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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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4
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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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 (*)