Bell Inequalities Induced by Pseudo Pauli Operators on Single Logical Qubits Weidong Tang1 1School of Mathematics and Statistics Shaanxi Normal University Xian 710119 China

2025-05-06 0 0 598.17KB 12 页 10玖币

侵权投诉

Bell Inequalities Induced by Pseudo Pauli Operators on Single Logical Qubits

Weidong Tang1, ∗

1School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China

In most Bell tests, the measurement settings are specially chosen so that the maximal quantum

violations of the Bell inequalities can be detected, or at least, the violations are strong enough

to be observed. Such choices can usually associate the corresponding Bell operators to a kind of

eﬀective observables, called pseudo Pauli operators, providing us a more intuitive understanding of

Bell nonlocality in some sense. Based on that, a more general quantum-to-classical approach for

the constructions of Bell inequalities is developed. Using this approach, one can not only derive

several kinds of well-known Bell inequalities, but also explore many new ones. Besides, we show

that some quadratic Bell inequalities can be induced from the uncertainty relations of pseudo Pauli

operators as well, which may shed new light on the study of uncertainty relations of some nonlocal

observables.

Some entangled states shared by distant observers

may exhibit a counterintuitive feature called the Bell

nonlocality, i.e., correlations produced by measurements

on space-like separated subsystems cannot be simulated

by any local hidden variable (LHV) model. This non-

classical feature can usually be shown by the violation

of a Bell inequality. So far, various Bell inequalities

have been proposed, including the original version[1], the

Clauser-Horne-Shimony-Holt (CHSH) inequality[2], the

Mermin inequality[3], the Collins-Gisin-Linden-Massar-

Popescu inequality[4] and etc. These Bell inequalities are

widely used in many quantum information tasks, such as

quantum games[5–7], device-independent quantum key

distribution[8–10], random number generation[11,12],

and self-testing[13–16].

From an overall perspective, there are mainly two ways

to construct Bell inequalities. The ﬁrst one is known as

the classical-to-quantum approach (or the local polytope

approach[17–20]), which is also regarded as the standard

construction of Bell inequalities. It exploits a mathe-

matical tool called the (local) polytope, which is the

convex hull of a ﬁnite number of vertices, to represent

the set of correlations admitting a local hidden variable

model, where the vertices correspond to local determin-

istic assignments (using 1 and 0 to describe the cor-

responding deterministic behaviors)[20], and the facets

can deﬁne a ﬁnite set of linear inequalities which are

called facet (or tight) Bell inequalities. A facet inequal-

ity is nontrivial if it can be violated on some quantum

systems. Finding such facet inequalities is the central

task in this approach. The constructions of the Collins-

Gisin-Linden-Massar-Popescu inequality[4], the Frois-

sard inequality[17], and the Collins-Gisin inequality[21]

are all of this type. The main shortcoming of this ap-

proach is the lack of eﬃciency, especially in the scenarios

involving a large number of parties (or measurements per

party, or outcomes for each measurement). By contrast,

the second way, also known as the quantum-to-classical

approach[16,22,23], seems to be more practical in con-

structing scalable Bell inequalities. The key of this ap-

proach relies on how to properly exploit quantum prop-

erties of the states with special symmetries. Therefore,

the involved states are usually chosen from the stabilizer

states, and their stabilizers would be invoked directly[24–

26] or indirectly[16,22] in the constructions of Bell in-

equalities.

Although the violation of the Bell inequality can be

used for revealing some quantum features such as the

nonlocality and entanglement, the inequality itself looks

more like a kind of mathematical tool since it is essen-

tially a constraint of classical correlations. The explo-

rations on more explicit physical signiﬁcance for the Bell

inequality are still very rare so far. In view of this, one

motivation of this work is to give a new physical expla-

nation for some (e.g. the maximal) quantum violations

of certain Bell inequalities. To realize that, a natural

thought is to look for some special connections between

the Bell operators and certain physical quantities. Once

such connections are found, they might shed new light on

the constructions of Bell inequalities. Then, to explore a

new approach for the constructions of Bell inequalities is

another motivation of this work.

To start, let us consider the two-qubit scenario since it

is a stepping stone to many multi-qubit generalizations,

and in which the typical Bell inequality is the CHSH

inequality. Below we will show how to associate the cor-

responding Bell operator to a physical quantity, and con-

versely, how to derive the CHSH inequality from such a

physical quantity.

Denote by X, Y, Z the Pauli matrices σx, σy, σz, and let

|0i,|1ibe the eigenstates of Zwith eigenvalues +1,−1

respectively. Consider a two-dimensional Hilbert space

spanned by two Bell states

|˜

0i=1

√2(|00i+|11i),|˜

1i=1

√2(|01i−|10i).(1)

Recall that a logical qubit can be encoded into several

physical qubits. In view of this, the state

|˜

ψi=α|˜

0i+β|˜

1i(|α|2+|β|2= 1),(2)

can be regarded as a special logical qubit, and {|˜

0i,|˜

1i}

is also called a logical qubit basis.

arXiv:2210.10977v1 [quant-ph] 20 Oct 2022

Besides, one can deﬁne three pseudo Pauli operators

on such a logical qubit as follows:

Z=|˜

0ih˜

0|−|˜

1ih˜

1|=1

2(Z1Z2+X1X2),

X=|˜

0ih˜

1|+|˜

1ih˜

0|=1

2(Z1X2−X1Z2),

Y=i(|˜

1ih˜

0|−|˜

0ih˜

1|) = 1

2(Y2−Y1).(3)

Clearly, their eigenvalues are all ±1, the same as those

of common Pauli operators. Apart from this deﬁnition,

Appendix A also shows us several examples based on

other logical qubit bases.

On the other hand, it is known that the CHSH inequal-

ity can be written as

|hBCHSHic|=|hA1B2+A1B0

2+A0

1B2−A0

1B0

2ic| ≤ 2,

(4)

where h icdenotes the classical expectation (by the LHV

model), and

BCHSH ≡A1B2+A1B0

2+A0

1B2−A0

1B0

2.(5)

is the corresponding Bell operator. Here A1, A0

1, B2, B0

are dichotomic observables measured by two distant ob-

servers. As is known, its quantum expectation satis-

ﬁes hBCHSHi ≤ 2√2, i.e., the maximal quantum viola-

tion of the CHSH inequality is 2√2. This bound can

only be attained by the maximally entangled states.

For example, a common choice is the Bell state |˜

0i=

(|00i+|11i)/√2. Accordingly, the observables to be mea-

sured are A1=X1, A0

1=Z1, B2= (X2+Z2)/√2 and

2= (X2−Z2)/√2, i.e., the Bell operator to be tested

in the real experiment (will referred to as experimental

Bell operator hereinafter) is

CHSH =X1

X2+Z2

√2+X1

X2−Z2

√2

+Z1

X2+Z2

√2−Z1

X2−Z2

√2.(6)

Note that the observables to be measured on each ob-

server’s side are a pair of complementary observables (e.g.

Xand Z), which is the price of testing the maximal vi-

olation of the CHSH inequality.

Rewrite Be

CHSH in terms of a combination of two sta-

bilizers of (|00i+|11i)/√2, i.e.,

CHSH =√2(X1X2+Z1Z2)≡ Be

CHSH(S),(7)

where X1X2and Z1Z2are the stabilizers. Besides, ac-

cording to Eq.(3) and Eq.(7), Be

CHSH can be further rep-

resented as

CHSH = 2√2˜

Z≡ Be

CHSH(L),(8)

which corresponds to an explicit physical quantity on a

single logical qubit! This quantity can be considered as

the z-component of an pseudo spin (up to a constant).

In fact, choosing other maximally entangled states and

proper measurement settings will induce similar corre-

spondences as well.

On the other hand, reversing the above discussion, i.e.,

CHSH(L)→ Be

CHSH(S)→ Be

CHSH → BCHSH, one can

construct a CHSH inequality. Note that the expression

of Be

CHSH(L) (or Be

CHSH(S)) indicates that the quantum

upper bound of Bis at least 2√2. The bound derived

directly from Be

CHSH(L) is a rough estimation for the

maximal quantum violation (will be referred to as rough

quantum upper bound below) of the ﬁnal Bell inequal-

ity. To show that 2√2 is also the exact quantum upper

bound, one can use a sum of square method[15,16,23],

see Appendix B. By contrast, the classical upper bound

can be derived by invoking the (classical) inequality

|x+y+z−xyz| ≤ 2 (−1≤x, y, z ≤1). Since 2 <2√2,

such a construction is successful.

Inspired by that, we can propose a systematic

(quantum-to-classical) approach to construct Bell in-

equalities. It can be simply described by the sequence

Be(L)→ Be(S)→ Be→ B,(9)

and an additional comparison for the classical upper

bound and the quantum one of B. As mentioned above,

a rough quantum upper bound can be given directly by

Be(L). In the demonstration of Bell nonlocality, if the

gap between this bound and the classical one is large

enough, one do not really have to calculate the exact

quantum upper bound (except for some speciﬁc tasks).

Fortunately, for many famous Bell inequalities, their ex-

act quantum upper bounds can be calculated by a sum

of square method, see Appendix B. If the classical upper

bound is less than the (rough or exact) quantum one, a

desired Bell inequality is constructed; otherwise, choose

another Be(L) and repeat the above process. Note that

from the matrix perspective, Be(L) = Be(S) = Be, but

their physical signiﬁcance may be diﬀerent.

We prefer to choose two fully entangled stabilizer states

(such as Bell states and some graph states) as logical

qubit bases in constructing Bell inequalities, since one

can easily detect quantum violations for many celebrated

Bell inequalities in such states, and besides, the prepa-

rations of stabilizer states are more attainable than non-

stabilizer ones by current experiments.

Usually we can choose Be(L) = βq~

k·~

˜σ, where βqis

a constant (rough quantum upper bound), ~

kis a unit

vector and ~

˜σ≡(˜

X, ˜

Y , ˜

Z). Besides, the expressions of

X, ˜

Y , ˜

Zinvoked in Be(S) rely on the choice of the log-

ical qubit basis, and each of them contains 2n−1terms

(since the bases are fully entangled stabilizer states, see

Appendix B), where nis the number of physical qubits

in each basis. Note that if Be(L) is properly chosen, Be

and Be(S) might take the same form. Apart from that,

another possible form of Becould be given by decom-

posing each term of Be(S) into a pair of complementary

observables (e.g from Eq.(7) to Eq.(6)). Since the alge-

braic structures of Beand Bare the same, one can easily

get the latter by applying a suitable replacement of op-

erators.

Example 1. — The CHSH inequality, the three-qubit

Mermin inequality and the three-qubit Svetlichny in-

equality. According to Eq.(9), to construct any of them,

we need to choose a suitable logical qubit basis and a

proper Be(L), which are listed in Table I, also see Ap-

pendix C for more detailed discussions. As mentioned

above, their maximal quantum violations hBimax can be

derived by the sum of square method. 

Example 2. — Graphical Bell inequalities induced

from the quantum error-correcting code [[5,1,3]][27]. Let

|˜

0i=|L5iand |˜

1i=Z0Z1Z2Z3Z4|L5ibe the bases,

which are also the bases for the coding space, where

|L5iis a 5-vertex loop graph state, which is stabilized

by gi⊕1=ZiXi⊕1Zi⊕2(i= 0,1,··· ,4), and ⊕stands

for addition modulo 5. Note that |L5ihL5|=Q4

i=0

I+gi

Then the pseudo Pauli operators can be written as

Z=1

16[

i=0

(ZiXi⊕1Zi⊕2−ZiYi⊕1Xi⊕2Yi⊕3Zi⊕4

+XiYi⊕2Yi⊕3)−X0X1X2X3X4],

X=1

16[

i=0

(−YiZi⊕1Yi⊕2+YiXi⊕1Zi⊕2Xi⊕3Yi⊕4

−ZiXi⊕2Xi⊕3) + Z0Z1Z2Z3Z4],

Y=1

16[

i=0

(−XiYi⊕1Xi⊕2+XiZi⊕1Yi⊕2Zi⊕3Xi⊕4

−YiZi⊕2Zi⊕3) + Y0Y1Y2Y3Y4].

(10)

Detailed calculations are shown in Appendix D. There

are three typical choices for Bell operators,

1(L) = 16 ˜

Z;Be

2(L) = 16√2˜

Z+˜

√2;

3(L) =16√3˜

Z+˜

X+˜

√3.(11)

They can induce three Bell inequalities: the Mermin-like

inequality, the Svetlichny-like inequality, and the Hyper-

Svetlichny-like inequality. Numerical calculations show

that their classical bounds satisfy

|hB1ic| ≤ 8; |hB2ic| ≤ 16; |hB3ic| ≤ 24.(12)

Here B1,2,3are ﬁnal Bell operators. One can get them

by substituting the expressions of pseudo Pauli opera-

tors into Eq.(11) and replacing Zi, Xi, Yiwith Ai, Bi, Ci

respectively. For example, B1=P4

i=0(AiBi⊕1Ai⊕2−

AiCi⊕1Bi⊕2Ci⊕3Ai⊕4+BiCi⊕2Ci⊕3)−B0B1B2B3B4,

where Ai, Bi, Ci∈[−1,1].

By contrast, the quantum upper bound for the

Mermin-like inequality can be easily derived. But for the

other two, we we only give their rough quantum upper

bounds by exploiting Eq.(11), namely,

hB1imax = 16; hB2imax ≥16√2; hB3imax ≥16√3.

(13)

Clearly, the gap between any above (exact or rough)

quantum upper bound and the corresponding classical

one is large enough for the detection of Bell nonlocality.

This example can be generalized to the (2k+ 1)-qubit

scenario (k≥2) as well. 

Example 3. — The chained (or multi-setting) Bell

inequality[28]. We still adopt the deﬁnition of |˜

0i,|˜

1i,˜

and ˜

Xby Eq.(1) and Eq.(3). Denote Zi(θ) := Zicos θ+

Xisin θ, where i∈ {1,2}and θ∈[0, π). One can rewrite

Zin Eq.(3) into another form:

Z=1

k=1

Z1(k−1

nπ)Z2(k−1

nπ); n≥2.(14)

Besides, ˜

X=˜

Z·(iY2) = 1

nPn

k=1 Z1(k−1

nπ)Z2(n+2k−2

2nπ).

We can choose

C(L) = 2ncos π

2n˜

Z. (15)

Then Be

C(S) = 2 cos π

2nPn

k=1 Z1(k−1

nπ)Z2(k−1

nπ)

and Be

C=Pn

k=1 Z1(k−1

nπ)Z2(2k−1

2nπ) +

Pn−1

k=1 Z1(k

nπ)Z2(2k−1

2nπ)−Z1(0) ⊗Z2(2n−1

2nπ). Here

we have exploited such a relation: for each 2 ≤k≤n,

Zi(k−1

nπ) = [Zi(2k−3

2nπ) + Zi(2k−1

2nπ)]/cos π

2n, and for

k= 1,Zi(0) = [Zi(1

2nπ)−Zi(2n−1

2nπ)]/cos π

2n.

Replacing Z1(k−1

nπ) and Z2(2k−1

2nπ) with Ak

1and Bk

respectively, one can get the ﬁnal Bell operator

BC=

k=1

Ak⊗Bk+

n−1

k=1

Ak+1 ⊗Bk−A1⊗Bn.(16)

Notice that |c1+c2+··· +c2n−1−Q2n−1

i=1 ci| ≤ 2n−2

(∀i∈ {1,2,··· ,2n−1},ci∈[−1,1]). Therefore,

|hBCic| ≤ 2n−2.(17)

According to Eq.(15), its quantum upper bound is at

least 2ncos π

2n, which is also the exact bound. One

can prove that by using the sum of square method, see

Ref.[15]. since 2ncos π

2n>2n[1 −π2/(8n2)] >2n−2,

Eq.(17) is a desired Bell inequality. 

To construct a multi-partite Bell inequality, apart from

a direct choice of stabilizer states as logical qubit bases,

another way is to use a special recursive method, which

is also a generalization of Mermin-Klyshko polynomial

technique[3,29,30]. To show that, ﬁrst we introduce a

linear expansion operation ∗:

(a)∗(α|0i+β|1i) = α∗(|0i) + β∗(|1i) = α|˜

0i+β|˜

1i;

(b)∗(uX +vY +wZ) = u∗(X) + v∗(Y) + w∗(Z)

=u˜

X+v˜

Y+w˜

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

BellInequalitiesInducedbyPseudoPauliOperatorsonSingleLogicalQubitsWeidongTang1,1SchoolofMathematicsandStatistics,ShaanxiNormalUniversity,Xi'an710119,ChinaInmostBelltests,themeasurementsettingsarespeciallychosensothatthemaximalquantumviolationsoftheBellinequalitiescanbedetected,oratleast,theviolatio...

展开>> 收起<<

Bell Inequalities Induced by Pseudo Pauli Operators on Single Logical Qubits Weidong Tang1 1School of Mathematics and Statistics Shaanxi Normal University Xian 710119 China.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Bell Inequalities Induced by Pseudo Pauli Operators on Single Logical Qubits Weidong Tang1 1School of Mathematics and Statistics Shaanxi Normal University Xian 710119 China

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: