Self-testing of diﬀerent entanglement resources via ﬁxed measurement settings Xinhui Li1Yukun Wang2 3Yunguang Han4yand Shi-Ning Zhu1 1National Laboratory of Solid State Microstructures School of Physics

2025-05-03 0 0 727.48KB 11 页 10玖币

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Self-testing of diﬀerent entanglement resources via ﬁxed measurement settings

Xinhui Li,1Yukun Wang,2, 3, ∗Yunguang Han,4, †and Shi-Ning Zhu1

1National Laboratory of Solid State Microstructures, School of Physics,

and Collaborative Innovation Center of Advanced Microstructure, Nanjing University, Nanjing, 210093, China

2Beijing Key Laboratory of Petroleum Data Mining,

China University of Petroleum, Beijing, 102249, China

3State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China

4College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

Self-testing, which refers to device independent characterization of the state and the measurement,

enables the security of quantum information processing task certiﬁed independently of the operation

performed inside the devices. Quantum states lie in the core of self-testing as key resources. However,

for the diﬀerent entangled states, usually diﬀerent measurement settings should be taken in self-

testing recipes. This may lead to the redundancy of measurement resources. In this work, we use

ﬁxed two-binary measurements and answer the question that what states can be self-tested with

the same settings. By investigating the structure of generalized tilted-CHSH Bell operators with

sum of squares decomposition method, we show that a family of two-qubit entangled states can be

self-tested by the same measurement settings. The robustness analysis indicates that our scheme

is feasible for practical experiment instrument. Moreover, our results can be applied to various

quantum information processing tasks.

I. INTRODUCTION

Bell nonlocality [1,2] is central to the understanding

of quantum physics. With the advent of quantum infor-

mation, Bell nonlocality has been studied as a resource

and applied to various quantum information processing

tasks, such as quantum key distribution [3,4], random-

ness expansion [5,6] and entanglement witness [7,8].

Moreover, if we assume quantum mechanics to be the

underlying theory, it is shown that certain extremal quan-

tum correlations uniquely identify the state and mea-

surements under consideration, a phenomenon known as

self-testing [9,10]. It is a concept of device indepen-

dence whose conclusion verdict relies only on the ob-

served statistics of measurement outcomes under the sole

assumptions of no-signaling and the validity of quan-

tum theory [11]. In 1990s, Popescu and Rohrlich et al.

pointed out that the maximal violation of the Clauser–

Horne–Shimony–Holt (CHSH) Bell inequality identiﬁes

uniquely the maximally entangled state of two qubit

[12,13]. In the last decades, self-testing has received

substantial attention. The scenarios for bipartite and

multipartite entangled states were presented in Refs. [14–

21]. The robustness analysis to small deviations from the

ideal case for self-testing these quantum states and mea-

surements were presented in Refs. [22–25], which made

self-testing more practical. Beyond these works focusing

on the single copy states, the parallel self-testing of tensor

product states have recently been studied. The ﬁrst par-

allel self-testing protocol was proposed for 2 EPR pairs

in [26]. The result was subsequently generalised for arbi-

trary n, via parallel repetition of the CHSH game in [28]

∗wykun06@gmail.com

†hanyunguang@nuaa.edu.cn

and via parallel repetition of the magic square game in

[29]. Self-testing of nEPR pairs via parallel repetition

of the Mayers-Yao self-test is given in [30].

In the most previous scenario, one measurement set-

ting is always competent to self-test one target state

up to local unitaries. For example, the tilted-CHSH

inequality can self-test two-qubit pure states |ψ(θ)i=

cos θ|00i+ sin θ|11iwith corresponding measurements

settings {σz, σx}⊗{cos µσz+ sin µσx,cos µσz−sin µσx},

meanwhile µis uniquely determined by θ. However, the

tasks of quantum information processing may involve

multiple states with diﬀerent entanglement degree [31].

The whole self-testing of quantum states results in an in-

creased consumption of the measurement resource, thus

strike the feasibility of practical realization. Therefore,

self-testing protocol with high practical performance is

meaningful and necessary. In this work, we focus on this

goal and provide a device independent scheme that cer-

tify a series of quantum states with reduced measure-

ment resource. Our results show that the generalized

tilted-CHSH operators allow the optimal measurements

for one party could rotate on Pauli x−zplane. Multi-

ple diﬀerent target states can be self-tested via a com-

mon measurement settings by choosing proper general-

ized tilted-CHSH operator. Hence, by utilizing a set of

Bell inequalities, we can self-test two-qubit states with

diﬀerent entanglement degree only based on two binary

measurements per party. Thus our scheme simpliﬁes the

measurement instruments and leads to less consumption

of measurement resources. Besides, our scheme demon-

strates satisfactory robustness in tolerance of noise. Fur-

ther, our scheme can serve for various quantum infor-

mation processing tasks with low measurement resources

cost, meanwhile provides secure certiﬁcation of the device

used in the task. The paper is structured as follows: In

Sec. II A, we give a brief description about the underly-

ing model and key deﬁnitions of our work. In Sec. II B,

arXiv:2210.12711v1 [quant-ph] 23 Oct 2022

we propose a scheme that self-tests diﬀerent two-qubit

entangled states via the same measurements using gener-

alized tilted-CHSH inequality. During this study, we de-

velop a family of self-testing criteria beyond the standard

tilted-CHSH inequality and prove these criteria using the

technique of sum-of-squares (SOS) decomposition. In

Sec. III, the robustness analysis is illustrated through an

example by the swap method and semideﬁnite program-

ming (SDP). In Sec. IV, the applications of our results

on quantum information processing tasks of device in-

dependent quantum key distribution, private query and

randomness generation are presented. In Sec.V, we sum-

marize the results and discuss the future research.

II. SELF-TESTING DIFFERENT ENTANGLED

STATES VIA TWO BINARY MEASUREMENTS

A. Self-testings

Consider the simplest scenario of two noncommunicat-

ing parties, Alice and Bob. Each has access to a black

box with inputs denoted respectively by x, y ∈ {0,1}and

outputs a, b ∈ {+1,−1}. One could model these boxes

with an underlying state |ψiAB and measurement pro-

jectors {Ma

x}x,a and Mb

yy,b, which commute for dif-

ferent parties. The state can be taken pure and the

measurements can be taken projective without loss of

generality, because the dimension of the Hilbert space

is not ﬁxed and the possible puriﬁcation and auxiliary

systems can be given to any of the parties. After suﬃ-

ciently many repetitions of the experiment one can es-

timate the joint conditional statistics, as known as the

behavior, p(a, b|x, y) = hψ|Ma

xMb

y|ψi. Self-testing refers

to a device-independent certiﬁcation way where the non-

trivial information on the state and the measurements

is uniquely certiﬁed by the observed behavior p(ab|xy),

without assumptions on the underlying degrees of free-

dom. Usually, self-testing can be deﬁned formally in the

following way.

Deﬁnition 0.1. We say that the correlations p(a, b|x, y)

allow for self-testing if for every quantum behavior (|ψi,

{Ma

x, Mb

y})compatible with p(a, b|x, y)there exists a local

isometry Φ=ΦA⊗ΦBsuch that

Φ|ψiAB |00iA0B0=|junkiAB ⊗ |ψiA0B0(1)

Φ(Ma

x|ψiAB |00iA0B0) = |junkiAB ⊗Ma

x|ψiA0B0,

where |00iA0B0is the trusted auxiliary qubits attached by

Alice and Bob locally into their systems, (|ψi,{Ma

x, Mb

y})

are the target system [10].

That is the correlations p(a, b|x, y)predicted by quan-

tum theory could determine uniquely the state and the

measurements, up to a local isometry.

B. Self-testing of entangled two-qubit states with

generalized tilted-CHSH inequality

In this section, we show that diﬀerent pure entangled

two-qubit states can be self-tested via ﬁxed measurement

setting with generalized tilted-CHSH inequality. The

candidate target states we considered are {|ψii}, with

|ψii= cos θi|00i+ sin θi|11i(2)

where θi∈(0,π

4]. It is already proved that pure en-

tangled two-qubit state can be self-tested using standard

tilted-CHSH inequality [14,24]. In the standard scheme,

one measurement setting is required for self-testing one

target state, which results in an increased consumption

of the measurement resource. Utilizing the property of

generalized generalized tilted-CHSH inequality, we show

that all these entangled states can be self-tested with the

given ﬁxed measurements, thus simpliﬁes the measure-

ment instruments. We have the following theorem.

Theorem 1. The family of entangled two-qubit states in

Eq. (2)can be self-tested by the same quantum measure-

ment settings in Eq. (3)with ﬁxed angle µ. The fact of

the self-testing result comes from the maximum quantum

violation of generalized tilted-CHSH inequalities in Eq.

(4).

The measurements in our scheme are chosen as,

A0=σz, B0= cos µσz+ sin µσx

A1=σx, B1= cos µσz−sin µσx(3)

with the ﬁxed angle µ∈(0,π

4].

The key idea of our self-testing scheme is that for a

given µin the unit measurement settings, a family of

Bell inequalities can be maximally violated by diﬀerent

entangled pairs respectively at the same time. Once the

form of the target source is conﬁrmed, the Bell inequality

which achieve their self-testing are determined based on

the observed statistics, p(a, b|x, y). More precisely, the

Bell inequalities have the form conditional on the input

i∈ {0,1,2, .., n}as

B[αi,βi]=βiA0+αi(A0B0+A0B1) + A1B0−A1B1,

(4)

named generalized tilted CHSH inequality [32], where

αi≥1. The maximal classical and quantum bounds

are C[αi,βi]= 2αi+βiand η[αi,βi]=p(4 + β2

i)(1 + α2

i),

respectively. It is already proved both theoretically and

numerically that pure entangled two-qubit state can be

self-tested using standard tilted-CHSH inequality with

α= 1 in Eq. (4) [14,24]. However, whether generalized

tilted-CHSH inequality can be used in self-testing is still

unknown.

We claim that the maximal quantum violation in Eq.

(4) uniquely certify the corresponding entangled pairs in

Eq. (2) and measurements in Eq. (3) with sin 2θi=

FIG. 1. Self-testing recipe. The ﬁxed untrusted measure-

ments setting are able to test diﬀerent input states. For a

given a target state |ψii, two observers randomly chose their

measurement xfor Alice, yfor Bob, and collect outcomes a

and bto construct the observation of B[αi,βi].

q4−α2

iβ2

4+β2

iand tan µ=sin 2θi

αi. Thus the family of pure

entangled two-qubit states are self-tested with the given

α,βand the ﬁxed two measurement settings µusing gen-

eralized tilted-CHSH inequality. The self-testing recipe

of our scheme is shown in Fig. 1. In the following, we

give the detailed proof of Theorem 1.

Proof. The proof of Theorem 1is divided into two

steps. First, we give two types of SOS decompositions for

generalized tilted-CHSH operator B[α,β](see Appendix

for details). Moreover, these SOS decompositions estab-

lish algebraic relations that are necessarily satisﬁed by

any quantum state and observables yielding maximal vi-

olation of the generalized tilted-CHSH inequality. Then

these algebraic relations are used in the isometry map to

provide the self-testing of any partially entangled two-

qubit state.

a. SOS decompositions for generalized tilted-CHSH

inequalities. The generalized tilted-CHSH inequalities

B[α,β]have the maximum quantum violation value η[α,β].

This implies that the operator b

B=η[α,β]I− B[α,β]is

positive semideﬁnite for all possible quantum states and

measurement operators Axand By. This can be proven

by providing a set of operators {Pi}which are polyno-

mial functions of Axand Bysuch that b

B[α,β]=PiP†

iPi,

holds for any set of measurement operators satisfying the

algebraic properties A2

x=I,B2

y=Iand [Ax, By]=0.

For convenience, we deﬁne three classes CHSH opera-

tors:

S0=A0(B0−B1) + 1

αA1(B0+B1),

S1=1

αA0(B0+B1)−A1(B0−B1),(5)

S2=A0(B0−B1)−αA1(B0+B1).

Then we can give two types of SOS decompositions for

generalized tilted-CHSH operator in Eq. (4). The ﬁrst

decomposition is given as

B[α,β]=1

∆+2η[α,β]{(b

B[α,β])2+α2(βA1−S0)2

+ (α2−1)[(−βA0+η[α,β]

α2+ 1 −A1(B0−B1))2

+ (−η[α,β]α

α2+ 1 A0+B0+B1)2]}.(6)

And the second one is

B[α,β]=α2

∆+2η[α,β]α2−1

α2[(−η[α,β]α

α2+ 1 A0+B0+B1)2

+ (−βA0+η[α,β]

α2+ 1 −A1(B0−B1))2]

+2A0−η[α,β]

2α(B0+B1) + β

2S1)2

α2(2A1−η[α,β]

2(B0−B1) + β

2S2)2(7)

where ∆ = 2(α2−1)qβ2+4

α2+1 .

For the special case α= 1 of standard tilted-CHSH

inequality, our result gives the following decomposition:

B[1,β]=1

2η[1,β]

[( b

B[1,β])2+ (βA1−S0)2],(8)

and

B[1,β]=1

2η[1,β]

[(2A0−η[1,β]

B0+B1

2+β

2S1)2

+ (2A1−η[1,β]

B0−B1

2+β

2S2)2],(9)

which reproduce the results in Ref. [14]. Thus we develop

a family of SOS decompositions for generalized tilted-

CHSH inequalities, which is beyond the standard form.

If one observes the maximal quantum violation of the

generalized tilted-CHSH inequality in Eq. (4) by any

state |ψiand measurements Ax,Byfor x, y ∈ {0,1}, then

each square of polynomial functions in two SOS decom-

positions acting on |ψiis equal to zero, i.e., Pi|ψi= 0.

Then we can obtain the anti-commutation relations for

the measurements operators acting on the underlying

state from the two SOS decompositions (6)–(7) as fol-

lowing (the details refer to Appendix)

(ZA−ZB)|ψi= 0,(10a)

(sin θXA(I+ZB)−cos θXB(I−ZA)) |ψi= 0.(10b)

Next we will show that these algebraic relations lead

to self-testing statement for any partially entangled two-

qubit state.

b. Self-testing of partially entangled states. Basing on

the Deﬁnition 0.1 of self-testing, one needs to construct

the isometry map such that the underlying system can

extract out the information of target state. The isometry

is a virtual protocol, all that must be done in laboratory

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Self-testingofdierententanglementresourcesviaxedmeasurementsettingsXinhuiLi,1YukunWang,2,3,YunguangHan,4,yandShi-NingZhu11NationalLaboratoryofSolidStateMicrostructures,SchoolofPhysics,andCollaborativeInnovationCenterofAdvancedMicrostructure,NanjingUniversity,Nanjing,210093,China2BeijingKeyLaborat...

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Self-testing of diﬀerent entanglement resources via ﬁxed measurement settings Xinhui Li1Yukun Wang2 3Yunguang Han4yand Shi-Ning Zhu1 1National Laboratory of Solid State Microstructures School of Physics

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