Certification of the maximally entangled state using non-projective measurements Shubhayan Sarkar1 1Center for Theoretical Physics Polish Academy of Sciences Aleja Lotników 3246 02-668 Warsaw Poland

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Certiﬁcation of the maximally entangled state using non-projective measurements

Shubhayan Sarkar1, ∗

1Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

In recent times, device-independent certiﬁcation of quantum states has been one of the inten-

sively studied areas in quantum information. However, all such schemes utilise projective measure-

ments which are practically difﬁcult to generate. In this work, we consider the one-sided device-

independent (1SDI) scenario and propose a self-testing scheme for the two-qubit maximally entan-

gled state using non-projective measurements, in particular, three three-outcome extremal POVM’s.

We also analyse the robustness of our scheme against white noise.

Introduction.— The existence of nonlocal correlations,

as was ﬁrst realised by Einstein, Podolski and Rosen

in 1935 [1] as a paradox and then subsequently by

Schr¨

odinger in the same year [2], is one of the most in-

triguing features of quantum theory. Consequently, Bell

[3,4] proposed a mathematical formulation to detect

whether quantum theory is inherently nonlocal or not,

and thus it is commonly referred to as Bell nonlocality.

Apart from its relevance in the foundations of physics,

Bell nonlocality has given rise to an enormous number

of applications in computation, communication and in-

formation theory [5].

A recent well profound application of nonlocality

is device-independent (DI) certiﬁcation where assum-

ing quantum theory and some other physically well-

motivated assumptions, the statistics obtained from a

black box are enough to validate the underlying mecha-

nism inside it. The strongest DI certiﬁcation is referred

to as self-testing. First introduced in [6], self-testing al-

lows one to certify the underlying quantum states and

the measurements, upto some freedom based on the

maximal violation of a Bell inequality [7]. In recent

times, there has been increased interest to ﬁnd proto-

cols to self-test various quantum systems due to their

applicability in various quantum information tasks. De-

spite the progress in designing schemes to self-test vari-

ous quantum states using projective measurements, for

instance, Refs. [8–18], there is no scheme that utilises

non-projective measurements.

The precision to experimentally generate sets of pro-

jective measurements, which are pre-requisite to ob-

serve any form of non-locality, reduces as the dimen-

sion of the system grows [for instance see [?]]. Thus,

a natural question arises whether noisy measurements

or non-projective measurements can also be used to self-

test quantum states. Further on to self-test any state or

measurement, one needs to observe the maximal viola-

tion of an inequality which touches the set of quantum

correlations, or simply the quantum set, at a particu-

lar point. It is also an open question in quantum foun-

dations whether a point on the boundary of the quan-

∗sarkar@cft.edu.pl

tum set in some scenarios can be saturated by only non-

projective measurements.

Self-testing quantum states using non-projective mea-

surements is not possible in the standard Bell scenario.

The reason being that the maximal violation of Bell in-

equalities can always be achieved by projective mea-

surements. Consequently, we consider another form of

nonlocality, known as quantum steering [19–21]. To wit-

ness quantum steering, one needs to consider the Bell

scenario with an additional assumption that one of the

parties is trusted. In the DI regime, this is referred to

as one-sided device-independent (1SDI) scenario. Cer-

tiﬁcation of quantum states and measurements in 1SDI

scenario has gained recent interest [22–28] as they are

more robust to noise and require detectors with lower

efﬁciencies when compared to fully DI scenarios [29,30].

In this work, we provide the ﬁrst scheme to certify the

two-qubit maximally entangled state



ϕ+=1

√2(|00⟩+|11⟩)(1)

using three three-outcome non-projective extremal mea-

surements in the 1SDI scenario. For this purpose, we

ﬁrst construct a steering inequality with two parties

such that each of them chooses three inputs and gets

three outputs. We then use the maximal violation of this

steering inequality to obtain the self-testing result. We

ﬁnally show that our scheme is highly robust when the

states and measurements are mixed with white noise.

Preliminaries—Before proceeding to the results, let us

ﬁrst describe the scenario and notions used throughout

this work.

Extremal positive operator valued measure (POVM). Any

measurement in quantum theory, usually referred to as

a POVM, is represented as M={Ma}where Maare the

measurement elements corresponding to the a-th out-

come of M. These elements are positive semi-deﬁnite

operators and PaMa=. Now, a POVM that can not

be expressed as a convex combination of other POVM’s

is deﬁned as an extremal POVM. As shown in [31] the

elements Maof any rank-one extremal POVM can be

expressed as Ma=λa|νa⟩⟨νa|where λa≥0 and the

elements are linearly independent.

Quantum steering scenario. In this work, we consider a

simple scenario to witness quantum steering, consisting

arXiv:2210.14099v2 [quant-ph] 5 Jun 2023

FIG. 1. Quantum steering scenario. Alice and Bob are spatially

separated and each of them receive a subsystem on which

they perform three three-outcome measurements. They are

not allowed to communicate during the experiment. Once it

is complete, they construct the joint probability distribution

{p(a,b|x,y)}.

of two spatially separated parties namely, Alice and Bob.

They locally perform measurements on their respective

subsystems which they receive from a preparation de-

vice. Bob can choose among three measurements de-

noted by Bysuch that y=0, 1, 2 each of which results in

three outcomes labelled by b=0, 1, 2. The measurement

performed by Bob might affect the received subsystem

with Alice which is denoted as σy

b∈ HAwhere σy

bare

positive semi-deﬁnite operators. The collection of these

operators σ={σy

bs.t.b=0, 1, 2, y=0, 1, 2}is called an

assemblage.

In quantum theory the operators σy

bare expressed for

any y,bas

σy

b=TrBh(A⊗Nb

y)ρABi(2)

where ρAB ∈ HA⊗HBis the state shared between Alice

and Bob and By={Nb

y}denote Bob’s measurements.

Alice is trusted here, which means that her measure-

ments are known or she can perform tomography on her

subsystem. If the shared state is not steerable, then the

assemblage has a local hidden state (LHS) model [19]

deﬁned as,

σy

b=X

p(λ)pλ(b|y)ρλ(3)

where Pλp(λ) = 1, pλ(b|y)are the probability distri-

butions over λand ρλare density matrices over HA. As

Alice can perform topographically complete measure-

ments on σb

y, in general, quantum steering is witnessed

by so-called “steering functional", a map from the as-

semblage {σb

y}to a real number [32].

Instead of checking the steerability of the assemblage,

quantum steering can be equivalently witnessed similar

to a Bell scenario where trusted Alice and untrusted Bob

performs the measurements Ax={Ma

x}and By={Nb

respectively and obtain the joint probability distribution

⃗

p={p(a,b|x,y)}where a,b,x,y=0, 1, 2. Here a,xde-

notes the output and input of Alice respectively. The

probabilities can be computed in quantum theory as,

p(a,b|x,y) = Tr h(Ma

x⊗Nb

y)ρABi=Tr Ma

xσy

b(4)

To witness quantum steering, a steering inequality Bcan

now be constructed from ⃗

pas

B(⃗

p) = X

a,b,x,y

ca,b|x,yp(a,b|x,y)≤βL(5)

where ca,b|x,yare real coefﬁcients and βLdenotes the

maximum value attainable using assemblages admit-

ting an LHS model (3). The probabilities one obtains

from such assemblages are expressed using (3) as

p(a,b|x,y) = X

p(λ)p(a|x,ρλ)p(a|x,λ). (6)

The above representation (6) will be particularly useful

to ﬁnd the LHS bound of the steering inequality pro-

posed in this work. In the DI framework, the above-

presented scenario is also referred to as the 1SDI sce-

nario [see Fig-1].

Self-testing. Inspired by [27], we now deﬁne self-

testing in the 1SDI scenario.

Deﬁnition 1. Consider the above 1SDI scenario with the

preparation device creating a state |ψ⟩AB. Alice and Bob per-

form measurements on this state and observe the joint prob-

ability distribution {p(a,b|x,y)}. Alice is trusted and her

measurements Axare ﬁxed and Bob’s measurements are rep-

resented as By={Nb

y}are arbitrary. Let us now con-

sider that the distribution {p(a,b|x,y)}is generated by an

ideal experiment with a state |˜

ψ⟩AB and Bob’s measurements

By={˜

y}. Then, the state |ψ⟩AB and measurements

Byare certiﬁed from {p(a,b|x,y)}if there exists a unitary

UB:HB→ HBsuch that

(A⊗UB)|ψ⟩AB =|˜

ψ⟩AB , (7)

and,

UBΠBNb

yΠBU†

B=˜

y, (8)

where ΠBis the projection onto the support of the local sup-

port ρB=TrA(|ψ⟩⟨ψ|AB).

Let us now proceed towards the results of this work.

Results.— We begin by constructing a steering in-

equality stated using the joint probability distribution ⃗

a,b,x=0

p(a,b̸=a|x,y=x)≤βL. (9)

Using the fact that Pa,bp(a,b|x,y) = 1 for all x,y, we

can simplify the above steering inequality as

W=3−

a,x=0

p(a,a|x,x)≤βL. (10)

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Certificationofthemaximallyentangledstateusingnon-projectivemeasurementsShubhayanSarkar1,∗1CenterforTheoreticalPhysics,PolishAcademyofSciences,AlejaLotników32/46,02-668Warsaw,PolandInrecenttimes,device-independentcertificationofquantumstateshasbeenoneoftheinten-sivelystudiedareasinquantuminformation...

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Certification of the maximally entangled state using non-projective measurements Shubhayan Sarkar1 1Center for Theoretical Physics Polish Academy of Sciences Aleja Lotników 3246 02-668 Warsaw Poland

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