Robust one-sided self-testing of two-qubit states via quantum steering Yukun Wang1 2Xinjian Liu1Shaoxuan Wang1Haoying Zhang1and Yunguang Han3 1Beijing Key Laboratory of Petroleum Data Mining
2025-05-03
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Robust one-sided self-testing of two-qubit states via quantum steering
Yukun Wang,1, 2 Xinjian Liu,1Shaoxuan Wang,1Haoying Zhang,1and Yunguang Han3, ∗
1Beijing Key Laboratory of Petroleum Data Mining,
China University of Petroleum, Beijing 102249, China
2State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China
3College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
(Dated: October 24, 2022)
Entangled two-qubit states are the core building blocks for constructing quantum communica-
tion networks. Their accurate verification is crucial to the functioning of the networks, especially
for untrusted networks. In this work we study the self-testing of two-qubit entangled states via
steering inequalities, with robustness analysis against noise. More precisely, steering inequalities are
constructed from the tilted Clauser-Horne-Shimony-Holt inequality and its general form, to verify
the general two-qubit entangled states. The study provides a good robustness bound, using both
local extraction map and numerical semidefinite-programming methods. In particular, optimal lo-
cal extraction maps are constructed in the analytical method, which yields the theoretical optimal
robustness bound. To further improve the robustness of one-sided self-testing, we propose a family
of three measurement settings steering inequalities. The result shows that three-setting steering in-
equality demonstrates an advantage over two-setting steering inequality on robust self-testing with
noise. Moreover, to construct a practical verification protocol, we clarify the sample efficiency of
our protocols in the one-sided device-independent scenario.
Usage: Secondary publications and information retrieval purposes.
I. INTRODUCTION
Quantum entangled states is the key resource of quan-
tum information technologies, such as quantum networks
[1], cryptography [2], computation [3], and metrology [4].
As we advance towards the second quantum revolution
[5], the characterization and certification of quantum de-
vices becomes an extremely important topic in the prac-
tical applications of quantum technologies [6,7].
To ensure the proper functioning of a quantum net-
work, it is essential to certify the entangled state de-
ployed in the network accurately and efficiently. Besides
the traditional quantum state tomography method, var-
ious methods have been proposed to improve the effi-
ciency and apply to different scenarios, such as direct
fidelity estimation [8], compressed sensing tomography
[9], and shadow tomography [10]. In the last few years,
quantum state verification (QSV) has attracted much at-
tention by achieving remarkably low sample efficiency
[11,12]. One drawback of quantum state verification
method is that it requires the perfect characterization
of the measurements performed by the quantum devices,
thus it is device dependent and not applicable to the un-
trusted quantum network. Self-testing [13,14] is a promi-
nent candidate of quantum state certification in device-
independent (DI) scenario, in which all quantum devices
are treated as black-boxes. Taking the advantage of Bell
nonlocality [15], many important results on self-testing
have been achieved, such as self-testing various quantum
entangled states [16–18], self-testing entangled quantum
measurement [19,20], and parallel self-testing [21,22].
∗hanyunguang@nuaa.edu.cn
Self-testing has wide applications in device-independent
quantum information tasks, such as device-independent
quantum random number generation [23,24], and quan-
tum key distribution [25,26].
Lying between standard QSV and self-testing, there
is semi-device-independent (SDI) scenario [27] in which
some parties are honest, while some others may be dis-
honest. The certification in this scenario can be called as
SDI self-testing or SDI state verification. This scenario
has wide applications in quantum information process-
ing, such as one-sided device-independent (1SDI) quan-
tum key distribution [28], quantum random number gen-
eration [29], verifiable quantum computation [30], and
anonymous communication [31–33]. Meanwhile the certi-
fication in the SDI scenario is closely related to the foun-
dational studies on quantum steering in the untrusted
quantum networks [34–37]. However, not much is known
about the quantum certification in the SDI scenario de-
spite its significance. In [30,38], the authors studied the
one-sided self-testing of maximally entangled two-qubit
state based on 2-setting quantum steering inequality. In
[39], the authors proposed various verification protocols
for Bell state based on multiple settings. For nonmaximal
entangled two-qubit states, the authors in [40] realized
the one-sided certification by combining fine-grained in-
equality [41] and analog CHSH inequalities [42], which is
more complicated compared with traditional self-testing.
In [27], the authors proposed tilted steering inequality
analogous to tilted-CHSH inequality [43] for one-sided
self-testing of two-qubit states. Then they generalized
the one-sided certification to general pure bipartite states
by adopting the subspace method in DI scenario [18]. In
Ref. [44], a class of steering inequalities concentrating
on the nonmaximal entangled bipartite-qudit state were
constructed, where they achieve the bipartite-qudit state
arXiv:2210.11243v2 [quant-ph] 21 Oct 2022
2
self-testing by performing only two measurements. While
in Ref. [45], steering inequalities with d+ 1 measure-
ment settings are used for self-testing the same states.
However, the robustness analysis there follows the norm
inequalities method in [16,38] (if it’s not missed), thus
the result is quite weak. For the multipartite case, the
studies of SDI certification are mainly focused on Green-
berger–Horne–Zeilinger (GHZ) states as the generaliza-
tion of Bell state [39,46,47].
In this paper, we focus on the robust one-sided self-
testing of two-qubit entangled states. We construct two
types of 2-setting steering inequalities for general two-
qubit entangled states based on tilted-CHSH inequality
and its general form. For the first type, analytical and op-
timal robustness bound is obtained using the local extrac-
tion channel method introduced in [48]. For the second
type, we get nearly linear robustness bound using numer-
ical method based on the swap trick [17] and semidefinite
programming (SDP). To put our work in perspective, we
compare the robustness result in the 1SDI scenario with
both DI and device-dependent scenario. Our result can
be applied to the certification of high dimensional quan-
tum devices as building blocks.
Furthermore, we construct three measurement settings
steering inequalities for general two-qubit states, which
is beyond the conventional one-sided self-testing based
on two settings. In [39], the authors studied the opti-
mal verification of Bell state and GHZ states in the 1SDI
scenario using multiple measurement settings. However,
their study is limited to the maximal entangled state in
bipartite case. Based on the 3-settings steering inequali-
ties, it is shown that the robustness bound can be further
improved. This opens the question that how much the
resistance to noise can be improved using multiple mea-
surement settings. Finally, to construct a practical veri-
fication protocol, we clarify the sample efficiency for our
protocols in the 1SDI scenario. It is shown that approx-
imately optimal sample efficiency can be obtained based
on the steering inequalities we constructed.
II. PRELIMINARY
A. Steering scenario and steering inequalities
Let us start by recalling the steering theory. Two dis-
tant parties, Alice and Bob, are considered, and between
them are many copies of state ρAB ∈HANHB. Bob
performs two measurements labeled by y, on his parti-
cle and obtains the binary outcome b. Meanwhile, Al-
ice receives the corresponding unnormalized conditional
states ρb|yand performs measurements randomly, labeled
by x, and obtains the binary outcome a. If Alice cannot
explain the assemblage of received states by assuming
pre-existing states at her location and some pre-shared
random numbers with Bob, she has to believe that Bob
has the steerability of her particle from a distance. To
determine whether Bob has steerability of her, Alice asks
Bob to run the experiment many times with her. Finally,
they obtain the measurement statistics. If the statistics
admit the description,
p(a, b|x, y;ρAB ) = X
λ
p(λ)p(a|x, ρλ)p(b|y, λ),(1)
then Alice knows Bob has not the steerability of her.
This non-steerable correlation models is the so called lo-
cal hidden variable (LHV)-LHS model [42]. The LHV-
LHS decomposition is based on the idea that Bob’s out-
comes are determined by a local hidden random λand
Alice’s outcomes are determined by local measurements
on quantum state ρλ.
The combination of the statistics will give a steering
inequality, where the LHV-LHS model can be used to es-
tablish local bounds for the steering inequality; violation
of such inequalities implies steering. In Ref. [36], the au-
thors introduced a family of steering inequalities for Bell
state
Sn≡1
n
n
X
k=1hˆσA
kBki ≤ Cn,(2)
Cnis the LHS bound
Cn= max
{Ak}(λmax 1
n
n
X
k=1
ˆσA
kBk!),(3)
where λmax(ˆ
O) denotes the largest eigenvalue of ˆ
O.
An approach to constructing this family of steering
inequalities is transforming from Bell inequalities. Bell
states are shown to maximally violate analog CHSH in-
equality [30,38,42]. For partial entangled two-qubit
states, the authors in Ref. [27] constructed tilted steer-
ing inequalities from tilted-CHSH inequalities [43]. In
this paper, we study the more general tilted steering
inequalities construction from tilted-CHSH inequalities
and study the robustness of one-sided self-testing based
on analog steering inequalities. Furthermore, we consider
to construct three measurement settings steering inequal-
ities for general two-qubit states.
B. SDI certification and local extraction channel
In this paper, we focus on one-sided self-testing two-
qubit entangled state based on the steering inequalities.
To this end, we first review the concept of self-testing.
Self-testing was originally known as a DI state verifi-
cation, where some observed statistics p(a, b|x, y) from
quantum devices can determine uniquely the underlying
quantum state and the measurements, up to a local isom-
etry. As an example, the maximal violation of CHSH
inequality uniquely identifies the maximally entangled
two-qubit state [14,16]. Usually, self- testing relies on
the observed extremal correlations, if the quantum sys-
tems that achieve the extremal correlations are unique
3
up to local isometries, we say the extremal correlations
p(a, b|x, y) self- test the target system {|¯
ψi,¯
Ma|x,¯
Nb|y}.
Denoting the local isometry as Φ = ΦAA0⊗ΦBB0, self-
testing can be formally defined as
Φ|ψiAB|00iA0B0=|junkiAB|¯
ψiA0B0
ΦMa|x⊗Nb|y|ψiAB|00iA0B0=|junkiAB ¯
Ma|x⊗¯
Nb|y|¯
ψiA0B0
(4)
Coming to the 1SDI scenario, only the existence of an
isometry ΦBon Bob’s side is required
Φ|ψiAB|0iB0=|junkiB⊗ | ¯
ψiAB0
ΦMb|y|ψiAB|0iB0=|junkiB⊗¯
Mb|y|¯
ψiAB0
(5)
where Mb|yacts on HB;¯
Mb|yacts on HB0.
In addition to the above ideal definition of self-testing,
it is essential to study the robustness of self-testing in
the imperfect case when the obtained data deviate from
the ideal value. There are two frameworks in the robust-
ness analysis of self-testing. The first approach is based
on the swap method by introducing an ancilla system.
The desired state can be swapped out of the real quan-
tum systemthen it could be calculated how far is it from
the target state. One way to calculate this closeness is
based on the analytic method involving mathematical in-
equalities techniques first proposed in [16]. The second
one is the numerical method based on semidefinite pro-
gramming combining NPA hierarchy [49]. Usually, the
numerical method gives much higher robustness.
The second approach is based on operator inequalities
first introduced in Ref. [48], which is now widely used
in the robustness analysis of self-testing. For self-testing
Bell state using CHSH inequality and self-testing GHZ
state using Mermin inequality, the operator inequalities
give nearly optimal bound. Robustness analysis of self-
testing with operator inequalities can recur to local ex-
traction map, which hinges on the idea that local mea-
surements can be used to virtually construct local ex-
traction channel to extract the desired state from the
real quantum system. The local extractability of target
ψAB from ρAB is quantified
Ξ(ρAB →ψAB) := max
ΛA,ΛB
F((ΛA⊗ΛB)(ρAB), ψAB),(6)
where the maximum is taken over all possible local chan-
nels constructed with local measurements. For the 1SDI
scenario, Alice’s side is trusted, thus the extraction chan-
nel in Alice’s side is ΛA=IA. The lower bound of
the fidelity between ρand the target state under the
observed steering inequality can be defined as one-sided
extractability
F(ρAB, ψAB) := inf
ρAB :S(ρ)≥Sobs
max
ΛB
F(ΛB(ρAB), ψAB),
(7)
where S(·) is the steering expression and Sobs is observed
violation. To derive a linear bound of the fidelity about
observed steering inequality violation, real parameters s
and τare required to be fixed such that F≥s·Sobs +τ.
This is equivalent to find ΛB(constructed by Bob’s local
measurement operators Mb
y) to make
K≥sS +τI(8)
where K:= (IA⊗Λ+
B)(ψAB) and Λ+refers to the dual
channel of quantum channel Λ. By taking the trace with
the input state ρAB on both sides of Eq. (8), one can
get F≥s·Sobs +τ, in view of hΛ+
B(ψAB), ρABi=
hψAB,ΛB(ρAB)i.
In the 1SDI scenario, Bob’s side is untrusted, thus Eq.
(8) is required to hold for Alice in two dimension and Bob
in arbitrary dimension. Since the measurements we con-
sidered in this paper is dichotomic, considering in qubit
space will be sufficient in Bob’s side.
III. ONE-SIDED SELF-TESTING BASED ON
2-SETTING STEERING INEQUALITIES
In device-independent scenario, general pure entangled
two-qubit state
|Φi= cos θ|00i+ sin θ|11i,(9)
has been proved to be self-tested [50,51] by the maximal
violation of tilted-CHSH inequalities [43], which can be
parametrized as
ˆ
Iα=αA0+A0B0+A0B1+A1B0−A1B1≤α+ 2,
(10)
where sin 2θ=q4−α2
4+α2. The maximum quantum value is
√8+2α2. The quantum measurements used to achieve
the maximal quantum violation are: {σz;σx}for Alcie,
and {cos µσz+ sin µσx; cos µσz−sin µσx}for Bob, where
tan µ= sin 2θand σx,z are Pauli X, Z measurements.
When α= 0, it corresponds to CHSH inequality and
the state can be self-tested as Bell state. The self-testing
criteria based on this tilted-CHSH inequalities is robust
against noise. The best robustness bound to date can
be found in [48,51], in which the authors introduced the
local extraction channel method. However, as claimed in
[48], the theoretical optimal upper bound is not achiev-
able. Theoretically, the optimal bound is tied to the max-
imum classical violation which starts to achieve nontrivial
fidelity. The nontrivial fidelity that demonstrates entan-
glement for the target state is F > cos2θ. They guessed
that it might be related to the fact that the quantum
value of the CHSH inequality does not reach its alge-
braic limit of 4. Here in 1SDI scenario, we will show that
the theoretical optimal bound can be achieved.
To achieve 1SDI self-testing criteria, we will construct
two types of 2-setting steering inequalities, which are
based on above tilted-CHSH inequality by taking the
measurements on Alice’s side as trusted.
4
A. One-sided self-testing based on standard
tilted-CHSH steering inequality
Taking the measurements on Alice’s side as trusted,
the standard tilted-CHSH inequality in Eq. (10) can be
transformed to the analog of tilted-CHSH steering in-
equality
ˆ
Sα=αA0+A0B0+A0B1+A1B0−A1B1
=αZ +Z(B0+B1) + X(B0−B1)
≤α+ 2,(11)
which maintains the maximum quantum violation SQ
α=
√8+2α2as in DI scenario. We prove that partial entan-
gled two-qubit states can be self-tested using this analog
tilted-CHSH steering inequality in 1SDI manner. The
proof is similar to DI self-testing using tilted-CHSH in-
equality, except that we can trust Alice’s measurements
now. The trustworthy of Alice’s side can simplify the
proof as an advantage. Another advantage is that theo-
retical optimal robustness bound can be obtained in 1SDI
scenario with this steering inequality. By contrast, the
optimal bound can not be achieved in DI self-testing with
tilted-CHSH inequality. In the following, we will show
both the analytical proof and the robustness analysis.
a. self-testing based on analog tilted-CHSH steering
inequality We provide the simple proof here. Though
Alice’s side are trustworthy, as definition only the exis-
tence of isometry in Bob’s side will efficient to determine
uniquely the state and the measurements. However, for
simplicity, we also introduce one isometry in Alice’s side,
which has been widely used in DI scenario, shown in Fig.
1. As shown in bellow, with sum of squares decomposi-
tion of positive semidefinite matrix [52], it’s easy to find
the algebraic relations that are necessarily satisfied by
target quantum state and measurements to complete the
proof.
After the isometry, the systems will be
Φ(|ψi) = 1
4[(I+ZA)(I+˜
ZB)|ψi|00i
+XA(I+ZA)(I−˜
ZB)|ψi|01i
+˜
XB(I−ZA)(I+˜
ZB)|ψi|10i
+XA˜
XB(I−ZA)(I−˜
ZB)|ψi|11i] (12)
To derive the underlying state |ψiis equivalent to the
target one, the algebraic relations between the operator
acting on the state should be given. We notice that
the analog tilted-CHSH steering inequality ˆ
Sαhave the
maximum quantum value SQ
α. This implies that the op-
erator b
Sα:= SQ
αI−ˆ
Sαshould be positive semidefinite
(PSD) for all possible quantum states and measurement
operators in Bob’s side. This can be proven by providing
a set of operators {Pi}which are polynomial functions
of Ax(ZA, XA) and Bysuch that b
Sα=PiP†
iPi, holds
for any set of measurement operators satisfying the al-
gebraic properties A2
x=I,B2
y=I. The decomposition
form of b
Sα=PiP†
iPiis called a sum of squares(SOS).
By SOS decomposition one can provide a direct certifi-
cate that the upper quantum bound of ˆ
Sαis SQ
αfrom its
PSD, as well as some relations between the projectors on
the states, which will be used to give self-testing state-
ment. This method was first introduced in [50] for the
family of CHSH-liked Bell inequalities. Given SOS de-
compostions, if one observes the maximal quantum vio-
lation of the steering inequality (CHSH-liked one) under
state |ψi, then each squared terms in SOS decomposi-
tions acting on |ψishould be zero, i.e., Pi|ψi= 0. Then
useful relations for the measurements operators acting on
underlying state can be obtained from these zero terms.
Similar to CHSH inequality scenario, two types of SOS
decompositions for analog tilted-CHSH operator in Eq.
(11) can be given. The first one is
b
Sα=1
2SQ
α{b
S2
α+ (αXA−S0)2}(13)
And the second one is
b
Sα=1
2SQ
α(2ZA− SQ
α
B0+B1
2+α
2S1)2
+ (2XA− SQ
α
B0−B1
2+α
2S2)2(14)
where
S0=ZA(B0−B1) + XA(B0+B1),
S1=ZA(B0+B1)−XA(B0−B1),(15)
S2=ZA(B0−B1)−XA(B0+B1).
Based on the maximal violation of analog tilted-CHSH
inequality, the existence of the SOS decomposition for b
Sα
implies :
ZA|ψi − ˜
ZB|ψi= 0,(16)
sin(θ)XA(I+˜
ZB)|ψi − cos(θ)˜
XB(I−ZA)|ψi= 0 (17)
where ˜
ZB:= B0+B1
2 cos µ, and ˜
XB:= B0−B1
2 sin µ. Then with the
algebraic relation of (16)-(17) and the fact that ZAXA=
−XAZA, the equation in Eq. (12) can be rewritten to
Φ(|ψi) = |junki[cos θ|00i+ sin θ|11i]
where |junki=1
2 cos θ(I+ZA)|ψi. This means the un-
derlying state are unique to the target one up to local
isometries, thus completes the self-testing statement.
b. self-testing robustness Here we mainly focus on
the self-testing of quantum states. For the self-testing of
quantum measurements, the analysis can be related to
quantum states according to Ref. [17]. The procedure
is similar, starting with ΦMB(|ψi) instead of Φ(|ψi). In
this case, the figure of merit should quantify how MB|ψi
is close to the ideal measurements acting on the target
state.
As introduced in Sec. II B, to obtain the better self-
testing robustness bound for the state, we should find the
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Robustone-sidedself-testingoftwo-qubitstatesviaquantumsteeringYukunWang,1,2XinjianLiu,1ShaoxuanWang,1HaoyingZhang,1andYunguangHan3,1BeijingKeyLaboratoryofPetroleumDataMining,ChinaUniversityofPetroleum,Beijing102249,China2StateKeyLaboratoryofCryptology,P.O.Box5159,Beijing,100878,China3CollegeofCompu...
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