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arXiv:quant-ph/0508210v2 25 Jan 2006

Bell inequalities stronger than the CHSH inequality

for 3-level isotropic states

Tsuyoshi Ito,1, ∗Hiroshi Imai,1,2, †and David Avis3, ‡

1Department of Computer Science, University of Tokyo

2ERATO-SORST Quantum Computation and Information Project, Japan Science and Technology Agency

3School of Computer Science, McGill University

(Dated: January 25, 2006)

We show that some two-party Bell inequalities with two-valued observables are stronger than the

CHSH inequality for 3⊗3 isotropic states in the sense that they are violated by some isotropic states

in the 3 ⊗ 3 system that do not violate the CHSH inequality. These Bell inequalities are obtained

by applying triangular elimination to the list of known facet inequalities of the cut polytope on nine

points. This gives a partial solution to an open problem posed by Collins and Gisin. The results

of numerical optimization suggest that they are candidates for being stronger than the I3322 Bell

inequality for 3⊗3 isotropic states. On the other hand, we found no Bell inequalities stronger than

the CHSH inequality for 2⊗2 isotropic states. In addition, we illustrate an inclusion relation among

some Bell inequalities derived by triangular elimination.

I.INTRODUCTION

Bell inequalities and their violation are an important

topic in quantum theory [1, 2]. Pitowsky [3, 4] introduced

convex polytopes called correlation polytopes which rep-

resent the set of possible results of various correlation

experiments. A Bell inequality is an inequality valid for

a certain correlation polytope. The correlation experi-

ments we consider in this paper are those between two

parties, where one party has mA choices of two-valued

measurements and the other party has mBchoices. The

Clauser-Horne-Shimony-Holt inequality [5] is an example

of a Bell inequality in this setting with mA= mB= 2.

Separable states satisfy all Bell inequalities with all

measurements by definition.

Werner disproved the converse: there exists a quantum

mixed state ρ which is entangled but satisfies all Bell in-

equalities. To overcome the difficulty of proving these two

properties of ρ, he investigated states of very high sym-

metry now called Werner states. Collins and Gisin [7]

compared the strengths of Bell inequalities by introduc-

ing a relevance relation between two Bell inequalities, and

they showed that a Bell inequality named I3322 is rele-

vant to the well-known CHSH inequality. Here relevance

means that there is a quantum mixed state ρ such that

ρ satisfies the CHSH inequality (with all measurements)

but ρ violates the I3322 inequality (with some measure-

ments). The state ρ they found has less symmetry than

the Werner states.

A test of relevance is a computationally difficult prob-

lem.For one thing, to test relevance, one must tell

whether a given state satisfies a given Bell inequality for

all measurements or not. This can be cast as a bilinear

semidefinite programming problem, which is a hard opti-

In a seminal paper [6],

∗Electronic address: tsuyoshi@is.s.u-tokyo.ac.jp

†Electronic address: imai@is.s.u-tokyo.ac.jp

‡Electronic address: avis@cs.mcgill.ca

mization problem. The “see-saw iteration” algorithm is

used to solve it in literature [1]. Although it is not guar-

anteed to give the global optimum, multiple runs with

different initial solutions seem sufficient for many cases.

Another difficulty is to choose the appropriate state ρ.

Collins and Gisin overcome this difficulty by restricting

states, which we will describe in Section IIC.

Collins and Gisin showed numerically that the I3322

Bell inequality is not relevant to the CHSH inequality

for 2-level Werner states.

lem [8]: “Find Bell inequalities which are stronger than

the CHSH inequalities in the sense that they are vio-

lated by a wider range of Werner states.” To answer

this problem, we test 89 Bell inequalities for 2- and 3-

level isotropic states by using the see-saw iteration al-

gorithm. Isotropic states are a generalization of 2-level

Werner states in that they are convex combinations of a

pure maximally entangled state and the maximally mixed

state. The high symmetry of the isotropic states allows

us to calculate the maximum violation of the CHSH in-

equality by 3-level isotropic states analytically. The 89

inequalities used in the test are the Bell inequalities that

involve at most five measurements per party in the list

of more than 200,000,000 tight Bell inequalities recently

obtained by Avis, Imai, Ito and Sasaki [9, 10] by us-

ing a method known as triangular elimination. We re-

strict computation to these 89 inequalities because the

optimization problem related to inequalities with many

measurements is difficult to solve. As a result, we find

five inequalities which are relevant to the CHSH inequal-

ity for 3-level isotropic states. They answer Collins and

Gisin’s problem where Werner states are replaced by 3-

level isotropic states. We give empirical evidence that the

five inequalities are also relevant to the I3322inequality.

To the best of our knowledge, no such Bell inequalities

were previously known.

They posed an open prob-

The rest of the paper is organized as follows. Section II

explains the necessary concepts. Section III discusses in-

clusion relation, which is used to prove irrelevance of a

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Bell inequality to another, and gives. the inclusion rela-

tion among the Bell inequalities we used in our experi-

ments. Section IV explains the method and the results

of our experiments to test relevance for 2- and 3-level

isotropic states. Section V concludes the paper and men-

tions some open problems.

II.PRELIMINARIES

A.Bell inequalities

We consider the following correlation experiment. Sup-

pose that two parties called Alice and Bob share a

quantum state ρ.Alice has mA choices A1,...,AmA

of two-valued measurements and Bob has mB choices

B1,...,BmB. We call the two possible outcomes of the

measurements 1 and 0. The result of this correlation ex-

periment can be represented by an (mA+mB+mAmB)-

dimensional vector q, where for 1 ≤ i ≤ mA and

1 ≤ j ≤ mB, the variables qi0, q0j and qij represent

the probability that the outcome of Aiis 1, that the out-

come of Bj is 1, and that two outcomes of both Aiand

Bjare 1, respectively.

An inequality aTq ≤ a0, where a is an (mA +

mB+ mAmB)-dimensional vector and a0 is a scalar, is

called a Bell inequality if it is satisfied for all separable

states ρ and all choices of measurements A1,...,AmA,

B1,...,BmB. The nontrivial Bell inequality with the

smallest values of mAand mBis the CHSH inequality [5]

−q10− q01+ q11+ q21+ q12− q22≤ 0(1)

for mA= mB= 2.

A Bell inequality is said to be tight if it cannot be writ-

ten as a positive sum of two different Bell inequalities.

The CHSH inequality is an example of a tight Bell in-

equality. Tight Bell inequalities are more useful as a test

of the nonlocality than the other Bell inequalities, since

if a state violates a non-tight Bell inequality aTq ≤ a0,

then the same state violates one of tight Bell inequalities

which sum up to aTq ≤ a0.

Throughout this paper, we denote a Bell inequality

aTq ≤ a0by

(A1) ··· (AmA)

a10

···

a11

···

...

a1mB··· amAmB

amA0

(B1)

...

(BmB) a0mB

a01

...

amA1

...

≤ a0,

following the notation by Collins and Gisin used in [7]

(with labels added to indicate which rows and columns

correspond to which measurements). For example, the

CHSH inequality (1) is written as

(A1) (A2)

−1

1

1

0

(B1) −1

(B2)

1

0

−1

≤ 0.

Another Bell inequality found by Pitowsky and

Svozil [11] and named I3322 inequality by Collins and

Gisin [7] is written as

(A1) (A2) (A3)

−1

11

11

1−1

00

(B1) −2

(B2) −1

(B3)

1

−1

0

0

≤ 0.(2)

Recently Avis, Imai, Ito and Sasaki [9, 10] proposed

a method known as triangular elimination that can be

used to generate tight Bell inequalities from known tight

inequalities for a well-studied related polytope, known

as the cut polytope. They obtained a list of more than

200,000,000 tight Bell inequalities by applying triangu-

lar elimination to a list [12] of tight inequalities for the

cut polytope on 9 points, CUT?

equalities which involve five measurements per party in

the list, and they are used in this paper. Among them

are the CHSH inequality, the positive probability (triv-

ial) inequality, the Imm22inequalities for m = 3,4,5, the

I(2)

3422inequality [7] and other unnamed Bell inequalities.

We label the 89 inequalities as A1 to A89. The list of

these inequalities is available online [13].

9. There are 89 Bell in-

B.Violation of a Bell inequality and bilinear

semidefinite programming

A test whether there exists a set of measurements vi-

olating a given Bell inequality in a given state can be

cast as a bilinear semidefinite programming problem as

follows. Let ρ be a density matrix in the d ⊗ d system

and aTq ≤ a0 be a Bell inequality. Each measurement

by Alice is represented by a positive operator valued mea-

sure (POVM) (Ei,I −Ei), where Eiis a Hermitian d×d

matrix such that both Eiand I−Eiare nonnegative def-

inite and I is the identity matrix of size d×d. Similarly,

each measurement by Bob is represented by a POVM

(Fj,I − Fj). For concise notation, we let E0= F0= I.

Then the test whether there exists a set of violating mea-

surements or not can be formulated as:

max

?

0≤i≤mA

0≤j≤mB

(i,j)?=(0,0)

aijtr(ρ(Ei⊗ Fj)) − a0

(3)

where E0= F0= I, Ei

O ? Ei,Fj? I.

T= Ei, Fj

T= Fj,

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Here the notation X ? Y means that Y − X is non-

negative definite. The optimal value of (3) is positive

if and only if there exist violating measurements, and if

so, the optimal solution gives the set of measurements

that is maximally violating the given Bell inequality in

the given state. If we fix one of the two groups of vari-

ables {E1,...,EmA} and {F1,...,FmB}, (3) becomes a

semidefinite programming problem on the other group of

variables. In this respect, (3) can be seen as a variation of

bilinear programming [14] with semidefinite constraints.

The optimization problem (3) is NP-hard, even for the

case d = 1, as follows from results in [15, Sections 5.1,

5.2].

If d = 2 and the inequality aTq ≤ a0 is the CHSH

inequality, then (3) can be solved analytically [16], hence

the Horodecki criterion, a necessary and sufficient con-

dition for a state ρ in the 2 ⊗ 2 system to satisfy the

CHSH inequality for all measurements. However, in gen-

eral, the analytical solution of (3) is not known. This

seems natural, given the difficulty of bilinear program-

ming. Section 2 of [14] describes a hill-climbing algo-

rithm which computes a local optimum by fixing one of

the two groups of variables and solving the subproblem

to optimize variables in the other groups repeatedly, ex-

changing the role of the two groups in turn. “See-saw

iteration” [1] uses the same method combined with the

observation that in the case of (3), each subproblem can

be solved efficiently by just computing the eigenvectors

of a Hermitian d × d matrix.

Thereexistsasetof

E1,...,EmAand F1,...,FmBwhich attains the maxi-

mum of (3). This fact is obtained from the proof of The-

orem 5.4 in [17] by Cleve, Høyer, Toner and Watrous.

Though they prove the case where ρ is also variable,

the relevant part in the proof is true even if the state is

fixed. See-saw iteration always produces projective mea-

surements as a candidate for the optimal measurements.

projectivemeasurements

C.Relevance relation

Collins and Gisin [7] introduced the notion of relevance

between two Bell inequalities and showed that the Bell

inequality (2) named I3322is relevant to the well-known

CHSH inequality. Here relevance means that there is a

quantum mixed state ρ such that ρ satisfies the CHSH

inequality (with any measurements) but ρ violates the

I3322inequality (with some measurements). They prove

the relevance of the I3322inequality to the CHSH inequal-

ity by giving an explicit example of a state ρ in the 2⊗2

system which satisfies the CHSH inequality for all mea-

surements, and which violates the I3322inequality for cer-

tain measurements.

Part of the difficulty of testing relevance comes from

how to choose an appropriate state ρ. Even if we only

consider the 2⊗2 system, the space of mixed states is 15-

dimensional. Collins and Gisin overcome this difficulty

by restricting the states to those parameterized by two

variables θ and α: ρ(θ,α) = α|ϕθ??ϕθ| + (1 − α)|01??01|,

where |ϕθ? = cosθ|00?+sinθ|11?. For any θ, the variable

α can be maximized by using the Horodecki criterion [16]

to give a state ρ(θ,αmax) on the boundary of the set of

the states which satisfy the CHSH inequality for all mea-

surements. Then they compute the maximum violation

of the I3322inequality by ρ(θ,αmax) for various values of

θ, and find a state satisfying the CHSH inequality but

not the I3322inequality.

III.INCLUSION RELATION

Before discussing relevance relations among Bell in-

equalities for isotropic states, we need an introduction

to inclusion relation among these inequalities, which is

used to distinguish “obvious” relevance relations from the

other relevance relations.

A.Definition of inclusion relation

Collins and Gisin [7] pointed out that the CHSH in-

equality is irrelevant to the I3322 inequality since if we

pick the I3322 inequality and fix two measurements A3

and B1 to the deterministic measurement whose result

is always 0, the inequality becomes the CHSH inequal-

ity.Generalizing this argument, Avis, Imai, Ito and

Sasaki [10] introduced the notion of inclusion relation

between two Bell inequalities. A Bell inequality aTq ≤ 0

includes another Bell inequality bTq ≤ 0 if we can obtain

the inequality bTq ≤ 0 by fixing some measurements in

the inequality aTq ≤ 0 to deterministic ones (i.e. mea-

surements whose result is always 1 or always 0).

Here we give a formal definition of the inclusion rela-

tion. Let aTq ≤ 0 be a Bell inequality with mA+ mB

measurements and bTq ≤ 0 another with nA+ nBmea-

surements, and assume mA ≥ nA and mB ≥ nB. The

inequality aTq ≤ 0 includes bTq ≤ 0 if there exists a

Bell inequality (a′)Tq ≤ 0 equivalent to the inequality

aTq ≤ 0 such that a′

0 ≤ j ≤ nB. Here equivalence means that the inequality

(a′)Tq ≤ 0 can be obtained from another aTq ≤ 0 by

zero or more applications of party exchange, observable

exchange and value exchange. See e.g. [18] or [7] for more

about equivalence of Bell inequalities. Readers familiar

with the cut polytope will recognize that inclusion is a

special case of collapsing [15, Section 26.4].

By using this notion, a Bell inequality aTq ≤ 0 is irrel-

evant to another Bell inequality bTq ≤ 0 if the inequality

bTq ≤ 0 includes the inequality aTq ≤ 0.

ij= bijfor any 0 ≤ i ≤ nAand any

B.Inclusion relation between known Bell

inequalities with at most 5 measurements per party

We tested the inclusion relation among the 89 tight

Bell inequalities described in Section IIA. Figure 1 on

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the last page shows the result. In the figure, the serial

number of each inequality is shown with the number of

measurements (omitted for inequalities with 5 + 5 mea-

surements) and its name (if there is one). An arc from

one inequality to another means that the former includes

the latter. Since the inclusion relation is transitive, the

arcs which are derived by other arcs are omitted. An as-

terisk (*) on the right of the serial number indicates the

inequality is a candidate for being relevant to I3322. Rele-

vancy was tested empirically using the method described

in Section IVC.

From the figure, one might be tempted to conjecture

that the CHSH inequality is included in all tight Bell

inequalities other than the positive probability inequal-

ity. However, this is not true. Enumeration of tight Bell

inequalities with four measurements by each party us-

ing the general convex hull computation package lrs [19]

takes an unrealistically long time, but in a partial list, we

have some counterexamples. In the notation by Collins

and Gisin, they are:

(A1) (A2) (A3) (A4)

0−1

−1

01

1−1

−1

(A1) (A2) (A3) (A4)

−1

00

−1

1−1

1−1

−1

0

−1

1

2

−1

2

−1

1

−1

(B1) −1

(B2)

(B3) −1

(B4) −1

1

0

1

≤ 0,(I(1)

4422)

0−1

−1

1

2

−1

−3

1

2

1

1

(B1)

(B2) −1

(B3) −1

(B4)

0

1

0

≤ 0.(I(2)

4422)

IV.RELEVANCE FOR 2- AND 3-LEVEL

ISOTROPIC STATES

A.Violation of a Bell inequality by isotropic states

Let |ψd? be a maximally entangled state in d⊗d system:

1

√d(|00? + |11? + ··· + |d − 1,d − 1?).

|ψd? =

The d-level isotropic state [20] (or U ⊗ U∗-invariant

state [21]) ρd(α) of parameter 0 ≤ α ≤ 1 is a state de-

fined by:

ρd(α) = α|ψd??ψd| + (1 − α)I

=α

d(|00? + |11? + ··· + |d − 1,d − 1?)

(?00| + ?11| + ··· + ?d − 1,d − 1|) +1 − α

d2

d2

I.

With α = 0, ρd(α) is a maximally mixed state I/d2,

which is separable and therefore satisfies all the Bell

inequalities for all measurements.

is known that ρd(α) is separable if and only if α ≤

1/(d + 1) [21]. With α = 1, ρd(α) is a maximally entan-

gled state |ψd??ψd|. Therefore ρd(α) represents a state in

the middle between a separable state and a maximally

entangled state for general α.

If two states ρ and ρ′satisfy a Bell inequality for all

measurements, then their convex combination tρ + (1 −

t)ρ′also satisfies the same Bell inequality for all mea-

surements. This means that for any d ≥ 2 and any

Bell inequality aTq ≤ 0, there exists a real number

0 ≤ αmax ≤ 1 such that ρd(α) satisfies the inequality

aTq ≤ 0 for all measurements if and only if α ≤ αmax.

A smaller value of αmaxmeans that the Bell inequality

is more sensitive for isotropic states.

More generally, it

B.Violation of the CHSH inequality by 3-level

isotropic states

In this section, we prove that the maximum violation of

the CHSH inequality by the 3-level isotropic state ρ3(α)

is given by max{0,α(3√2 + 1)/9 − 4/9}. As a corollary,

the threshold αmaxfor the CHSH inequality with d = 3

is equal to αmax= 4/(3√2 + 1) = 0.76297427932.

As we noted in Section IIB, we can restrict E1, E2, F1

and F2 to projective measurements in the optimization

problem (3). We consider the rank of measurements E1,

E2, F1and F2. Since the CHSH inequality is not violated

if any one of E1, E2, F1and F2has rank zero or three, we

only need to consider the case where the four measure-

ments E1, E2, F1and F2have rank one or two. Instead

of considering all the combinations of ranks of the mea-

surements, we fix their rank to one and consider the in-

equalities obtained by exchanging outcomes “0” and “1”

of some measurements in the CHSH inequality. (In terms

of the cut polytope, this transformation corresponds to

switching [15, Section 26.3] of inequalities. See [10] for

details.) For example, suppose that E1and F1have rank

two and E2and F2have rank one in the optimal set of

measurements. Then instead of the CHSH inequality in

the form (1), we exchange the two outcomes of measure-

ments E1 and F1 in the inequality, and obtain (in the

Collins-Gisin notation):

(A1) (A2)

01

(B1) 0

(B2) 1

1−1

−1−1

≤ 1, (4)

with the four measurements of rank one. We have 24=

16 possibilities for the ranks of the four measurements

and corresponding 16 inequalities transformed from (1).

These inequalities are identical to either (1) or (4) if it

is relabelled appropriately. Therefore, we can assume

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the four measurements have rank one at the expense of

considering the inequality (4) in addition to (1).

We compute the maximum violation V (α) (resp.

V′(α)) of the inequality (1) (resp. (4)) under the assump-

tion that the four measurements have rank one. In the

maximally mixed state ρ3(0) = I9/9, the violations of

the two inequalities are constant regardless of the actual

measurements, and they are:

V (0) = −q10− q01+ q11+ q12+ q21− q22

= −1/3 − 1/3 + 1/9 + 1/9 + 1/9 − 1/9 = −4/9,

V′(0) = q20+ q02+ q11− q12− q21− q22− 1

= 1/3 + 1/3 + 1/9 − 1/9 − 1/9 − 1/9 − 1 = −5/9.

Since the violations of the inequalities are constant in the

state ρ3(0), the maximum violation in the state ρ3(α)

is achieved by the optimal set of measurements in the

state ρ3(1), V (α) = αV (1) + (1 − α)V (0) and V′(α) =

αV′(1) + (1 − α)V′(0). Therefore, what remains is to

compute the values of V (1) and V′(1).

To obtain the value of V (1), let Ei= |ϕ1i??ϕ1i|, Fj=

|ϕ2j??ϕ2j|, |ϕ1i? = xi0|0? + xi1|1? + xi2|2? and |ϕ2j? =

yj0|0? + yj1|1? + yj2|2?. Note that x1, x2, y1 and y2

are unit vectors in C3. Using them, the violations of the

inequality (1) is equal to

−2

3+13(|x1·y1|2+|x1·y2|2+|x2·y1|2−|x2·y2|2), (5)

If we fix y1 and y2 arbitrarily, then optimization of x1

and x2in (5) can be performed separately. Since (5) de-

pends only on the inner products of the vectors and not

the vectors themselves, we can replace the vectors x1

and x2with their projection onto the subspace spanned

by y1 and y2.This means that we can consider the

four vectors x1, x2, y1 and y2are vectors in C2whose

lengths are at most one. Then the Tsirelson inequal-

ity [22, 23] tells the maximum of |x1· y1|2+ |x1· y2|2+

|x2·y1|2−|x2·y2|2is equal to√2+1, and the vectors giv-

ing this maximum are |ϕ11? = cos(π/4)|0? + sin(π/4)|1?,

|ϕ12? = |0?, |ϕ21? = cos(π/8)|0? + sin(π/8)|1? and

|ϕ22? = cos(3π/8)|0? + sin(3π/8)|1?.

(1) is V (1) = (√2 − 1)/3 = 0.138071, and V (α) =

(1 − α)(−4/9) + α(√2 − 1)/3 = α(3√2 + 1)/9 − 4/9.

By a similar argument, we can compute the value of

V′(1). Using the same definition for x1, x2, y1and y2,

the violation of the inequality (4) is given by

The violation of

−4

3+13(|x1·y1|2−|x1·y2|2−|x2·y1|2−|x2·y2|2). (6)

The maximum of (6) is equal to −1, and it is achieved by

setting |ϕ11? = |ϕ21? = |0?, |ϕ12? = |1? and |ϕ22? = |2?.

Therefore V′(1) = −1 and V′(α) = −14α/9 − 5/9 < 0.

This means the inequality (4) is never violated under the

assumption that the four measurements have rank one.

Removing the assumption of the ranks of the mea-

surements, we obtain that the maximum violation of

the CHSH inequality in the state ρ3(α) is given by

max{0,V(α),V′(α)} = max{0,α(3√2 + 1)/9 − 4/9}.

C.Computation of violation of Bell inequalities

with at most 5 measurements per party

We performed preliminary experiments to compute an

upper bound on the value of αmaxwith d = 2 and d = 3

for the 89 inequalities described in Section IIA.

see-saw iteration algorithm finds a candidate for the op-

timal solution of (3). When 0 ≤ α ≤ 1 is given, we can

use this search algorithm to tell whether αmax < α (if

violating measurements are found) or αmax≥ α (other-

wise), if we ignore the possibility that the hill-climbing

search fails to find the global optimum. This allows us to

compute the value of αmaxby binary search. In reality,

the hill-climbing search sometimes fails to find the global

optimum, and if it finds violating measurements then it

surely means αmax < α, whereas if it does not find vi-

olating measurements then it does not necessarily mean

αmax≥ α. Therefore, the value given by binary search

is not necessarily the true value of αmax but an upper

bound on it.

The

In each step of the binary search, we performed a see-

saw iteration with 1,000 random initial measurements

and picked the solution giving the maximum in the

1,000 trials. To compute eigenvalues and eigenvectors

of 3 × 3 Hermitian matrix, we used LAPACK [24] with

ATLAS [25, 26]. All computations were performed using

double-precision floating arithmetic. Due to numerical

error, the computation indicates a small positive vio-

lation even if the state does not violate the inequality.

Therefore, we only consider violation greater than 10−13

significant.

For d = 2, the computation gave an upper bound

0.70711 for all inequalities except for the positive proba-

bility inequality. (For the positive probability inequality

we have αmax = 1 since it is satisfied by any quantum

state.) It is known that in the case d = 2, the CHSH

inequality is satisfied if and only if α ≤ 1/√2 = 0.70711

from the Horodecki criterion [16]. These results suggest

that there may not be any Bell inequalities relevant to

the CHSH inequality for 2-level isotropic states, indicat-

ing the negative answer to Gisin’s problem [8] in the case

of 2-level system.

We performed the same computation for d = 3. This

time some Bell inequalities gave a smaller value of αmax

than the CHSH inequality did. Some of them gave a

small value of αmax simply because it includes another

such inequality. Filtering them out, we identified five

inequalities which are candidates for being relevant to the

CHSH inequality for the 3-level isotropic states. Rows

and columns in bold font indicate that they correspond

to nodes added by triangular elimination.

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TABLE I: Upper bound of the value of αmax obtained by the experiments.

αmax

Bell inequalityOriginal cut polytope inequality

7

6

8 (Par(7), parachute ineq.)

89

2 (Pentagonal ineq.)

2 (Pentagonal ineq.)

1 (Triangle ineq.)

0.7447198434 A28

0.7453308276 A27

0.7553800191 A5

0.7557816805 A56

0.7614396336 A8

0.7629742793 A3 (I3322)

0.7629742793 A2 (CHSH)

1A1 (Positive probability) 1 (Triangle ineq.)

A28:

(A1) (A2) (A3) (A4) (A5)

−2

101

011

11−1

110

1

−10

−1−100

(B1) −2

(B2) −1

(B3) −1

(B4)

(B5)

1

1

0

1

−1

0

0

0

0

0

−1

0

≤ 0,

A27:

(A1) (A2) (A3) (A4) (A5)

−1

111

10−1

0−1

−1

100

00−1

0

−1

1

1

1

−1

(B1) −2

(B2)

(B3) −1

(B4) −1

(B5) −1

0

1

1

0

0

0

1

01

≤ 0,

A5:

(A1) (A2) (A3) (A4)

00−1

1

0

1

1

−1

(B1) −2

(B2) −1

(B3) −1

(B4)

1

1

10

1

1

0

−1

1

−1

−1

0

0

≤ 0,

A56:

(A1) (A2) (A3) (A4) (A5)

−1

01−1

10−1

−1

111

002

00−2

1

1

1

−1

2

−2

(B1) −1

(B2)

(B3)

(B4) −2

(B5) −2

0

0

2

2

0

0

0

−1−1

≤ 0,

A8:

(A1) (A2) (A3) (A4)

0−1

11

11

−11

−10

0

−1

−2

1

1

0

1

1

0

(B1) −1

(B2) −2

(B3)

(B4)

(B5)

−1

1

0

0

0

0

0

0

≤ 0.

Adding the CHSH and the I3322 inequalities, we per-

formed the experiments with 50,000 initial solutions with

the seven inequalities. Table I summarizes the results we

obtained. In Table I, the column labeled “Original cut

polytope inequality” shows the facet inequality of CUT?

to which triangular elimination is applied. The number

corresponds to the serial number of the facet in cut9.gz

of [12]. For the CHSH inequality, the obtained upper

bound 0.76298 is consistent with the theoretical value

4/(3√2 + 1) = 0.762974 proved in Section IVB. The

I3322inequality gave the same upper bound as the CHSH

inequality. Besides, in the optimal measurements with α

near 4/(3√2 + 1), the matrices E3and F1are zero, cor-

responding to the fact that the I3322inequality includes

the CHSH inequality. This is consistent with Collins and

Gisin’s observation [7] in the 2 ⊗ 2 system that the I3322

inequality is not better than the CHSH inequality for

states with high symmetry.

Five Bell inequalities A28, A27, A5, A56 and A8 gave

a smaller value of αmax than 4/(3√2 + 1). The set of

measurements giving optimal violation for these Bell in-

equalities with α slightly larger than the computed value

of αmaxis given in the Appendix.

These Bell inequalities are relevant to the CHSH in-

equality. As a result, Bell inequalities including any of

them are also relevant to the CHSH inequality. More-

over, if the true value of αmax for the I3322 inequality

is 4/(3√2 + 1), then these five Bell inequalities are also

relevant to the I3322 inequality. We make the following

conjecture.

Conjecture 1. The state ρ3(4/(3√2 + 1)) satisfies the

I3322 inequality for all measurements. In other words,

αmax= 4/(3√2 + 1) for the I3322 inequality in the case

of d = 3.

9

To support this conjecture, we searched for the opti-

mal measurements for the I3322 inequality in the states

ρ3(α) with α = α+= 0.7629742794 > 4/(3√2 + 1) and

α = α− = 0.7629742793 < 4/(3√2 + 1), using see-saw

iteration algorithm with random initial solutions. With

α = α+, 100 out of 633 trials gave a violation greater

than 10−13, whereas with α = α−, none of 50,000 trials

gave a violation greater than 3×10−15. Considering nu-

merical error in computation, we consider that this result

Page 7

7

can be seen as an evidence that the I3322inequality be-

haves differently in the state ρ3(α) depending on whether

α is greater or less than 4/(3√2 + 1).

V.CONCLUDING REMARKS

We used numerical optimization to show that certain

Bell inequalities are relevant to the CHSH inequality

for isotropic states. No Bell inequalities relevant to the

CHSH inequality were found for 2-level isotropic states.

This supports Collins and Gisin’s conjecture in [7] that no

such Bell inequalities exist. For 3-level isotropic states,

however, five Bell inequalities relevant to the CHSH in-

equality were found. The results of numerical experi-

ments were given to support the conjecture that they are

also relevant for the I3322inequality.

The violation of the CHSH inequality by 3-level

isotropic states was shown by using Tsirelson’s inequal-

ity.Cleve, Høyer, Toner and Watrous [17] generalize

Tsirelson’s inequality to Bell inequalities corresponding

to “XOR games,” which do not depend on individual

variables qi0,q0j,qij but only involves combinations in

the form xij = qi0+ q0j − 2qij.

I3322 inequality is not such an inequality, and we can-

not use the result there to prove the theoretical value

of αmax for the I3322 inequality.

inequalities relevant to the CHSH inequality for 3-level

isotropic states, the inequality A8, which can be written

as −?

x35+x41−x42≤ 0, is the only one that corresponds to an

XOR game. An important open problem is to generalize

Cleve, Høyer, Toner and Watrous’s result to cover Bell

inequalities which do not correspond to XOR games.

Unfortunately, the

Among the five Bell

i=1,2

?

j=1,2,3xij+x13− x23+x14− x34+ x25−

Acknowledgments

The first author is supported by the Grant-in-Aid for

JSPS Fellows.

APPENDIX A: OPTIMAL MEASUREMENTS COMPUTED FOR EACH INEQUALITIES

A28:

E1= I − |ϕ11??ϕ11|,

|ϕ11? = 0.819512|0? + (−0.181891 − 0.067213i)|1? + (0.239561 + 0.483124i)|2?,

E2= |ϕ12??ϕ12|,

|ϕ12? = 0.391928|0? + (0.546808 − 0.330668i)|1? + (−0.064601 + 0.658695i)|2?,

E3= |ϕ13??ϕ13|,

|ϕ13? = 0.585206|0? + (0.266618 − 0.150612i)|1? + (0.721307 − 0.208519i)|2?,

E4= |ϕ14??ϕ14|,

|ϕ14? = 0.696701|0? + (0.109760 + 0.562926i)|1? + (0.269399 − 0.336302i)|2?,

E5= I − |ϕ15??ϕ15|,

|ϕ15? = 0.745551|0? + (0.060720 − 0.038486i)|1? + (0.610743 + 0.256863i)|2?,

F1= I − |ϕ21??ϕ21|,

|ϕ21? = 0.665942|0? + (0.124951 + 0.288249i)|1? + (0.306094 − 0.603430i)|2?,

F2= I − |ϕ22??ϕ22|,

|ϕ22? = 0.794583|0? + (−0.503910 − 0.071325i)|1? + (−0.075809 − 0.322300i)|2?,

F3= I − |ϕ23??ϕ23|,

|ϕ23? = 0.738612|0? + (0.143632 − 0.211840i)|1? + (0.594179 + 0.189467i)|2?,

F4= |ϕ24??ϕ24|,

|ϕ24? = 0.314299|0? + (0.087381 + 0.592536i)|1? + (0.427166 + 0.600009i)|2?,

F5= I − |ϕ25??ϕ25|,

|ϕ25? = 0.745551|0? + (0.060720 + 0.038486i)|1? + (0.610743 − 0.256863i)|2?

A27:

E1= |ϕ11??ϕ11|,

|ϕ11? = 0.512740|0? + (0.141298 − 0.367921i)|1? + (0.118341 − 0.753500i)|2?,

E2= I − |ϕ12??ϕ12|,

|ϕ12? = 0.429346|0? + (0.490358 + 0.190555i)|1? + (−0.588595 − 0.438697i)|2?,

E3= I − |ϕ13??ϕ13|,

|ϕ13? = 0.649098|0? + (−0.034498 + 0.390106i)|1? + (0.648622 + 0.067734i)|2?,

E4= |ϕ14??ϕ14|,

|ϕ14? = 0.782874|0? + (−0.199336 − 0.104823i)|1? + (−0.579621 + 0.020651i)|2?,

E5= |ϕ15??ϕ15|,

|ϕ15? = 0.504711|0? + (0.266955 − 0.029362i)|1? + (−0.176172 − 0.801313i)|2?,

F1= |ϕ21??ϕ21|,

|ϕ21? = 0.477430|0? + (−0.243408 + 0.631106i)|1? + (−0.024181 + 0.560297i)|2?,

F2= |ϕ22??ϕ22|,

|ϕ22? = 0.521997|0? + (0.270933 − 0.132987i)|1? + (0.586914 + 0.540334i)|2?,

F3= |ϕ23??ϕ23|,

|ϕ23? = 0.631718|0? + (0.176373 + 0.079451i)|1? + (−0.678537 + 0.321093i)|2?,

F4= |ϕ24??ϕ24|,

|ϕ24? = 0.839814|0? + (−0.361305 − 0.101706i)|1? + (−0.207777 − 0.332648i)|2?,

F5= |ϕ25??ϕ25|,

|ϕ25? = 0.634648|0? + (−0.135288 + 0.308277i)|1? + (−0.492423 + 0.491328i)|2?

A5:

E1= I − |ϕ11??ϕ11|,

|ϕ11? = 0.079911|0? + (0.347597 − 0.352563i)|1? + (0.852394 + 0.148034i)|2?,

E2= |ϕ12??ϕ12|,

|ϕ12? = 0.466812|0? + (0.336458 − 0.338316i)|1? + (0.063365 − 0.741896i)|2?,

E3= I − |ϕ13??ϕ13|,

|ϕ13? = 0.700997|0? + (−0.090375 + 0.325520i)|1? + (0.625759 − 0.053829i)|2?,

E4= |ϕ14??ϕ14|,

|ϕ14? = 0.569742|0? + (−0.703808 − 0.061209i)|1? + (−0.405767 − 0.107957i)|2?,

F1= |ϕ21??ϕ21|,

|ϕ21? = 0.611974|0? + (0.261472 + 0.553836i)|1? + (−0.402289 + 0.297574i)|2?,

F2= |ϕ22??ϕ22|,

|ϕ22? = 0.743739|0? + (−0.644052 − 0.121119i)|1? + (−0.050055 − 0.121959i)|2?,

F3= |ϕ23??ϕ23|,

|ϕ23? = 0.327181|0? + (−0.492820 + 0.363796i)|1? + (−0.442899 + 0.567075i)|2?,

F4= I − |ϕ24??ϕ24|,

|ϕ24? = 0.558366|0? + (0.295353 − 0.157594i)|1? + (0.593099 + 0.473699i)|2?

Page 8

8

A56:

E1= |ϕ11??ϕ11|,

|ϕ11? = 0.764669|0? + (0.520735 − 0.023147i)|1? + (0.314448 − 0.211429i)|2?,

E2= I − |ϕ12??ϕ12|,

|ϕ12? = 0.523087|0? + (−0.660068 + 0.130414i)|1? + (0.115043 + 0.510340i)|2?,

E3= I − |ϕ13??ϕ13|,

|ϕ13? = 0.651881|0? + (0.010176 − 0.025750i)|1? + (−0.599260 + 0.463866i)|2?,

E4= I − |ϕ14??ϕ14|,

|ϕ14? = 0.480244|0? + (0.435821 − 0.476742i)|1? + (0.370530 + 0.463520i)|2?,

E5= I − |ϕ15??ϕ15|,

|ϕ15? = 0.484893|0? + (0.214118 + 0.403736i)|1? + (0.401826 + 0.628144i)|2?,

F1= |ϕ21??ϕ21|,

|ϕ21? = 0.704822|0? + (0.050276 − 0.044858i)|1? + (−0.676460 + 0.202702i)|2?,

F2= I − |ϕ22??ϕ22|,

|ϕ22? = 0.279921|0? + (−0.406294 + 0.685472i)|1? + (0.534341 + 0.034308i)|2?,

F3= I − |ϕ23??ϕ23|,

|ϕ23? = 0.580814|0? + (0.563163 + 0.064963i)|1? + (0.561359 − 0.161735i)|2?,

F4= I − |ϕ24??ϕ24|,

|ϕ24? = 0.522791|0? + (−0.366663 − 0.240466i)|1? + (−0.161766 − 0.712921i)|2?,

F5= I − |ϕ25??ϕ25|,

|ϕ25? = 0.575083|0? + (0.352241 + 0.118045i)|1? + (−0.170766 − 0.708598i)|2?

A8:

E1= |ϕ11??ϕ11|,

|ϕ11? = 0.589845|0? + (0.252414 − 0.592962i)|1? + (−0.067286 + 0.481911i)|2?,

E2= |ϕ12??ϕ12|,

|ϕ12? = 0.571429|0? + (−0.328221 − 0.214531i)|1? + (0.352103 + 0.629079i)|2?,

E3= I − |ϕ13??ϕ13|,

|ϕ13? = 0.789596|0? + (0.397845 + 0.124284i)|1? + (0.373987 + 0.250887i)|2?,

E4= I − |ϕ14??ϕ14|,

|ϕ14? = 0.588353|0? + (−0.068306 − 0.217513i)|1? + (−0.748446 + 0.204184i)|2?,

F1= |ϕ21??ϕ21|,

|ϕ21? = 0.500028|0? + (−0.062398 + 0.498087i)|1? + (−0.351826 − 0.611724i)|2?,

F2= |ϕ22??ϕ22|,

|ϕ22? = 0.416357|0? + (−0.421270 + 0.580072i)|1? + (0.375055 − 0.414762i)|2?,

F3= |ϕ23??ϕ23|,

|ϕ23? = 0.555120|0? + (−0.275989 − 0.322007i)|1? + (0.606921 − 0.378986i)|2?,

F4= I − |ϕ24??ϕ24|,

|ϕ24? = 0.771642|0? + (0.389862 + 0.263652i)|1? + (0.160470 − 0.396628i)|2?,

F5= I − |ϕ25??ϕ25|,

|ϕ25? = 0.759855|0? + (0.022187 + 0.015402i)|1? + (0.430543 − 0.486336i)|2?

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9

A1(1+1,"Positive probability")

A2(2+2,"CHSH")

A3(3+3,"I_3322")A4(3+4,"I_3422^2")

A5(4+4)*

A6(4+4),A8(4+5)*,A10(4+5),

A17(4+5),A18(4+5),A21(4+5),

A26,A39,A61,A65,A80

A7(4+4,"I_4422")A9(4+5)

A11(4+5),A13(4+5),

A14(4+5),A19(4+5),A32,

A36,A46,A60,A62,A79

A12(4+5)

A15(4+5),A20(4+5),A42,

A43,A51,A53,A64,A67,

A68,A72,A74,A86,A87

A16(4+5)*,A24*,A33*,A34*,

A35*,A40*,A44*,A45*,A47*,A50*,

A54*,A57*,A63*,A70*,A85*

A22(4+5)

A23*A25,A27*

A28*,A29,A30,A31,A37,A38,

A41,A48,A49,A52,A56*,A58,A73,

A75,A76,A77,A78,A81,A83

A55*,A66* A59*,A71*,A82*A69,A88,A89("I_5522")A84

FIG. 1: Inclusion relation among 89 Bell inequalities, with at most 5 measurements per party, obtained by triangular elimination from facets of CUT?

on the right of the serial number indicates that the inequality is relevant to the CHSH inequality for 3 ⊗ 3 isotropic states and that it is a candidate for being relevant

to I3322.

9. An asterisk (*)