB
i
, (78)
where (9) have been used to find the
2-comb projector on A, P
A
(1)
≺A
(2)
2
=
(1−A
3
+A
2
A
3
−A
1
A
2
A
3
+A
0
A
1
A
2
A
3
)
, as well as
the 1-comb projector on B, P
B
1
=
(1−B
1
+B
0
B
1
)
.
Notice that
A
3
W
AB
=
1
8
1 +
1
√
2
σ
A
0
z
σ
A
2
z
σ
B
1
z
,
and
B
1
W
AB
=
1
8
1 +
1
√
2
σ
A
0
x
σ
A
2
z
σ
A
3
z
σ
B
0
z
,
while all the other combinations of prescripts
are equivalent to the action of D
AB
on the
matrix, i.e.
A
3
B
1
W
AB
=
B
0
B
1
W
AB
=
A
2
A
3
W
AB
= . . . =
1
8
= D
AB
h
W
AB
i
. This
allows us to simplify the projective condition
(78) into
W
AB
= W
AB
−
1−A
3
−B
1
W
AB
−D
AB
h
W
AB
i
,
(79)
which is effectively verified:
W
AB
=
A
3
W
AB
+
B
1
W
AB
−D
AB
h
W
AB
i
=
1
4
+
1
8
√
2
h
σ
A
0
x
σ
A
2
z
σ
A
3
z
σ
B
0
z
+ σ
A
0
z
σ
A
2
z
σ
B
1
z
i
−
1
8
=
1
8
1 +
1
√
2
h
σ
A
0
x
σ
A
2
z
σ
A
3
z
σ
B
0
z
+ σ
A
0
z
σ
A
2
z
σ
B
1
z
i
.
(80)
Accepted in Quantum 2021-01-03, click title to verify. Published under CC-BY 4.0. 23
The same way, operator (28),
W =
1
8
1 +
1
√
2
h
σ
A
2
z
σ
A
3
z
σ
L
2
x
σ
B
0
z
+ σ
A
2
z
σ
L
2
z
σ
B
1
z
i
,
(81)
written here without superscript, will now be
proven to be a valid (M)PM on 2 nodes A and B,
where L
2
is extending the input system of party
A. It is straightforward to check that it is a PSD
matrix with Tr [W ] = 4 = d
A
3
d
B
1
, hence veri-
fying (12a) and (12c). The projective condition
(12b) is expressed as
W =
I [W ] −
(1−A
3
+L
2
A
2
A
3
)(1−B
1
+B
0
B
1
)
(W ) + D[W ] =
A
3
+B
1
−A
3
B
1
+A
3
B
0
B
1
−B
0
B
1
+L
2
A
2
A
3
B
1
−L
2
A
2
A
3
(W ) .
(82)
Computing each term on the right hand side, we
find
A
3
W =
1
8
1 +
1
√
2
σ
A
2
z
σ
L
2
z
σ
B
1
z
,
B
1
W =
1
8
1 +
1
√
2
σ
A
2
z
σ
A
3
z
σ
L
2
x
σ
B
0
z
,
A
3
B
1
W =
A
3
B
0
B
1
W =
L
2
A
2
A
3
B
1
W =
L
2
A
2
A
3
W =
B
0
B
1
W =
1
8
.
(83)
Therefore, (82) becomes
W =
A
3
W +
B
1
W −
A
3
B
1
W
=
1
4
+
1
8
√
2
h
σ
A
2
z
σ
L
2
z
σ
B
1
z
+ σ
A
2
z
σ
A
3
z
σ
L
2
x
σ
B
0
z
i
−
1
8
=
1
8
1 +
1
√
2
h
σ
A
2
z
σ
A
3
z
σ
L
2
x
σ
B
0
z
+ σ
A
2
z
σ
L
2
z
σ
B
1
z
i
,
(84)
proving that the projective condition (12b) is in-
deed verified for W.
Accepted in Quantum 2021-01-03, click title to verify. Published under CC-BY 4.0. 24