Characterize arbitrary quantum networks in the noisy intermediate-scale quantum era Zhen-Peng Xu School of Physics and Optoelectronics Engineering Anhui University 230601 Hefei China and

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Characterize arbitrary quantum networks in the noisy intermediate-scale quantum era

Zhen-Peng Xu∗

School of Physics and Optoelectronics Engineering, Anhui University, 230601 Hefei, China and

Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany

(Dated: October 24, 2023)

Quantum networks are of high interest nowadays. In short, it describes the distribution of quan-

tum sources represented by edges to diﬀerent parties represented by nodes in the network. Bundles

of tools have been developed recently to characterize quantum states from the network in the ideal

case. However, features of quantum networks in the noisy intermediate-scale quantum (NISQ) era

invalidate most of them and call for feasible tools. By utilizing purity, covariance and topology of

quantum networks, we provide a systematic approach to tackle with arbitrary quantum networks

in the NISQ era, which can be noisy, intermediate-scale, random and sparse. One application of

our method is to witness the progress of essential elements in quantum networks, like the quality of

multipartite entangled sources and quantum memory.

Numerous works have advertised from diﬀerent scales

the advent of quantum network technology, as small as

the storage of a single entangled pair [1], and as broad

as quantum internet [2–4]. Apart from the theoretical

importance, quantum networks appear naturally in prac-

tice, especially in quantum key distribution [5, 6], quan-

tum network metrology [7–9] and quantum distributed

computation [10]. A recent move is into the characteriza-

tion of diﬀerent quantum correlations arising from quan-

tum networks [11–23].

A quantum network can be abstracted as a hyper-

graph, where each node stands for a local lab and each

hyperedge represents a quantum source that distributes

particles only to labs associated with the correspond-

ing nodes, see Fig. 1 for examples. A correlated quan-

tum network (CQN) allows the pre-shared classical pro-

tocol, i.e., global classical correlation [12], an indepen-

dent quantum network (IQN) allows not. Despite recent

progress [11–23], the study of quantum network states is

still in its cradle. Past research has focused mainly on

IQN, bundles of tools [17–19, 24, 25] have been added into

the current toolbox. However, they become incapable to

detect the underlying structure of CQN even when only

a small amount of global classical correlation appears. In

comparison, few methods [11–15] exist for CQN, which

either work only for special kinds of states like symmetric

states [13–15], limited quantum networks like the trian-

gle network [11, 12] or complete n-partite network with

(n−1)-partite sources [26–28]. However, an undeniable

fact is that we progress toward the noisy intermediate-

scale quantum (NISQ) era, as pointed out sagaciously by

Preskill [29]. The global classical correlation exists then

frequently in real applications, which can even elicit from

the ﬂap of a butterﬂy’s wings in Brazil [30].

Apart from the unavoidable global noise, quantum net-

works in the NISQ era share at least other three fea-

tures: intermediate-scale, random, and sparse. Though

the size of quantum networks in the NISQ era is limited,

it can be not small, considering that IBM has unveiled

a quantum chip with 433 qubits [31] already. The ran-

2 3

(a) (b) (c)

FIG. 1. Three quantum networks, where each node stands for

one local lab, one edge in real line represents a genuine bipar-

tite entangled quantum source shared by diﬀerent labs in the

corresponding nodes, and one edge in dashed line represents a

separable quantum source. In practice, the quantum network

in (a) can degenerate to the one in (b), even to the one in (c).

domness in the network [32] can originate from the ran-

dom establishment of quantum links with quantum re-

peaters [33, 34], and also the decoherence of established

links as considered in waiting time [35]. Degeneration

of a triangle quantum network until a classical network

is illustrated in Fig. 1. Since genuine multipartite en-

tanglement is hard to prepare and to maintain [36, 37],

the realistic quantum networks will be sparse. Tools for

quantum networks in the NISQ era regarding those fea-

tures are still missing.

In this work, we characterize correlated quantum net-

works in the NISQ era by employing the purity of the

state and covariance of the measured data. Those meth-

ods are operational in the sense that only the available

experiment data is employed, without knowing the exact

underlying quantum state. Purity of the network state

plays an essential role here, as the classical correlation in

a state can be captured by its purity. Pretty recent re-

search shows that the purity of a multipartite state can be

evaluated eﬃciently with only local operations [38, 39],

which ﬁts the network scenario. The methods developed

here are feasible for noisy intermediate-scale or big quan-

tum networks. Interestingly, they work even for a collec-

tion of networks with diﬀerent kinds of topology, which

can cover the random network models, especially the ones

with probabilistic genuine bipartite sources as in the con-

arXiv:2210.13751v2 [quant-ph] 23 Oct 2023

sideration of quantum repeaters [33]. Thus, they answer

one corresponding open question in the review paper on

nonlocality in quantum networks [22]. We can also ap-

ply our methods to a part of the network instead of the

whole, which ﬁts the sparse structure of the network in

the NISQ era and reduces the diﬃculty of computation.

GHZ state under decoherence.— As a warming-up ex-

ercise we discuss the Greenberger-Horne-Zeilinger (GHZ)

state of nqubits under decoherence,

ρ(α) = (1 −α)|GHZ+⟩⟨GHZ+|+α|GHZ−⟩⟨GHZ−|,(1)

where |GHZ±⟩= (|0···0⟩±|1···1⟩)/√2, and α∈[0,1/2]

describes the degree of decoherence. Despite its simplic-

ity, this example allows us to introduce our main ideas.

If all the nparties implement the same measurement

Z=|0⟩⟨0|−|1⟩⟨1|, then two possible combinations of

outcomes happen equally with probability 1/2, i.e., ei-

ther all of the outcomes are 0, or all of them are 1.

To simulate this statistical behaviour without genuine n-

partite entanglement, the state for simulation can only be

ρc= [|0···0⟩⟨0···0|+|1···1⟩⟨1···1|]/2, since no other

0−1 string appear as a combination of outcomes. Such

a simulation invalidates known methods with only sta-

tistical data [19, 26–28]. It costs at least one classical

bit of randomness, as the Shannon entropy or the Von

Neumann entropy of ρcis 1. However, the Von Neumann

entropy of the state ρ(α) is −[αlog α+(1−α) log(1−α)],

which is strictly less than 1 for α∈[0,1/2). This means

that we cannot simulate the statistical behaviour and the

Von Neumann entropy of ρ(α) simultaneously by a quan-

tum network with at most (n−1)-partite sources.

The Von Neumann entropy is one way to measure the

purity of the state, capturing partially the classical cor-

relations in the state. To continue, we examine ﬁrstly

diﬀerent measures of purity and choose a suitable one

for our following methods. For a given state ρin the

d-dimensional space, the common measures of its pu-

rity [40–42] are R´enyi α-purity log2d−log2(Tr(ρα))/(1−

α), which converges to the Von Neumann entropy as

αtends to 1, and linear entropy purity Tr(ρ2)−1/d.

Through the whole text, we take τ= Tr(ρ2) to quantify

the purity, which determines R´enyi 2-purity and linear

entropy purity. The advantage of τover other quanti-

ﬁers, like the Von Neumann entropy, is that it ﬁts the

covariance of experimental data well in our approach, as

both of them contain the information of ρ2. As for the

estimation of purity of a multipartite state with diﬀerent

measures, it can be done eﬃciently with only local opera-

tions [38, 39], which are feasible in the network scenario.

Noisy quantum networks.— Noise is unavoidable for

the quantum network states in the NISQ era, either the

local noise or the global one. Quantum networks with dif-

ferent noise models can all be classiﬁed as CQNs. Firstly,

we develop the covariance matrix decomposition method

for CQN, where a key step is to separate the part related

to global classical correlation out in the whole covariance

matrix. For a given hypergraph G(V, E) and a state ρ

from CQN of G, the state ρcan be decomposed as

ρ=Xkpkρk, ρk=Oi∈VC(k)

iOe∈Eη(k)

e,(2)

where {pk}kwith Pkpk= 1 and pk>0 is the global clas-

sical correlation, C(k)

iis a local channel for the i-th party,

η(k)

eis an entangled state distributed from the source la-

beled by the hyperedge e.

For simplicity, we assume each party has only one mea-

surement, and denote Mithe measurement for the i-th

party. Then we introduce three kinds of covariance matri-

ces, Γ, Γ(k)and Γ(c), whose elements in the i-th row and

j-th column are Γij , Γ(k)

ij and Γ(c)

ij , respectively, where

Γij =⟨MiMj⟩−⟨Mi⟩⟨Mj⟩,⟨Mi⟩= Tr(ρMi),

Γ(k)

ij =⟨MiMj⟩k− ⟨Mi⟩k⟨Mj⟩k,⟨Mi⟩k= Tr(ρkMi),

Γ(c)

ij =Xkpk⟨Mi⟩k⟨Mj⟩k− ⟨Mi⟩⟨Mj⟩.(3)

The covariance matrix Γ is the one that can be observed

directly in experiments. The covariance matrices {Γ(k)}k

are hidden in the experimental data when we assume that

the randomness of the sampling {pk, ρk}kis inaccessible.

The covariance matrix Γ(c)can be viewed as a classical

covariance matrix, since it is only about the distribution

of classical data {⟨M1⟩k,...,⟨Mn⟩k}k. Throughout the

whole paper, we only consider the dichotomic measure-

ments with outcomes ±1. A pivotal observation is that

the classical covariance matrix Γ(c)can be separated out

from the observed one Γ perfectly, i.e.,

Γ = XkpkΓ(k)+ Γ(c),(4)

whose proof can be found in Sec. A in Supplemental Ma-

terial (SM) [43]. Since {Γ(k)}kare about network states

from IQN, the existing method in Ref. [19] can be em-

ployed to impose constraints on them. However, if there

is no limitation of Γ(c), the observed covariance matrix Γ

can still have arbitrary relation with the network topol-

ogy G. As it turns out, the purity of the state implies

a nontrivial condition on Γ(c), leading to a semi-deﬁnite

programming (SDP) to determine whether a state can

arise from CQN with a given topology.

Observation 1. For a given state ρfrom the CQN with

the network topology G(V, E), measurements {Mi}i∈V,

which result in the covariance matrix Γ, it holds that

Γ = Xe∈EΥe+T, ΠeΥeΠe= Υe⪰0,

T⪰0,maxi∈VTii ≤β, Tr(T)≤l1β, (5)

where l1is the maximal eigenvalue of Pi∈VMi⊗Mi,

β= 2√1−τ2,Tii is the i-th diagonal term of T,Πe=

Pi∈ePiwith Pito be the projection onto i-th row.

2 3

FIG. 2. The decomposition of the covariance matrix Γ of

a noisy state from the triangle network, where each matrix

contains 9 elements, the elements in the blank area are 0.

The block structure of each Υeimposes a constraint on itself.

The critical step is to obtain constraints of Γ(c)from available

information of the quantum network, like purity.

To apply the criterion in this observation, we need

ﬁrstly estimate the purity of the state ρ, and then im-

plement the measurements {Mi}iand obtain the covari-

ance matrix from the experimental data. It should work

for arbitrary network topology with around 50 nodes in

practice. This observation can be understood as follows.

The term Pe∈EΥecorresponds to PkpkΓ(k), as each

Γ(k)has a similar decomposition [19]. The variable T

plays the role of Γ(c)and inherits all its constraints. A

detailed proof is provided in Sec. B in SM [43]. The ap-

plication of Observation 1 to the triangle quantum net-

work is illustrated in Fig. 2. We remark that the rank

of the state determines the R´enyi-0 purity which reads

log2(d/r). By considering the rank ralso, we can set

β= min{r(1 −τ),2√1−τ2}as a tighter bound.

Revisit GHZ state under decoherence.— We take the

state ρ(α) in Eq. (1) and measurements Zfor all parties

as an example to illustrate Observation 1. The covariance

matrix Γ of ρ(α) contains always only 1 as its elements.

If Γ is from the statistics of a state in a network without

n-partite sources, then Γ should satisfy the decomposi-

tion in Eq. (5), where Gis the hypergraph with nnodes

and includes all subsets with (n−1) elements as hyper-

edges. Notice that, the rank of Γ is 1, and Γ ⪰Υe⪰0,

which implies that each Υeshould be proportional to Γ.

Since Υealways contains element 0 as exempliﬁed in

Fig. 2, we have Υe= 0, for all e∈E. Consequently,

T= Γ and maxiTii = 1. The rank of the state ρ(α) is

however 2 and the purity is τ= 1 −2α+ 2α2. Thus,

β= 1 happens only for α= 1/2, in which case ρ(α)

is fully separable. This leads to the conclusion that ρ(α)

can arise from a network without n-partite sources if and

only if α= 1/2. Our criterion is therefore tight.

Intermediate-scale networks.— The advantage of co-

variance matrix decomposition is that it requires only

experimental data of few measurements and information

of purity. However, the computation becomes heavy for

intermediate-scale networks with around 500 nodes, due

to the complexity of SDP in the method.

Here we propose another approach to solve this is-

sue, which can even take care of the randomness fea-

ture in the NISQ era. Firstly, we introduce the fact that

Pij |Mij | ≤ rTr(M) for a semideﬁnite matrix Mwhose

rank is r, and take the triangle network as an example.

According to the decomposition of Γ in Observation 1,

Pij |Γij | ≤ Pij Pe[|Υe,ij |+|Γ(c)

ij |]≤Pe2 Tr(Υe) +

3 Tr(Γ(c)), where the last inequality is from the block

structures of Υe’s and Γ(c)as in Fig. 2. Consequently,

Pij |Γij | ≤ 2 Tr(Γ)+Tr(Γ(c)) by applying the ﬁrst equal-

ity in Eq. (5) again. For the general network topology

G(V, E) with V={1, . . . , n}and kto be the maximal

size of hyperedges in E, we have

Xi,j |Γij | ≤ kTr(Γ) + (n−k) Tr(Γ(c)).(6)

In practice, we can replace Tr(Γc) in Eq. (6) by any es-

timation of it, like the analytical upper bound in Eq. (5)

results from series of relaxations [43]. A good estimation

plays a vital role in the eﬃciency of the inequality here,

same as in the criterion in Observation 1. Nowadays, it is

still hard to prepare genuine multiparite entangled states

for a large system [36]. Thus, kis usually much smaller

than nin Eq. (6), i.e., small sources in a big network.

Random networks.— The establishment of genuine

multipartite entanglement among remote labs is usually

random as in the scenario of quantum repeaters [33]. The

established one can still degenerate to less-partite ones

randomly due to decoherence. This urges us to intro-

duce the concept of random network, where the genuine

multipartite entanglement in each source exists proba-

bilistically. As an example, we consider a genuine tripar-

tite entangled source, whose degeneration is captured by

the triangle network in Fig. 2, assumed to be with proba-

bility p. The network state ρhas then the decomposition

ρ=pP3

i=1 qiρi+(1−p)ρ0, where ρ0is the original tripar-

tite state and other ρi’s are independent triangle network

states, Piqi= 1 and qi≥0. This leads to the covari-

ance matrix Γ = pP3

i=1 qiΓ(i)+ (1 −p)Γ(0) + Γ(c), where

Γ(i)’s are the covariance matrices for ρi’s, and Γ(c)is the

classical one. As argued before, ˜

Γ := P3

i=1 qiΓ(i)has the

decomposition Pe∈EΥeas in Fig. 2, which implies that

i,j=1 |˜

Γij | ≤ 2 Tr(˜

Γ). Consequently,

Xi,j |Γij | ≤ 2pTr(˜

Γ) + 3[(1 −p) Tr(Γ(0)) + Tr(Γ(c))]

= 2 Tr(Γ) + [(1 −p) Tr(Γ(0)) + Tr(Γ(c))]

≤2 Tr(Γ) + 3(1 −p) + Tr(Γ(c)),(7)

where the last inequality is from the fact that any vari-

ance should be no more than 1 as the outcomes of the

measurements are ±1.

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Characterizearbitraryquantumnetworksinthenoisyintermediate-scalequantumeraZhen-PengXu∗SchoolofPhysicsandOptoelectronicsEngineering,AnhuiUniversity,230601Hefei,ChinaandNaturwissenschaftlich-TechnischeFakult¨at,Universit¨atSiegen,Walter-Flex-Straße3,57068Siegen,Germany(Dated:October24,2023)Quantumnetw...

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Characterize arbitrary quantum networks in the noisy intermediate-scale quantum era Zhen-Peng Xu School of Physics and Optoelectronics Engineering Anhui University 230601 Hefei China and

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