Covariance-Matrix-Based Criteria for Network Entanglement

Abstract

Quantum networks offer a realistic and practical scheme for generating multiparticle entanglement and implementing multiparticle quantum communication protocols. However, the correlations that can be generated in networks with quantum sources and local operations are not yet well understood. Covariance matrices, which are powerful tools in entanglement theory, have been also applied to the network scenario. We present simple proofs for the decomposition of such matrices into the sum of positive semi-definite block matrices and, based on that, develop analytical and computable necessary criteria for preparing states in quantum networks. These criteria can be applied to networks where nodes share at most one source, such as all bipartite networks.

Full text

Citation: Hansenne, K.; Gühne, O.
Covariance-Matrix-Based Criteria for
Network Entanglement. Entropy 2023,
25, 1260. https://doi.org/10.3390/
e25091260
Academic Editors: Marcelo Terra
Cunha, Ana Cristina Sprotte Costa,
Cristhiano Duarte and Diogo O.
Soares-Pinto
Received: 19 July 2023
Revised: 14 August 2023
Accepted: 20 August 2023
Published: 24 August 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
entropy
Article
Covariance-Matrix-Based Criteria for Network Entanglement
Kiara Hansenne * and Otfried Gühne
Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany
*Correspondence: kiara.hansenne@uni-siegen.de
Abstract:
Quantum networks offer a realistic and practical scheme for generating multiparticle
entanglement and implementing multiparticle quantum communication protocols. However, the
correlations that can be generated in networks with quantum sources and local operations are not
yet well understood. Covariance matrices, which are powerful tools in entanglement theory, have
been also applied to the network scenario. We present simple proofs for the decomposition of such
matrices into the sum of positive semi-definite block matrices and, based on that, develop analytical
and computable necessary criteria for preparing states in quantum networks. These criteria can be
applied to networks where nodes share at most one source, such as all bipartite networks.
Keywords: quantum networks; network entanglement; covariance matrices
1. Introduction
Entanglement is a central element in quantum theory and the subject of famous
debates at the beginning of the 20th century [
1
,
2
] and gained the status of a resource
with the advent of quantum information theory some decades later (see Refs. [
3
,
4
] for
reviews). Entanglement between two parties has been widely studied and characterised,
but much less is known regarding multiparticle entanglement. Indeed, when more than
two parties are involved, the structure of entanglement becomes more complex, with non-
equivalent classes of entanglement appearing [
3
,
4
]. Apart from the foundational interest
in understanding the structure of multiparticle entanglement, the significance lies in the
fact that it also is a resource for many quantum information applications, such as quantum
conference key distribution [
5
], quantum error correcting codes [
6
], or high-precision
metrology [
7
]. However, generating and manipulating genuine multipartite entangled
states experimentally is a difficult task, particularly when the number of entangled parties
is large (see Ref. [
4
] and references therein). To circumvent this issue, the arguably more
experimentally friendly concept of quantum networks has been introduced [
8
,
9
]. In the
network setup, the goal is to generate a global
N
-partite state using a set of sources
(represented by edges in a (hyper)graph) that distribute (connect) subsystems of entangled
states to the different parties of the network (the nodes of the (hyper)graph). Strictly, we
require sources to distribute particles to at most
N−
1 nodes, and the parties might be
allowed to apply a local operation to their system. Figure 1details an example of a tripartite
network with bipartite sources.
The power and limitations of such networks have already been studied in Refs. [
10
–
19
],
however, it is still unclear which useful quantum states can actually be prepared using
them. In the most general definition, the parties and the sources can additionally share a
global classical random variable, and we say that such networks arise from local operations
and shared randomness (LOSR). However, it is also realistic to consider models where
there is no access to such a global variable. The main aim of this paper is to show that
covariance matrices (CMs) can be used to derive strong criteria for entanglement in the
various network scenarios.
First introduced for continuous variable systems [
20
,
21
], covariance matrices possess
useful properties and have previously been used to characterise bipartite and multiparticle
Entropy 2023,25, 1260. https://doi.org/10.3390/e25091260 https://www.mdpi.com/journal/entropy

Entropy 2023,25, 1260 2 of 19
entanglement [
22
,
23
]. Recently, they have also been used to derive necessary criteria for
network scenarios [
11
,
14
,
24
,
25
]. In Ref. [
11
], the authors formulate a necessary condition
for a probability distribution to arise from measurements performed on a quantum network
state. The condition states that the covariance matrix of the probability distribution can
be decomposed into the sum of positive semi-definite (PS D) block matrices, and can be
formulated as a semi-definite program (SDP). We call this decomposition into a sum of PSD
block matrices the block decomposition of a covariance matrix. This result was applied in
Ref. [
14
] to derive practical analytical criteria for networks with dichotomic measurements
and for networks with bipartite sources. More recently, similar SDPs were developed in
Ref. [
25
] for the case of LOSR networks, with extra assumptions on rank and purity. Finally,
aiming at generality, the authors of Ref. [
24
] showed that in the case of no-common-double-
source (NCDS) networks (in no-common-double-source networks, two nodes can hold
subsystems from at most one common source), the block decomposition criterion holds for
all generalised probabilistic theories.
Figure 1.
Basic triangle network. Each source distributes subsystems to the three nodes, Alice, Bob,
and Charlie. They each end up with a bipartite system
X=X1X2
(
X=A
,
B
,
C
), on which they could
apply a local operation.
In this paper, we propose an alternative proof to the block decomposition of the CM of
the triangle network state derived in Ref. [
11
]. From it, we obtain an analytical, computable
necessary criterion for a state to arise from a triangle network. This criterion can also be
used to upper-bound the maximal fidelity a triangle network state can have to a given target
state, for instance the GHZ state. We discuss the fact that these bounds are still valid for
networks with LOSR and, finally, show how this result can be extended to NCDS networks.
2. Network Entanglement
The general triangle network situation involves Alice, Bob, and Charlie wanting to
share a tripartite (entangled) state, but they only have access to bipartite sources, as shown
in Figure 1. This situation differs from the usual consideration of tripartite entanglement,
where the parties have access to a tripartite state generated by a global source. In addition,
the parties in the triangle network considered here do not have access to classical commu-
nication, which prevents them from executing teleportation or entanglement swapping
protocols. Although classical communication is usually considered a cheap resource in
quantum communication protocols, it does require time. The classical information must be
communicated across the network, which can introduce undesirable latency, particularly in
a context where quantum memories are still sub-optimal and expensive.
In this manuscript, we will focus on different triangle network scenarios: the basic
triangle network (BT N), where bipartite sources are shared among the parties; the triangle
network with local unitaries (UTN), where Alice, Bob, and Charlie are allowed to perform
unitary operations on their local systems; and finally, the triangle network with local
channels (CTN), where, as the name indicates, local channels are performed by the parties.

Entropy 2023,25, 1260 3 of 19
In the BT N, three (entangled) bipartite source states (
$a
,
$b
, and
$c
) are prepared and
each subsystem is sent to a node according to the distribution in Figure 1. Alice, Bob, and
Charlie own the bipartite systems
A1A2=A
,
B1B2=B
, and
C1C2=C
, respectively. The
global state of the system ABC reads
$BTN =$b⊗$c⊗$a. (1)
Notice that the order of the subsystems is not
ABC
for the right-hand side, it is organised
following the partition
C2A1A2B1B2C1
. The reduced states of Alice, Bob, and Charlie are
separable bipartite states. This scenario has, for instance, been studied in the context of
pair entangled network states [
16
]. In this work, we will assume that the sources all send
d×d
-dimensional states, while keeping in mind that all the results can easily be extended
to unequal dimensions.
In the following two scenarios, we allow the parties to perform operations on their
local systems. First, we only give Alice, Bob, and Charlie the possibility of performing a
unitary operation on their system, namely,
UA
,
UB
, and
UC
, respectively. This leads to the
following global state
$UTN = (UA⊗UB⊗UC)$BTN(U†
A⊗U†
B⊗U†
C). (2)
Alice, Bob, and Charlie no longer necessarily hold separable bipartite states. We note
that here again, there is no tripartite interaction between the parties. Second, we drop
the unitary restriction on the local operations, meaning that Alice, Bob, and Charlie may
now apply channels on their local systems, represented by completely positive and trace
preserving maps EA,EB, and EC, respectively. In that case, the global state reads
$CTN =EA⊗EB⊗ EC($BTN ). (3)
We note that if the dimensions match, then
{$BTN }⊂{$UTN}⊂{$CTN}
, but in general
CTN
networks can be defined in broader scenarios, since the maps
EA
,
EB
, and
EC
may reduce
the dimension.
These definitions naturally extend to networks with a higher number of parties or
sources. A special instance is the previously-mentioned NCDS networks: in this case, any
two parties share subsystems from at most one source. For instance, all bipartite networks
are NCDS.
Finally, we could also allow the whole system to be coordinated by a global classical
random variable
λ
. In the most general situation, this would result in states of the form
$∆=∑λpλ$(λ)
CTN
. These networks are called LOSR networks. One direct consequence is that
the set of states
{$∆}
is convex, whereas Equations (1)–(3) lead to non-convex state sets. As
already pointed out in Refs. [
10
,
17
], in the case of unbounded source dimensions, it suffices
to consider that either the state or the parties have sole access to the global variable.
3. Covariance Matrices
In this paper, the tools used to analyse network entanglement are covariance matrices,
which characterise states through the covariance of some given observables. In practice,
the CM
Γ
is constructed for a state
$
and a set of observables
{Mi}
, and has the following
matrix elements
[Γ({Mi},$)]mn =hMmMni$− hMmi$hMni$(4)
with
hXi$=tr(X$)
being the expectation value of the observable
X
when the state of
the system is given by
$
. As in network scenarios the parties can only access their local
systems, it is sensible to choose observables
Ai
,
Bj
, and
Ck
that only act on Alice’s, Bob’s,
and Charlie’s sides, respectively. Explicitly, we have
Ai⊗1B⊗1C
,
1A⊗Bj⊗1C
, and
1A⊗1B⊗Ck
, and we will use the notation
{Ai
,
Bj
,
Ck}={Ai⊗1B⊗1C}i∪{1A⊗Bj⊗

Entropy 2023,25, 1260 4 of 19
1C}j∪{1A⊗1B⊗Ck}k
. In that case, the CM of a tripartite state
$
has the following block
structure:
Γ({Ai,Bj,Ck},$) = 

ΓAγEγF
γT
EΓBγG
γT
FγT
GΓC

(5)
where
ΓA=Γ({Ai}
,
$(A))
is the CM of the reduced state
$(A)
. For a state
$
on a system
XY
,
we denote by
trY($) = $(X)
the reduced state of the subsystem
X
. The matrices
ΓB
and
ΓC
have analogous expressions. The elements of the off-diagonal block
γE
are given by the
real numbers
[γE]mn =hAm⊗Bni$− hAmi$hBni$, (6)
with identity operators padded where needed (note that Equation
(6)
can be defined
equivalently by taking the expectation values on
$(AB)
). Again, the matrices
γF
and
γG
can
be expressed in a similar way.
4. Basic Triangle Network
In this section, we derive the explicit structure of CMs of BTN states. Let us first define
what we will call the reduced observable
A(2)
i
of
Ai
, which describes an effective observable
on the system A2. It is given by
A(2)
i=trA1Ai[$(A1)
BTN ⊗1A2]. (7)
Note that
Ai
acts on both
A1
and
A2
, so
A(2)
i
is an operator acting on states of
A2
, where
the effect of
$b=$(A1C2)
BTN
has been taken into account. We define
B(1)
j
similarly and will
use the notation
{A(2)
i
,
B(1)
j}={A(2)
i⊗1B1}i∪{1A2⊗B(1)
j}j
. The off-diagonal blocks of
Equation (5) can be expressed using the reduced observables, that is,
[γE]mn =hA(2)
m⊗B(1)
ni−hA(2)
mihB(1)
ni. (8)
To see this, we notice that the reduces state
$(AB)
BTN
is a product state with respect to the
partition
A1|A2B1|B2
and use a local basis decomposition of the observables
Am
and
Bn
(see Appendix A). All expectation values of Equation
(8)
are taken with respect to the state
$(A2B1)
BTN , which is nothing but $c.
This representation means that
γE
can be computed using only the reduced observ-
ables on the state
$(A2B1)
BTN
. This is a direct consequence of the fact that the marginal states
of Alice, Bob, and Charlie are product states, which will no longer be the case in the next
scenarios. Let us now introduce our first proposition.
Proposition 1
(Block decomposition for CMs of BTN states)
.
The CM of a BTN state with local
observables {Ai,Bj,Ck}can be decomposed as
ΓBTN =Γ({Ai,Bj,Ck},$B TN )
=

ΓA2γE0
γT
EΓB10
0 0 0

| {z }
Tc
+

ΓA10γF
0 0 0
γT
F0ΓC2


| {z }
Tb
+

0 0 0
0ΓB2γG
0γT
GΓC1


| {z }
Ta
+

RA0 0
0RB0
0 0 RC


| {z }
R
(9)
where the matrices Ta, Tb, and Tcare CMs for the state-dependent reduced observables, i.e.,
Tc=Γ({A(2)
i,B(1)
j},$(A2B1)
BTN ). (10)
and analogously for Tband Ta. The matrix R is positive semi-definite.

Entropy 2023,25, 1260 5 of 19
Using Equation
(8)
, it is only left to show that
RA=ΓA−ΓA1−ΓA2
is PSD, as well as
RB
and
RC
. To achieve this, we show that
hx|RA|xi
can always be written as the trace of a
product of PSD matrices. The proof is given in Appendix B.
We want to emphasize that the results presented in this manuscript are valid only
in the context of finite-dimensional Hilbert spaces. A potential future research direction
is to investigate how these results can be extended to the infinite-dimensional case. As
mentioned in the introduction, CMs are also well suited for continuous variable systems.
Armed with this, we can now derive the structure of the covariance matrix of a
BTN state when the observables are full sets of local orthogonal observables, namely,
{Ai}={G(A1)
α⊗G(A2)
β}
, where
{G(Ak)
α}
is a set of
d2
orthogonal observables acting on
states of
Ak
such that
trG(Ak)
αG(Ak)
α0=dδαα0
(
k=
1, 2). This is performed in a similar
way for the systems
B
and
C
. When the situation is explicit enough, we will drop the
superscripts. In the case of qubits, the Pauli operators
σx
,
σy
, and
σz
, together with the
2
×
2 identity operator
1
, are an obvious choice. With such sets of observables, a direct
computation (see Appendix C) shows that
RX=ΓX−ΓX1−ΓX2
=Γ{Gα},$(X1)
BTN ⊗Γ{Gβ},$(X2)
BTN ,X=A,B,C(11)
and, therefore, Ris trivially PSD in the case of full sets of orthogonal observables.
The structure of the matrices
Ta
,
Tb
, and
Tc
can also be further explored. First, let us
compute the reduced observables
A(2)
i=trGα$(A1)
BTN Gβ=a(1)
αGβ. (12)
where the coefficients
a(1)
α=trGα$(A1)
BTN 
are nothing but the (real) Bloch coefficients of the
reduced states. In Appendix D, we show that
ΓA2=|~
a(1)ih~
a(1)|⊗Γ({Gβ},$(A2)
BTN )(13)
and that
γE=|~
a(1)ih~
b(2)|⊗γ({Gβ,Gα},$(A2B1)
BTN )(14)
with
|~
a(1)i= (a(1)
0
,
. . .
,
a(1)
d2−1)T∈Rd2
and similarly for
|~
b(2)i
. The matrix
γ({Gβ
,
Gα}
,
$(A2B1)
BTN )
is the off-diagonal block of the CM with the same observables and state. Finally, we can write
Tc=|~
a(1)⊕~
b(2)ih~
a(1)⊕~
b(2)|?Γ$(A2B1)
BTN , (15)
where
?
is the “block-wise” Kronecker product, called the Khatri–Rao product [
26
,
27
].
Formally, if
A
and
B
are block matrices, the
i
,
j
th block of their Khatri–Rao product,
(A?B)i,j
,
is the Kronecker product of the
i
,
j
th block of
A
and
B
,
Ai,j⊗Bi,j
. For instance, if
A
and
B
are 2 ×2 block matrices,
A=A0,0 A0,1
A1,0 A1,1,B=B0,0 B0,1
B1,0 B1,1, (16)
we obtain
A?B=A0,0 ⊗B0,0 A0,1 ⊗B0,1
A1,0 ⊗B1,0 A1,1 ⊗B1,1(17)
(see Ref. [27] for more details).
Finally, one has
Proposition 2.
The CM of a B TN state, using complete sets of orthogonal observables acting locally
can be decomposed as

Entropy 2023,25, 1260 6 of 19
ΓBTN =|~
a(1)⊕~
b(2)ih~
a(1)⊕~
b(2)|?Γ$(A2B1)
BTN +|~
b(1)⊕~
c(2)ih~
b(1)⊕~
c(2)|?Γ$(B2C1)
BTN 
+|~
b(1)⊕~
c(2)ih~
b(1)⊕~
c(2)|?Γ$(B2C1)
BTN +diagnΓ$(X1)
BTN ⊗Γ$(X2)
BTN ,X=A,B,Co.(18)
Therefore, in order to test compatibility with the BTN scenario for a given state, one
can check if its CM can be written like the right-hand side of the above equation. While this
might be cumbersome to test, we notice that the matrix
ΓBTN −R
is also PS D, which can
also be used to check compatibility in the following way:
Proposition 3 (Positivity condition).The matrix
Ξ($BTN) = Γ{GA1
α⊗GA2
β,GB1
γ⊗GB2
δ,GC1
e⊗GC2
ζ},$BTN
−diagnΓ{GX1
α},$(X1)
BTN ⊗Γ{GX2
β},$(X2)
BTN ,X=A,B,Co(19)
is positive semi-definite.
We note that neither term of the right-hand side of Equation
(19)
contains the reduced
observables, which makes Ξeasy to compute.
An advised reader might point out that in order to verify if a given state is compatible
with the BT N scenario, it suffices to test whether
$BTN =$(A2B1)
BTN ⊗$(B2C1)
BTN ⊗$(C2A1)
BTN
, up to
reordering of the subsystems. We stress that although this simple equation does answer the
question, it requires the knowledge of the full density operator, whereas CM-based criteria
only need expectation values of some chosen observables in order to be evaluated.
To close this section on BTN, we present a few examples. First, we note that the
lowest-dimensional achievable states are sixty-four-dimensional states (six qubits, or three
ququarts (a ququart (sometimes ququad) is a four-dimensional quantum system)), and that
the local dimensions cannot be prime numbers. Therefore, we start with the three-ququart
GHZ state,
|GH Z4i=1
√2(|000i+|333i), (20)
which we mix with white noise
$GH Z4(v) = v|GH Z4ihG HZ4|+ (1−v)164
64 , (21)
where
v
is the visibility. The corresponding
Ξ
matrix is PS D only for
p=
0, meaning
that the GHZ state cannot be prepared in a BTN network even with a very high amount
of white noise. The same result is obtained when applied to the four-level GHZ state,
1/2(|000i+|111i+|222i+|333i).
Proposition 3may also be applied to three-ququart Dicke states, which are defined by
|D3,4,ki=N∑
i1+i2+i3=k|i1i2i3i,k=1, . . . , 9, (22)
with
N
being a normalisation factor. When mixed with white noise, they cannot be prepared
in the BT N scenario when
p6=
0 and
p6=
1 for
k=
1, and when
p6=
0 for
k=
2,
. . .
, 7. More
generally, by directly applying the result of Proposition 2, we can check whether the CM
of a BT N state can be written like the right-hand side of Equation
(18)
. Performing this for
|D3
1i
, we conclude that this state cannot be generated in the BTN scenario. On the other
hand, the CM of the maximally mixed state 1
/64 has such a decomposition.
The nature of interesting states that can be prepared in the BTN scenario remains an
open question. An obvious approach would be to distribute three Bell pairs across the
network, resulting in a three-ququart genuine multipartite entangled triangle state. Getting

Entropy 2023,25, 1260 7 of 19
ahead of the next sections, where it will be permitted to apply local transformations on
systems
A
,
B
, and
C
, it is less straightforward to see what operations could be applied after
distributing, for instance, Bell pairs.
5. Triangle Network with Local Operations
Let us now consider the situation where Alice, Bob, and Charlie can perform unitaries
on their respective systems. As described in Section 2, the global state now reads
$UTN =(UA⊗UB⊗UC)$BT N(U†
A⊗U†
B⊗U†
C).(23)
First, we note that in general, for any set of observables
{Mi}
, any unitary
U
, and any state
$, there exists an orthogonal matrix Osuch that [23]
Γ({Mi},U$U†) = Γ({U†MiU},$) = OTΓ({Mi},$)O. (24)
Note that not all orthogonal transformations of CMs correspond to a unitary transformation
on the system. From that, we obtain the following proposition:
Proposition 4.
Consider the CM of a UTN state with observables
Ai
,
Bj
,
Ck
that only act on the
systems
A
,
B
,
C
, respectively. There exist an orthogonal matrix
O=OA⊕OB⊕OC
and a BTN
state $BTN such that the CM of $UTN can be written as
ΓUTN ≡Γ({Ai,Bj,Ck},$UTN) = OTΓB TNO(25)
with ΓBTN as in Equation (9).
Thus, the CM of
$UTN
can always be decomposed as a sum of positive semi-definite
matrices with the following block decomposition
ΓUTN =

  0
  0
0 0 0

| {z }
OTTcO
+

0
000
0


| {z }
OTTbO
+

0 0 0
0 
0 


| {z }
OTTaO
+

0 0
00
0 0 


| {z }
OTRO
. (26)
We may also look at this situation by noticing that the CMs of UTN states can be written as
ΓUTN =Γ({U†
AAiUA,U†
BBjUB,U†
CCkUC},$BTN )
=TU
c+TU
b+TU
a+RU,(27)
with TU
cbeing the CMs of $(A2B1)
BTN with the following reduced observables
A(2)
U,i≡(U†
AAiUA)(2)
=∑
α,β
trU†
AAiUAGα⊗GβtrGα$A1Gβ
(28)
and
B(1)
U,j
built in the same way. The matrices
TU
b
and
TU
a
are defined similarly. The matrix
RU
A
is equal to
AU−EU
A−FU
A
. One issue with this formulation is that, if one wants to test
whether a given state is compatible with the UTN scenario, the unitaries
UA
,
UB
, and
UC
and the state
$BTN
corresponding to the decomposition of
$UTN
are in general not known,
thus there is no way to explicitly know the reduced observables.
We are now interested in triangle networks in which the local operations can be
any quantum channel, i.e., no longer restricted to unitary operations. By making use
of the Stinespring dilation theorem [
28
], we can show that CTN states also lead to CMs
that posses a block decomposition. As a matter of fact, any channel can be implemented

Entropy 2023,25, 1260 8 of 19
by performing a unitary transformation on the system together with an ancilla, and then
tracing out the ancilla. The covariance matrix of any state
$
after a channel
E
(i.e.,
E($)
) with
observables
{Mi}
, therefore, has the same expression as taking the CM of the state together
with an ancilla and applying the corresponding unitary
U
, that is,
U($⊗$ancilla)U†
, with
observables
{Mi⊗1ancilla}
. We can also see this by noticing that the CM of a reduced state
is just a principal submatrix of the CM of the global state. Applying this to each node of the
triangle network, we obtain the following proposition:
Proposition 5
(Block decomposition for CMs of CTN states)
.
Let
ΓCTN
be the covariance matrix
of
$CTN =EA⊗ EB⊗ EC($BTN )
with local observables
Ai
,
Bj
, and
Ck
as in Equation
(5)
. There
exist matrices ΥX
i(X =A,B,C and i =1, 2) such that
ΓCTN =

ΥA
2γE0
γT
EΥB
10
0 0 0

|{z }
PSD
+

ΥA
10γF
0 0 0
γT
F0ΥC
2


| {z }
PSD
+

0 0 0
0ΥB
2γG
0γT
GΥC
1


|{z }
PSD
. (29)
Comparing to Equation
(26)
, we consider that we distributed the black blocks to the
first three matrices. Although the proof techniques differ notably, the block decomposition
has already been presented in the first work on CMs of network states [
11
]. As demonstrated
in that same work, Proposition 5can be evaluated as an SDP. However, we are seeking
practical analytical methods and criteria to determine if a state cannot be prepared in
a network setting. In the next section, we present such a criterion that follows from
Proposition 5.
Let us briefly discuss how this proposition applies to LOSR triangle network states,
which can be expressed as
$∆=∑λpλ$λ
CTN
. We recall that in this set up, the sources and the
local operations may be coordinated by a classical random variable
λ
. From the concavity
property in Ref. [
23
], we know that the difference between the covariance matrix
Γ($∆)
and
the weighted sum of CMs
∑λpλΓ($λ
CTN)
is a PS D matrix. Using Proposition 5, we directly
obtain that there exist block matrices such that
Γ($∆)≥

  0
  0
0 0 0

| {z }
PSD
+

0
000
0


| {z }
PSD
+

0 0 0
0 
0 


| {z }
PSD
. (30)
While a similar trick can lead to powerful necessary criteria for separability in the case of
entanglement [
23
], it is not the case here. This is because the extreme points in the case of
LOSR triangle network states are not well characterised, as already discussed in Supplemen-
tary Note 1 of Ref. [
17
]. Indeed, while it is known that the extreme points are of the form
EA⊗EB⊗ EC(|cihc|⊗|bihb|⊗|aiha|)
, it is not clear whether they are necessarily pure: on
the one hand, the author of Ref. [
13
] showed that no three-qubit genuine multipartite en-
tangled (GME) pure state can be generated in a triangle network, and on the other hand, the
authors of Ref. [
10
] managed to find a state in the LOSR triangle network that has a fidelity
to the GHZ state of 0.5170. From Ref. [
29
], we know that states with such fidelity must be
GME, thus, there exist extremal points of the set of three-qubit LOSR network states that are
GME mixed states. Finally, we notice that pure GME states can exist in higher-dimensional
triangle networks: For instance, the three-ququart state
|φ+iA2B1⊗|φ+iB2C1⊗|φ+iC2A1
is
GME for the partition A1A2|B1B2|C1C2[16].
When additional properties of the states are known, such as the purity or the rank,
SDP-based criteria for LOSR networks can be obtained, as shown in Ref. [25].

Entropy 2023,25, 1260 9 of 19
6. Covariance Matrix Criterion for Triangle Network States
As seen in the previous section, CMs of CTN states with local observables {Ai,Bj,Ck}
possess a block decomposition. From Proposition 5, we obtain inequalities for any unitarily
invariant norm k·k,
2kγEk≤kA2k+kB1k, (31)
for which we can take the trace norm and obtain
2kγEktr +2kγFktr +2kγGktr ≤tr(A1+A2+B1+B2+C1+C2). (32)
This gives us a direct necessary criterion for triangle network states.
Proposition 6
(CMand trace norm criterion for triangle network states)
.
Let
Γ
be the CM of
a triangle network state $CTN =EA⊗ EB⊗ EC($B TN )with local observables {Ai,Bj,Ck}. Then,
tr(Γ)≥2kγEktr +2kγFktr +2kγGktr (33)
has to hold, with γE,γF, and γGas in Equation (5).
Now, we apply the trace norm criterion to exclude states from the triangle network
scenario.
First, we notice that, contrarily to BT N states, three-qubit states can be generated in the
CTN scenario. Therefore, we first consider the three-qubit GHZ state that we mix with white
noise, i.e.,
$GH Z(v) = v|GHZihGH Z|+ (1−v)18
8. (34)
By taking the three-qubit observable set
SGH Z ={σz11
,
1σz1
,
11σz}
for the CM, the CM
and trance norm criterion excludes
$GH Z(v)
for
v>1/2
. The CM of the Wstate
1/√3(|
100
i+
|
010
i+|
001
i)
with observables
SW={σx11
,
σy11
,
1σx1
,
1σy1
,
11σx
,
11σy}
excludes it for
v>3/4
by the same method. Note that we omitted the tensor product signs for readability.
Further than that, we can put a bound on the fidelity of a triangle state to the GHZ state.
Consider an arbitrary state
$=F|GH ZihGHZ|+ (
1
−F)˜
$
, where
hGH Z|˜
$|GH Zi=
0.
From Proposition 6with
SGH Z
, we show that
F
cannot be larger than 3
−√5'
0.76. We
note that this result was already obtained in Ref. [
17
] using similar methods, and that by
exploiting symmetries, this upper bound on the fidelity can be improved to
1/√2'
0.71,
which is, to our knowledge, the best analytical bound so far.
It is worth realising that upper bounds on the fidelity of triangle states to a given target
state
|Ψi
also hold in the case of LO SR networks. Indeed, LOSR network states
$∆
are states
that can be written as a convex combination of CTN states, and thus,
max$∆hΨ|$∆|Ψi=
max$CTN hΨ|$CTN|Ψi
. From this, we can conclude that from Proposition 6it follows that any
state with a fidelity to the GHZ state higher than 3
−√5'
0.71 is excluded also from the
LOSR triangle network scenario.
Before closing this section, a brief comment is in order. At first glance, it may seem
that by only using the
σz
correlations of a three-qubit state, we could exclude it from the set
of LOSR network states and, thus, learn about its entanglement. However, this would be
problematic because all separable three-qubit states are in the set of LOS R triangle network
states, and all
σz
correlations can be simulated by separable states. However, this is not the
logic of the argument above: Proposition 6puts a bound on the extremal points of LOSR
triangle network states, which then by convexity holds for all LO SR states. In order to draw
a conclusion for a given state, knowledge of the fidelity to some target state is required,
which requires additional measurements than the σzcorrelations alone.
7. NCDS Networks
In this section, we show that the block decomposition of covariance matrices of net-
work states can also hold for larger networks. Indeed, if we consider networks where two

Entropy 2023,25, 1260 10 of 19
nodes share parties from at most one common source (NCDS networks), the triangle network
results can be extended. Examples of such networks are networks with bipartite sources.
More explicitly, consider an
N
-node NCDS network with a set of sources
S
. The number
of sources is given by
|S|
, and each source
s∈S
is the set of nodes the source connects. Let
ΓNCDS
be the CM of a global state of such a network with observables
{Ax|i:x=
1,
. . .
,
N}
,
where
Ax|i
is the
i
th observable that only acts on the node
x
. Then,
ΓNCDS
has a block form
analogous to Equation
(5)
, where the diagonal blocks are labelled
Γx
and the off-diagonal
block are γxy =γT
yx (x6=y,x,y∈ {1, . . . , N}). Formally, we state that
Proposition 7
(Block decomposition for CMs of NCDS network states)
.
There exist matrices
Υs
x
(
s∈S
,
x∈s
) such that
ΓNCDS
can be decomposed as a sum of
|S|
positive semi-definite block
matrices
Ts
(
s∈S
) where the off-diagonal blocks of each
Ts
are
γxy
for
{x
,
y} ⊂ s
and 0for
{x,y} 6⊂ s, and where the diagonal blocks are Υs
x.
For a technical proof, see Appendix E. In there, we prove that in the case of basic (i.e.,
without local operations) networks with no common double source (BNCDS networks), the
proposition holds. Following a similar line of reasoning to the proofs for triangle networks,
the proposition naturally extends to NCDS networks with local operations.
Let us consider an easy example for the sake of clarity. Figure 2shows a five-partite
network consisting of two tripartite sources
$a
and
$b
, and one bipartite
$c
. The set of
sources is given by S={a,b,c}={{1, 2, 3},{3, 4, 5},{1, 5}}.
Figure 2.
Five-partite network with two tripartite sources
$a
and
$b
, and one bipartite source
$c
. The
parties 1, 2, 3, 4, and 5 may perform a local channel Eion their system i(i=1, . . . , 5).
Following the notation of Proposition 7, there must exist eight matrices
Υa
1
,
Υa
2
,
Υa
3
,
Υb
3
,
Υb
4,Υb
5,Υc
1, and Υc
5such that the CM of the global network state
$NCDS =E1⊗E2⊗ E3⊗ E4⊗ E5($a⊗$b⊗$c)(35)
may be decomposed as
ΓNCDS =






Υa
1  0 0
Υa
20 0
  Υa
30 0
0 0 0 0 0
0 0 0 0 0






| {z }
Ta
+






0 0 0 0 0
0 0 0 0 0
0 0 Υb
3 
0 0 Υb
4
0 0   Υb
5






| {z }
Tb
+






Υc
1000 
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
000Υc
5






| {z }
Tc
, (36)
where the off-diagonal blocks are simply the ones from ΓNCDS.
We see directly that we can extend Proposition 6as

Entropy 2023,25, 1260 11 of 19
Proposition 8
(CMand trace norm criterion for NCDS network states)
.
Let
ΓNCDS
be as
above. Then,
tr(ΓNCDS )≥2N
∑
x>y=1
γxy 
tr (37)
has to hold.
We note that this criterion does not take network topology into account: it treats a
network with a single
N−
1-partite source the same way it treats a line network with
N−
1
bipartite sources. While this is interesting as it can exclude states from all networks, we
also expect it to be weaker than criteria designed for specific network topologies. On top
of that, Proposition 8only takes into account that the principal submatrices of each
Ts
are
positive semi-definite, not that the matrices themselves are PSD.
As an example, let us consider an
N
-qubit GHZ state,
|GH ZNi=1/√2(|0i⊗N+|1i⊗N)
,
of visibility
v
mixed with white noise. As observables, we take
σ(x)
z
for each qubit
x
. The
resulting CM will have diagonal elements equal to one, whereas the off-diagonal elements
will be
v
. Applying the previous proposition, we exclude
N
-partite GHZ states mixed with
white noise from any NCDS network scenario for
v>1
N−1. (38)
With N=3, we recover the result of the example for the triangle network.
Nevertheless, we are forced to observe that the criterion only considers two-body
correlation, therefore, cannot fully capture the entanglement in the target states. To see this,
let us look at the four-qubit cluster state
|Cli=|+0+0i+|+0−1i+|−1−0i+|−1+1i
(up to normalisation). Its generators are
σxσz11
,
σzσxσz1
,
1σzσxσz
, and
11σzσx
, where the
only two-body correlations are given by
σxσz11
and
11σzσx
. A possible set of observables
is S={σ(1)
x,σ(2)
z,σ(3)
z,σ(4)
x}and we obtain
Γ(S,|Cli) = 



1100
1100
0011
0011



. (39)
The trace criterion is satisfied and, thus, we cannot exclude
|Cli
from NCDS network
scenarios by means of Proposition 8. Moreover, we directly see that the matrix has a block
decomposition, namely,
1 1
1 1⊕1 1
1 1
, which a priori could arise from a network with
two bipartite sources. However, we know from Ref. [
17
] that the four-qubit cluster state
cannot be generated in bipartite networks.
8. Conclusions
In this work, we presented alternative proofs to the block decomposition of covariance
matrices for network states. From these, we derived analytical criteria to certify that some
states cannot be generated through quantum networks as we define them in Equation
(3)
.
This means that those excluded states either require global sources that connect all nodes,
classical communication, non-local operations, or shared randomness to be generated.
Concerning the latter resource, we also showed that Propositions 6can be used to upper-
bound the fidelity to some target states that LOSR network states can have. Furthermore,
we stress that our criteria are analytical and computable.
Regarding extensions of our work, it would be worthwhile to investigate whether the
proof of Proposition 7can be extended to networks beyond NCDS networks. As shown
in Ref. [
11
], the latter is indeed possible, which implies that Propositions 7and 8hold for
general networks as well.

Entropy 2023,25, 1260 12 of 19
Finally, as the field of network entanglement and its potential applications in quan-
tum information theory continues to grow, it may be valuable to investigate additional
avenues for identifying compatible network states. Specifically, an area of interest is finding
sufficient criteria for network states, as current results only provide necessary criteria. By
developing such criteria, we may be able to learn more about states that can be generated
in networks without communication and about their potential usefulness, for instance, for
quantum conference key agreement. In this context, it is interesting to also consider noisy
networks: this would translate to imposing additional conditions on the sources states, e.g.,
by making them travel through depolarisation channels or by constraining their purity.
Author Contributions:
Conceptualization, K.H. and O.G.; Formal analysis, K.H.; Writing—original
draft, K.H.; Writing—review & editing, O.G.; Supervision, O.G. All authors have read and agreed to
the published version of the manuscript.
Funding:
This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation, Projects No. 447948357 and No. 440958198), the Sino-German Center
for Research Promotion (Project No. M-0294), the German Ministry of Education and Research (Project
QuKuK, BMBF Grant No. 16KIS1618K), and the House of Young Talents of the University of Siegen.
Institutional Review Board Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Here, we prove that the off-diagonal blocks of the CM of a triangle network state can
be expressed using the reduced observables, that is,
[γE]mn =hA(2)
m⊗B(1)
ni−hA(2)
mihB(1)
ni, (A1)
with
A(2)
m=trA1Am$(A1)
BTN ⊗1A2(A2)
and similarly for B(1)
n.
Proof of Equation (8).
Let us decompose
Am
and
Bn
, respectively, in orthogonal bases
{Gα⊗Gβ}and {Gγ⊗Gδ}satisfying tr(GαGα0)=dδαα0as
Am=1
d2∑
α,β
trGα⊗GβAmGα⊗Gβ(A3)
and
Bn=1
d2∑
γ,δ
tr(Gγ⊗GδBn)Gγ⊗Gδ, (A4)
and notice that the reduced states of $BTN are product states:
$(AB)
BTN =$(A1)
BTN ⊗$(A2B1)
BTN ⊗$(B2)
BTN , (A5)
$(A)
BTN =$(A1)
BTN ⊗$(A2)
BTN , (A6)
$(B)
BTN =$(B1)
BTN ⊗$(B2)
BTN . (A7)
Combining this, [γE]mn straightforwardly decomposes as
hA(2)
m⊗B(1)
ni$(A2B1)
BTN − hA(2)
mi$(A2)
BTN hB(1)
ni$(B1)
BTN
, (A8)
and the proof is complete.

Entropy 2023,25, 1260 13 of 19
Appendix B
We prove here one of our central results, namely, that the CM of a BTN state can be
decomposed into the sum of CMs with reduced observables, i.e.,
Proposition A1 (Block decomposition for CMs of BTN states).The
CM of a BTN state with local observables {Ai,Bj,Ck}can be decomposed as
ΓBTN =Γ({Ai,Bj,Ck},$B TN )
=

ΓA2γE0
γT
EΓB10
0 0 0

| {z }
Tc
+

ΓA10γF
0 0 0
γT
F0ΓC2


| {z }
Tb
+

0 0 0
0ΓB2γG
0γT
GΓC1


| {z }
Ta
+

RA0 0
0RB0
0 0 RC


| {z }
R
(A9)
where the matrices Ta, Tb, and Tcare CMs for the state-dependent reduced observables, i.e.,
Tc=Γ({A(2)
i,B(1)
j},$(A2B1)
BTN ). (A10)
and analogously for Tband Ta. The matrix R is positive semi-definite.
Proof of Proposition A1.Following Equation (A9), the matrix RAis given by
RA=ΓA−ΓA1−ΓA2, (A11)
where the entries of ΓA2are
[ΓA2]mn =hA(2)
mA(2)
ni−hA(2)
mihA(2)
ni, (A12)
which is a CM for the reduced observables, evaluated on the state
$(A2)
BTN
only. Let us now
show that such a matrix
RA
is positive semi-definite by showing that
hx|RA|xi ≥
0 for an
arbitrary complex vector |xi. Using the definition
M=∑
i
xiAi(A13)
and the fact that
M(1)=∑
i
xiA(1)
iand M(2)=∑
i
xiA(2)
i, (A14)
we have that
hx|RA|xi=hM†Mi−hM†ihMi−h(M(1))†M(1)i − hM†ihMi
−h(M(2))†M(2)i−hM†ihMi
=hM†Mi+hM†ihMi−h(M(1))†M(1)i − h(M(2))†M(2)i.
(A15)
Since
M
acts on
A1A2
, we can use a Schmidt-like decomposition for the bipartition
A1|A2
,
M=∑
i
Pi⊗Qi(A16)
and use the fact that $(A1A2)
BTN is a product state. We then arrive at
hx|RA|xi=∑
ij hP†
iPjihQ†
iQji+hP†
iihPjihQ†
iihQji − hP†
iPjihQ†
iihQji − hP†
iihPjihQ†
iQji
=trΓ(P)TΓ(Q),
(A17)

Entropy 2023,25, 1260 14 of 19
where
[Γ(P)]ij =hP†
iPji− hP†
iihPji(A18)
and similarly
Γ(Q)
are CMs of the observables
{Pi}
in the state
$A1
BTN
and
$A2
BTN
, respectively.
These matrices are positive semi-definite, so we have
trΓ(P)TΓ(Q)≥
0, which finishes
the proof.
Appendix C
In the main text, we write (see Equation (11))
RX=ΓX−ΓX1−ΓX2
=Γ{Gα},$(X1)
BTN ⊗Γ{Gβ},$(X2)
BTN ,X=A,B,C.(A19)
A proof is given by direct calculation.
Proof of Equation (11).
We show the statement for
X=A
. The matrices
ΓA
,
ΓA1
, and
ΓA2
have, respectively, the following matrix elements
[ΓA]αβ|α0β0=h(Gα⊗Gβ)(Gα0⊗Gβ0)i−hGα⊗GβihGα0⊗Gβ0i, (A20)
[ΓA1]αβ|α0β0=hGαGα0ihGβihGβ0i − hGαihGα0ihGβihGβ0i, (A21)
[ΓA2]αβ|α0β0=hGαihGα0ihGβGβ0i − hGαihGα0ihGβihGβ0i, (A22)
where the expectation values are taken on the state
$(A)
BTN
, with identity operators padded
where needed. So, the matrix elements of RAare
[RA]αβ|α0β0=h(Gα⊗Gβ)(Gα0⊗Gβ0)i−hGαGα0ihGβihGβ0i − hGαihGα0ihGβGβ0i
+hGαihGα0ihGβihGβ0i
=(hGαGα0i − hGαihGα0i)(hGβGβ0i − hGβihGβ0i)
=[Γ({Gα},$(A1)
BTN ]αα0[Γ({Gβ},$(A2)
BTN ]β β0
=[Γ({Gα},$(A1)
BTN )⊗Γ({Gβ},$(A2)
BTN )]αβ|α0β0
(A23)
since [A⊗B]ij|i0j0=Aii0Bjj0, and, therefore,
RA=Γ({Gα},$(A1)
BTN )⊗Γ({Gβ},$(A2)
BTN ). (A24)
Remark A1.
We note that in general, for a product state
$=$1⊗$2
and product observables
{Ak⊗Bl}, it holds that
Γ({Ak⊗Bl},$) = |~
aih~
a|⊗Γ({Bl},$2) + Γ({Ak},$1)⊗|~
bih~
b|+Γ({Ak},$1)⊗Γ({Bl},$2), (A25)
where
|~
ai
and
|~
bi
are the vectors with entries
hAki$1
and
hBli$2
, respectively. In the case of complete
sets of orthogonal observables, |~
aiand |~
biare the Bloch vectors of $1and $2, respectively.
Appendix D
In this appendix, we want to show that Equations (13) and (14) hold, which we recall
to be
ΓA2=|~
a(1)ih~
a(1)|⊗Γ({Gβ},$(A2)
BTN )(A26)
and
γE=|~
a(1)ih~
b(2)|⊗γ({Gβ,Gα},$(A2B1)
BTN )(A27)

Entropy 2023,25, 1260 15 of 19
respectively, with~
a(1)≡(a(1)
0, . . . , a(1)
d2−1)T∈Rd2and similarly for~
b(2).
Proof of Equations (13) and (14). A direct calculation shows that
[ΓA2]mn ≡[ΓA2]αβ|α0β0=ha(1)
αa(1)
α0GβGβ0i−ha(1)
αGβiha(1)
α0Gβ0i
=a(1)
αa(1)
α0hGβGβ0i−hGβihGβ0i
=a(1)
αa(1)
α0[Γ({Gβ},$(A2)
BTN )]ββ0
=[|~
a(1)ih~
a(1)|⊗Γ({Gβ},$(A2)
BTN )]αβ|α0β0
(A28)
and that
[γE]mn ≡[γE]αβ|α0β0=a(1)
αb(2)
βhGβ⊗Gαi−hGβihGαi
=[|~
a(1)ih~
b(2)|]αβ [γ({Gβ,Gα},$(A2B1)
BTN )]α0β0
=[|~
a(1)ih~
b(2)|⊗γ({Gβ,Gα},$(A2B1)
BTN )]αβ|α0β0.
(A29)
Appendix E
Let us first recall notation from the main text. We have an
N
-node NCDS network with
a set of sources
S
. The number of sources is given by
|S|
, and each source
s∈S
is the set
of nodes the source connects where the nodes themselves are labelled by
x∈ {
1,
. . .
,
N}
.
The sources states are denoted
$s
,
s∈S
. Each party
x
obtains
nx
qudits from
nx
different
sources and any two distinct parties share at most one source.
Let
ΓNCDS
be the CM of a global state of such a network with observables
{Ax|i:x=
1,
. . .
,
N}
, where
Ax|i
is the
i
th observable that only acts on the node
x
. We show in this
appendix that
Proposition A2
(Block decomposition for CMs of NCDS network states)
.
There exist matrices
Υs
x
(
s∈S
,
x∈s
) such that
ΓNCDS
can be decomposed as a sum of
|S|
positive semi-definite block
matrices
Ts
(
s∈S
), where the off-diagonal blocks of each
Ts
are
γxy
for
{x
,
y} ⊂ s
and 0for
{x,y} 6⊂ s, and where the diagonal blocks are Υs
x.
As mentioned in the main text, we first prove the proposition for basic networks with
no common double source (BNCDS networks). The extension to NCDS networks without
local operations follows using similar tricks to the triangle network scenario.
To achieve this, we extend Remark A1 to Nparties
Lemma A1.
Let
$=$1⊗ · ·· ⊗ $N
be a product state and
{A(1)
i1⊗ · ·· ⊗ A(N)
iN}
be a set of
product observables. The covariance matrix reads
Γ{A(1)
i1⊗ ·· ·⊗ A(N)
iN},$=
N
O
α=1|~
aαih~
aα|+Γ({A(α)
iα,$α)−
N
O
α=1|~
aαih~
aα|, (A30)
where |~
aαiis the vector with entries hA(α)
iαi$α.
Proof. The CM has matrix elements

Entropy 2023,25, 1260 16 of 19
[Γ({A(1)
i1⊗ ·· ·⊗ A(N)
iN},$)]i1...in|j1...jn
=h(A(1)
i1⊗ ·· ·⊗ A(N)
iN)(A(1)
j1⊗ ·· ·⊗ A(N)
jN)i$−hA(1)
i1⊗ ·· ·⊗ A(N)
iNi$hA(1)
j1⊗ ·· ·⊗ A(N)
jNi$
=
N
∏
α=1hAiαAjαi$α−
N
∏
α=1hAiαi$αhAjαi$α
(A31)
and
Ω=Γ{A(1)
i1⊗ ·· ·⊗ A(N)
iN},$=
N
O
α=1|~
aαih~
aα|+Γ({A(α)
iα,$α)−
N
O
α=1|~
aαih~
aα|(A32)
has matrix elements
Ωi1...in|j1...jn=
N
∏
α=1hAiαi$αhAjαi$α+hAiαAjαi$α−hAiαi$αhAjαi$α−
N
∏
α=1hAiαi$αhAjαi$α,(A33)
which is exactly Equation (A31).
We are now ready to prove the block decomposition of a CM of a BNCDS state with
product observables, that is, we furthermore require that the observables are of the form
Ax|i=Ax1|i⊗ · ·· ⊗ Axnx|i
, with
Ax1|i
acting on the first qudit of the party
x
, labelled
x1
,
and similarly for the others.
Lemma A2.
Let
$
be a BNCDS network state. Let
Ax|i=Ax1|i⊗ ·· ·⊗ Axnx|i
, with
Ax1|i
acting
on the first qudit of the party x, labelled x1, and similarly for the others. Then,
Γ{Ax|i:x=1, . . . , N},$=∑
s∈S
Γ{ARED
xα|i:xα∈s},$s+
N
M
x=1
Rx, (A34)
where Rxare PSD matrices and
ARED
xα|iα=Axα|iα
nx
∏
β6=α,β=1hAxβ|iβi$(xβ).(A35)
We note that the matrices
Γ{ARED
xα|i:xα∈s},$s
are padded with blocks of zeros where needed,
such that they are partitioned in
N×N
blocks with the
i
th diagonal block corresponding to the
ith party.
Proof.
From the fact that each subset of observables
{Ax|i}
only acts on one party of the
network, it directly follows that ΓNCDS has a block structure,
ΓNCDS =




Γ1γ12 . . . γ1N
γT
12 Γ2. . . γ2N
.
.
..
.
.....
.
.
γT
1NγT
2N. . . ΓN





. (A36)
Let us investigate the structure of
Γx
(
x∈ {
1,
. . .
,
N}
) for a BNCDS network state
$BNCDS. We recall that
Γx=Γ{Ax1|i⊗ ·· ·⊗ Axnx|i}i,$(x)
BNCDS(A37)

Entropy 2023,25, 1260 17 of 19
where
$(x)
BNC DS =tr ˆ
x($BNC DS )
,
ˆ
x={
1,
. . .
,
N} \ {x}
. For the sake of readability, we will
drop the subscript BNCDS until the end of the proof. As
$(x)
is a product state,
Γx
can be
decomposed following Lemma A1, i.e.,
Γx=
N
O
α=1|~
xαih~
xα|+Γ({Axα|iα},$(xα))−
N
O
α=1|~
xαih~
xα|, (A38)
with hAxα|iαi$(xα)being the vector elements of |~
xαi. Therein, the summands
Γ({Axα|iα},$(xα))
nx
O
β6=α,β=1|~
xβih~
xβ|(A39)
can be written as
Γ{ARED
xα|iα},$(xα), (A40)
with
ARED
xα|iα=Axα|iα
nx
∏
β6=α,β=1hAxβ|iβi$(xβ). (A41)
Now, we analyse the off-diagonal blocks have matrix elements
γxy ij =hAx|i⊗Ay|ji$xy − hAx|ii$xhAy|ji$y. (A42)
They are trivially equal to zero when the nodes
x
and
y
are not connected as in that case,
$(xy)=$(x)⊗$(y)
. On the other hand, if they are connected, it is by one source exactly.
Without loss of generality, we assume that
x1
and
y1
are connected by the same source, and
the state can be written as
$(xy)=$(x1y1)nx
O
α=2
$(xα)
ny
O
β=2
$(yβ). (A43)
Therefore, Equation (A42) reads
γxy ij =hAx1|i1⊗Ay1|j1i$(x1y1)− hAx1|i1i$(x1)hAy1|i1i$(y1)nx
∏
α=2hAxα|iαi$(xα)
ny
∏
β=2hAyβ|iβi$(yβ), (A44)
which, with the reduced observables of Equation (A41) can be written as
γxy ij =hARED
x1|i1⊗ARED
y1|j1i$(x1y1)−hARED
x1|i1i$(x1)hARED
y1|i1i$(y1). (A45)
Finally, putting everything together, we obtain
Γ{Ax|i:x=1, . . . , N},$=∑
s∈S
Γ{ARED
xα|i:xα∈s},$s+
N
M
x=1
Rx, (A46)
where
Rx=
nx
O
α=1|~
xαih~
xα|+Γ({Axα|iα},$α)−
nx
O
α=1|~
xαih~
xα|−
nx
∑
α=1

Γ({Axα|iα},$α)
nx
O
β6=α,β=1|~
xβih~
xβ|
(A47)
is positive semi-definite.
Now that we have the explicit structure of CMs for product observables on BNCDS
network states, it directly follows that in this case, the CMs have a block decomposition
as described in Proposition A2. We use the following lemma to argue that the block
decomposition holds for any set of local observables:

Entropy 2023,25, 1260 18 of 19
Lemma A3.
Let
Γ{Ni}n
i=1},$
be a CM. Let
C
be a real matrix such that
Mj=∑n
i=1Cij Ni
,
j=1, . . . , m. Then
Γ{Mj}m
j=1,$=CTΓ({Ni}n
i=1,$)C. (A48)
Proof. A direct calculation gives
hΓ{Mj}m
j=1,$ikl =
n
∑
i,j=1hCik AiCjl Aji$− hCik Aii$hCj l Aji$
=∑
i,j
CT
ki[Γ({Ni}n
i=1,$)]ijCjl ,(A49)
which proves the claim.
Combining all those results, we are now ready to prove Proposition A2.
Proof of Proposition A2.
From Lemma A2, we know that the block decomposition holds
for BNCDS network states with product observables. When those product observables are
chosen to be a complete set of observables, Lemma A3 shows that the block decomposition
holds for any observable set acting on BNCDS network states. Finally, an analogous reason-
ing to the cases of UTN and CTN leads to the conclusion that the block decomposition holds
for states of NC DS networks with local operations.
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