Article

Bell inequalities stronger than the Clauser-Horne-Shimony-Holt inequality for three-level isotropic states

McGill University, Montréal, Quebec, Canada
Physical Review A (Impact Factor: 2.81). 04/2006; 73(4). DOI: 10.1103/PhysRevA.73.042109
Source: arXiv
ABSTRACT
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.

Full-text

Available from: David Avis
arXiv:quant-ph/0508210v2 25 Jan 2006
Bell inequalities stronger than the CHSH inequality
for 3-level isotropic states
Tsuyoshi Ito,
1,
Hiroshi Ima i,
1, 2,
and David Avis
3,
1
Department of Computer Science, University of Tokyo
2
ERATO-SORST Quantum Computation and Information Project, Japan Science and Technology Agency
3
School of Computer Science, McGill University
(Dated: January 25, 2006)
We show that some two-party Bell inequalities with two-valued observables are stronger th an 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 th e list of known facet inequalities of the cut polytope on nine
point s. This gives a partial solution to an open problem posed by Collins and Gisin. The results
of numerical optimization suggest that t hey are candidates for being stronger than the I
3322
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. INTRODUC TION
Bell inequalities and their vio lation 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 correla tion
exp eriments. A Bell inequality is an inequality valid for
a certain correlation polytope. The correlation ex peri-
ments we consider in this paper are those between two
parties, where one party has m
A
choices of two-valued
measurements and the other party has m
B
choices. The
Clauser-Horne-Shimony-Holt inequality [5] is an exa mple
of a Bell inequality in this se tting with m
A
= m
B
= 2.
Separable states satisfy all Bell inequalities with all
measurements by definition. In a seminal paper [6],
Werner disproved the converse: there exists a quantum
mixed state ρ which is entangled but sa tisfies 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 I
3322
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 measure ments)
but ρ violates the I
3322
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 ca st as a bilinear
semidefinite programming problem, which is a hard opti-
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 iteratio n” 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 II C.
Collins and Gisin showed numerically that the I
3322
Bell inequality is no t relevant to the CHSH inequality
for 2-level Werner states. They posed an open prob-
lem [8]: Find Bell inequalities which are stronger than
the CHSH inequalities in the s e ns e 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 o f 2 -level
Werner states in that they are c onvex combinations of a
pure maxima lly 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 tria ngular 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 sta tes . They answer Collins and
Gisin’s problem where Werner states are replaced by 3-
level isotropic states. We g ive empirical evidence that the
five inequalities are also relevant to the I
3322
inequality.
To the best of our knowledge, no such Bell inequalities
were previously known.
The res t of the paper is organized as follows. Section II
explains the necessary concepts. Section III discuss e s in-
clusion relation, which is used to prove irrelevance of a
Page 1
2
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 r e le vance for 2- and 3-level
isotropic states. Section V concludes the paper and men-
tions so me 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 m
A
choices A
1
, . . . , A
m
A
of two-valued measurements and Bob has m
B
choices
B
1
, . . . , B
m
B
. We call the two possible outcomes o f the
measurements 1 and 0. The result of this correlation ex-
periment can be represented by an (m
A
+ m
B
+ m
A
m
B
)-
dimensional vector q, where for 1 i m
A
and
1 j m
B
, the variables q
i0
, q
0j
and q
ij
represent
the probability that the outcome of A
i
is 1, that the out-
come of B
j
is 1, and that two outcomes of both A
i
and
B
j
are 1, re spe c tively.
An inequality a
T
q a
0
, where a is an (m
A
+
m
B
+ m
A
m
B
)-dimensional vector and a
0
is a scalar, is
called a Bell inequality if it is satisfied for all separable
states ρ and all choices of measurements A
1
, . . . , A
m
A
,
B
1
, . . . , B
m
B
. The nontrivial Bell inequality with the
smallest values of m
A
and m
B
is the CHSH inequality [5]
q
10
q
01
+ q
11
+ q
21
+ q
12
q
22
0 (1)
for m
A
= m
B
= 2.
A Bell inequality is said to be t ight 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 a
T
q a
0
,
then the same state violates one of tight Bell inequalities
which sum up to a
T
q a
0
.
Throughout this paper, we denote a Bell inequality
a
T
q a
0
by
(A
1
) ··· (A
m
A
)
a
10
··· a
m
A
0
(B
1
) a
01
a
11
··· a
m
A
1
.
.
.
.
.
.
.
.
.
.
.
.
(B
m
B
) a
0m
B
a
1m
B
··· a
m
A
m
B
a
0
,
following the notation by Collins and Gisin used in [7]
(with labels added to indicate which r ows and columns
correspo nd to which measurements). For example, the
CHSH inequality (1) is written as
(A
1
) (A
2
)
1 0
(B
1
) 1 1 1
(B
2
) 0
1 1
0.
Another Bell inequality found by Pitowsky and
Svozil [11] and named I
3322
inequality by Collins and
Gisin [7] is written as
(A
1
) (A
2
) (A
3
)
1 0 0
(B
1
) 2 1 1 1
(B
2
) 1
1 1 1
(B
3
) 0
1 1 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 inequa lities 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
9
. There are 89 Bell in-
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 I
mm22
inequalities for m = 3, 4, 5, the
I
(2)
3422
inequality [7] a nd other unnamed Bell inequalities.
We label the 89 inequalities as A1 to A89. The lis t of
these inequalities is available online [13].
B. Violation of a Bell inequality and bi linear
semidefinite programming
A test whether there exis ts 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 a
T
q a
0
be a Bell inequality. Each measurement
by Alice is represe nted by a positive operator valued mea-
sure (POVM) (E
i
, I E
i
), where E
i
is a Hermitian d ×d
matrix such that both E
i
and I E
i
are nonnega tive def-
inite and I is the identity matrix of size d ×d. Similarly,
each measurement by Bob is represented by a POVM
(F
j
, I F
j
). For concise notation, we let E
0
= F
0
= I.
Then the test whether there exists a set of violating mea-
surements or not can be fo rmulated as:
max
X
0im
A
0jm
B
(i,j)6=(0,0)
a
ij
tr(ρ(E
i
F
j
)) a
0
(3)
where E
0
= F
0
= I,
E
i
T
= E
i
, F
j
T
= F
j
,
O E
i
,F
j
I.
Page 2
3
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 mea surements, and if
so, the optimal solution gives the set of measurements
that is ma ximally violating the g iven Bell inequality in
the given state. If we fix one of the two groups of vari-
ables {E
1
, . . . , E
m
A
} and {F
1
, . . . , F
m
B
}, (3) b e c omes a
semidefinite programming problem on the other gr oup 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 a
T
q a
0
is the CHSH
inequality, then (3) can be solved a nalytically [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, give n the difficulty of bilinear prog ram-
ming. Section 2 of [14] describes a hill-climbing algo-
rithm which computes a local optimum by fixing one of
the two groups of va riables and solving the subproblem
to optimize variables in the other groups r e peatedly, 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.
There exists a set of projective measurements
E
1
, . . . , E
m
A
and F
1
, . . . , F
m
B
which attains the maxi-
mum of (3). This fact is obtained from the proof of The-
orem 5.4 in [1 7] by Cleve, Høyer, Toner and Watrous.
Though they prove the case where ρ is also variable,
the r e levant part in the proof is tr ue even if the state is
fixed. See-saw iteration always produces projective mea-
surements as a candidate for the optimal measurements.
C. Relevance relation
Collins and Gisin [7] introduced the notion of relevance
betwee n two Bell inequalities a nd showed that the Bell
inequality (2) named I
3322
is relevant to the well-known
CHSH inequality. Here relevance means that there is a
quantum mixed state ρ such that ρ sa tisfies the CHSH
inequality (with any measure ments) but ρ violates the
I
3322
inequality (with some measurements). They prove
the relevance of the I
3322
inequality 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 I
3322
inequality for cer-
tain measurements.
Part of the difficulty of testing rele vance comes from
how to choose an appropriate state ρ. Even if we only
consider the 22 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|ϕ
θ
ihϕ
θ
|+ (1 α)|01ih01|,
where |ϕ
θ
i = cos θ|00i+ sin θ|11i. For any θ, the variable
α can be ma ximized by using the Horodecki criterion [16]
to give a state ρ(θ, α
max
) on the boundary of the set of
the states which satisfy the CHSH inequality fo r all mea-
surements. Then they co mpute the maximum violation
of the I
3322
inequality by ρ(θ, α
max
) for various values of
θ, and find a state satisfying the CHSH inequality but
not the I
3322
inequality.
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 I
3322
inequality since if we
pick the I
3322
inequality and fix two measurements A
3
and B
1
to the deterministic measurement whose result
is always 0, the inequality becomes the CHSH inequal-
ity. Generalizing this argument, Avis, Ima i, Ito and
Sasaki [10] introduced the no tion of inclusion relatio n
betwee n two Bell inequalities. A Bell inequality a
T
q 0
includes another Bell inequality b
T
q 0 if we can obta in
the inequality b
T
q 0 by fixing some measurements in
the inequality a
T
q 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 a
T
q 0 be a B ell inequality with m
A
+ m
B
measurements and b
T
q 0 another with n
A
+ n
B
mea-
surements, and assume m
A
n
A
and m
B
n
B
. The
inequality a
T
q 0 includes b
T
q 0 if there exists a
Bell inequality (a
)
T
q 0 equivalent to the inequality
a
T
q 0 such that a
ij
= b
ij
for any 0 i n
A
and any
0 j n
B
. Here equivalence means that the inequality
(a
)
T
q 0 can be obtained from another a
T
q 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
sp e c ial case of collapsing [15, Section 2 6.4].
By using this notion, a Bell inequality a
T
q 0 is irrel-
evant to another Bell inequality b
T
q 0 if the inequality
b
T
q 0 includes the inequality a
T
q 0.
B. Inclusion relation b etween known Bell
inequalities with at most 5 me asurements per party
We tested the inclusion relation among the 89 tight
Bell inequalities described in Section II A. Figure 1 on
Page 3
4
the last page shows the result. In the figure, the serial
number of each inequa lity 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 a re derived by other arcs are omitted. An as -
terisk (*) o n the right of the serial number indicates the
inequality is a candidate for being relevant to I
3322
. Rele-
vancy was tested empirically using the method described
in Sectio n IV C.
From the figure, one might be tempted to conjecture
that the CHSH ineq uality 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 gener al convex hull computation package lrs [19]
takes an unrealistically long time, but in a partial list, we
have some counterexamples. In the nota tion by Collins
and Gisin, they are:
(A
1
) (A
2
) (A
3
) (A
4
)
0 1 1 1
(B
1
) 1 1 1 0 2
(B
2
) 0
0 1 1 1
(B
3
) 1
1 1 1 1
(B
4
) 1
1 1 2 1
0, (I
(1)
4422
)
(A
1
) (A
2
) (A
3
) (A
4
)
1 0 1 3
(B
1
) 0 0 0 1 1
(B
2
) 1
1 1 1 2
(B
3
) 1
1 1 2 1
(B
4
) 0
1 1 1 1
0. (I
(2)
4422
)
IV. RELEVANCE FOR 2- AND 3-LEVEL
ISOTROPIC STATES
A. Violation of a Bell ineq uality by isotropic states
Let |ψ
d
i be a maximally entangled state in dd system:
|ψ
d
i =
1
d
(|00i + |11i + ···+ |d 1, d 1i).
The d-level isotropic state [20] (or U U
-invariant
state [21]) ρ
d
(α) of parameter 0 α 1 is a state de-
fined by:
ρ
d
(α) = α|ψ
d
ihψ
d
| + (1 α)
I
d
2
=
α
d
(|00i + |11i + ···+ |d 1, d 1i)
(h00| + h11| + ···+ hd 1, d 1|) +
1 α
d
2
I.
With α = 0, ρ
d
(α) is a ma ximally mixed state I/d
2
,
which is separable and therefore satisfies all the Bell
inequalities fo r all measur e ments. More genera lly, it
is known that ρ
d
(α) is s e parable if and only if α
1/(d + 1) [21]. With α = 1, ρ
d
(α) is a maximally entan-
gled state |ψ
d
ihψ
d
|. Therefore ρ
d
(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 + (1
t)ρ
also satisfies the same Bell inequality for all mea-
surements. This means that for any d 2 and any
Bell inequality a
T
q 0, there exists a real number
0 α
max
1 such that ρ
d
(α) satisfies the inequality
a
T
q 0 for all measurements if and only if α α
max
.
A smaller value of α
max
means that the Bell inequality
is mor e sensitive for isotropic states.
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 s tate ρ
3
(α)
is given by max{0, α(3
2 + 1)/9 4/9}. As a corollar y,
the threshold α
max
for the CHSH inequality with d = 3
is equal to α
max
= 4/(3
2 + 1) = 0.762974 27932.
As we noted in Section II B, we c an restr ict E
1
, E
2
, F
1
and F
2
to projective measurements in the optimization
problem (3). We consider the rank of measurements E
1
,
E
2
, F
1
and F
2
. Since the CHSH inequality is not v iolated
if any one of E
1
, E
2
, F
1
and F
2
has rank z e ro or three, we
only need to consider the case where the four measure-
ments E
1
, E
2
, F
1
and F
2
have rank one or two. Instead
of consider ing all the combinations of ranks of the mea-
surements, we fix their rank to one and c onsider the in-
equalities obtained by exchanging outcomes “0” and “1”
of so me 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 E
1
and F
1
have rank
two and E
2
and F
2
have 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 E
1
and F
1
in the inequality, a nd obtain (in the
Collins-Gisin no tation):
(A
1
) (A
2
)
0 1
(B
1
) 0 1 1
(B
2
) 1
1 1
1, (4)
with the four meas urements of rank one. We have 2
4
=
16 po ssibilities 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 a ssume
Page 4
5
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 ρ
3
(0) = I
9
/9, the violations of
the two inequalities are constant regardless of the actual
measurements, and they are:
V (0) = q
10
q
01
+ q
11
+ q
12
+ q
21
q
22
= 1/3 1/3 + 1/9 + 1/9 + 1/9 1/9 = 4/9,
V
(0) = q
20
+ q
02
+ q
11
q
12
q
21
q
22
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 ρ
3
(0), the maximum violation in the state ρ
3
(α)
is achieved by the optimal set of measurements in the
state ρ
3
(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 E
i
= |ϕ
1i
ihϕ
1i
|, F
j
=
|ϕ
2j
ihϕ
2j
|, |ϕ
1i
i = x
i0
|0i + x
i1
|1i + x
i2
|2i and |ϕ
2j
i =
y
j0
|0i + y
j1
|1i + y
j2
|2i. Note that x
1
, x
2
, y
1
and y
2
are unit vectors in C
3
. Using them, the violations of the
inequality (1) is equal to
2
3
+
1
3
(|x
1
·y
1
|
2
+ |x
1
·y
2
|
2
+ |x
2
·y
1
|
2
|x
2
·y
2
|
2
), (5)
If we fix y
1
and y
2
arbitrarily, then optimization of x
1
and x
2
in (5) can be performed separately. Since (5) de-
pends only on the inner pr oducts of the ve c tors and not
the vectors themselves, we can replace the vectors x
1
and x
2
with their projection o nto the subs pace spanned
by y
1
and y
2
. This means that we can consider the
four vectors x
1
, x
2
, y
1
and y
2
are vectors in C
2
whose
lengths are at most one. Then the Tsirelson inequal-
ity [22, 23] tells the maximum of |x
1
· y
1
|
2
+ |x
1
· y
2
|
2
+
|x
2
·y
1
|
2
|x
2
·y
2
|
2
is equa l to
2+1, and the vectors giv-
ing this maximum are |ϕ
11
i = cos(π/4)|0i + sin(π/4)|1i,
|ϕ
12
i = |0i, |ϕ
21
i = cos(π/8)|0i + s in(π/8)|1i and
|ϕ
22
i = cos(3π/8)|0i + sin(3π/8)|1i. The violation of
(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 x
1
, x
2
, y
1
and y
2
,
the violation of the inequality (4) is given by
4
3
+
1
3
(|x
1
·y
1
|
2
|x
1
·y
2
|
2
|x
2
·y
1
|
2
|x
2
·y
2
|
2
). (6)
The maximum of (6) is equal to 1, and it is achieved by
setting |ϕ
11
i = |ϕ
21
i = |0i, |ϕ
12
i = |1i and |ϕ
22
i = |2i.
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 as sumption of the ranks of the mea-
surements, we obtain that the max imum violation of
the CHSH inequality in the state ρ
3
(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 pr e liminary experiments to compute an
upper bound on the value of α
max
with d = 2 and d = 3
for the 89 inequalities described in Section II A. The
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 alg orithm to tell whether α
max
< α (if
violating measurements are found) or α
max
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 α
max
by binary s e arch. In reality,
the hill-climbing search sometimes fails to find the global
optimum, and if it finds violating measurements then it
surely means α
max
< α, whereas if it does not find vi-
olating measurements then it doe s not necessarily mean
α
max
α. Therefore, the value given by binary search
is not necessarily the true value of α
max
but an upper
bound on it.
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 L APACK [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 viola te the inequality.
Therefore, we only consider violation greater than 10
13
significant.
For d = 2, the computation gave a n upper bound
0.70711 for all inequalities e xcept for the positive proba-
bility inequality. (For the positive probability inequality
we have α
max
= 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-le vel system.
We performed the same computation fo r d = 3 . This
time some Bell inequalities gave a smaller value of α
max
than the CHSH inequality did. Some of them gave a
small value of α
max
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 a dded by triangular elimination.
Page 5
6
TABLE I: Upper bound of the value of α
max
obtained by th e experiments.
α
max
Bell inequality O riginal cut polytope inequality
0.7447198434 A28 7
0.7453308276 A27 6
0.7553800191 A5 8 (Par(7), parachute ineq.)
0.7557816805 A56 89
0.7614396336 A8 2 (Pentagonal ineq.)
0.7629742793 A3 (I
3322
) 2 (Pentagonal ineq.)
0.7629742793 A2 (CHSH) 1 (Triangle ineq.)
1 A1 (Positive probability) 1 (Triangle ineq.)
A28:
(A
1
) (A
2
) (A
3
) (A
4
) (A
5
)
2 1 1 0 0
(B
1
) 2 1 0 1 1 1
(B
2
) 1
0 1 1 1 1
(B
3
) 1
1 1 1 0 0
(B
4
) 0
1 1 0 1 0
(B
5
) 0
1 1 0 0 0
0,
A27:
(A
1
) (A
2
) (A
3
) (A
4
) (A
5
)
1 0 0 1 1
(B
1
) 2 1 1 1 0 0
(B
2
) 0
1 0 1 1 1
(B
3
) 1
0 1 1 1 1
(B
4
) 1
1 1 0 1 0
(B
5
) 1
1 0 0 1 0
0,
A5:
(A
1
) (A
2
) (A
3
) (A
4
)
0 0 1 1
(B
1
) 2 1 1 1 0
(B
2
) 1
1 1 0 1
(B
3
) 1
1 1 1 1
(B
4
) 0
0 1 1 0
0,
A56:
(A
1
) (A
2
) (A
3
) (A
4
) (A
5
)
1 0 0 2 2
(B
1
) 1 0 1 1 1 0
(B
2
) 0
1 0 1 1 0
(B
3
) 0
1 1 1 1 2
(B
4
) 2
1 1 1 1 2
(B
5
) 2
0 0 2 2 0
0,
A8:
(A
1
) (A
2
) (A
3
) (A
4
)
0 1 2 0
(B
1
) 1 1 1 1 1
(B
2
) 2
1 1 1 1
(B
3
) 0 1 1 0 0
(B
4
) 0 1 0 1 0
(B
5
) 0 0 1 1 0
0.
Adding the CHSH and the I
3322
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 fa c e t inequality of CUT
9
to which triangula r elimination is applied. The number
correspo nds 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 IV B. The
I
3322
inequality gave the sa me upper bound as the CHSH
inequality. Besides, in the optimal measurements with α
near 4/(3
2 + 1), the matrices E
3
and F
1
are zero, cor -
responding to the fact that the I
3322
inequality includes
the CHSH inequality. This is consistent with Collins and
Gisin’s observation [7] in the 2 2 system that the I
3322
inequality is not better than the CHSH inequality for
states with high symmetry.
Five Bell inequalities A2 8, A27 , A5, A56 and A8 gave
a smaller value of α
max
than 4/(3
2 + 1). The set of
measurements giving optimal violation for these Bell in-
equalities with α slightly larger than the computed va lue
of α
max
is 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 α
max
for the I
3322
inequality
is 4/(3
2 + 1), then these five Bell inequalities are also
relevant to the I
3322
inequality. We ma ke the following
conjecture.
Conjecture 1. The state ρ
3
(4/(3
2 + 1)) satisfies the
I
3322
inequality for all measurements. In other words,
α
max
= 4/(3
2 + 1) for the I
3322
inequality in the case
of d = 3 .
To support this conjecture, we searched for the opti-
mal measurements for the I
3322
inequality in the states
ρ
3
(α) with α = α
+
= 0.7629742794 > 4/(3
2 + 1) a nd
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 gr e ater
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 res ult
Page 6
7
can be seen as an evidence that the I
3322
inequality be -
haves differently in the state ρ
3
(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 sta tes . 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 s tates,
however, five Bell inequalities relevant to the CHSH in-
equality were found. The results of numerical experi-
ments were give n to support the conjecture that they are
also relevant for the I
3322
inequality.
The violation of the CHSH inequality by 3-level
isotropic states was shown by using T sirelson’s inequal-
ity. Cleve, Høyer, To ner and Watrous [17] generalize
Tsirelson’s inequality to Bell inequalities correspo nding
to “XOR games,” which do not depend on individual
variables q
i0
, q
0j
, q
ij
but only involves combinations in
the form x
ij
= q
i0
+ q
0j
2q
ij
. Unfortunately, the
I
3322
inequality is not such a n ineq uality, and we can-
not use the result there to prove the theore tical va lue
of α
max
for the I
3322
inequality. Among the five Bell
inequalities relevant to the CHSH inequality for 3-level
isotropic states, the inequality A8, which can be written
as
P
i=1,2
P
j=1,2,3
x
ij
+ x
13
x
23
+ x
14
x
34
+ x
25
x
35
+x
41
x
42
0, is the only one that corresponds to an
XOR game. An impor tant open pr oblem is to generalize
Cleve, Høyer, Toner and Watrous’s result to cover Bell
inequalities which do not correspond to XOR games .
Acknowledgments
The first author is supported by the Grant-in-Aid for
JSPS Fellows.
APPENDIX A: OPTIMAL MEASUREMENTS COMPUTED FOR EACH INEQUALITIES
A28:
E
1
= I |ϕ
11
ihϕ
11
|, |ϕ
11
i = 0.819512|0i + (0.181891 0.067213i)|1i + (0.239561 + 0.483124i)|2i,
E
2
= |ϕ
12
ihϕ
12
|, |ϕ
12
i = 0.391928|0i + (0.546808 0.330668i)|1i + (0.064601 + 0.658695i)|2i,
E
3
= |ϕ
13
ihϕ
13
|, |ϕ
13
i = 0.585206|0i + (0.266618 0.150612i)|1i + (0.721307 0.208519i)|2i,
E
4
= |ϕ
14
ihϕ
14
|, |ϕ
14
i = 0.696701|0i + (0.109760 + 0.562926i)|1i + (0.269399 0.336302i)|2i,
E
5
= I |ϕ
15
ihϕ
15
|, |ϕ
15
i = 0.745551|0i + (0.060720 0.038486i)|1i + (0.610743 + 0.256863i)|2i,
F
1
= I |ϕ
21
ihϕ
21
|, |ϕ
21
i = 0.665942|0i + (0.124951 + 0.288249i)|1i + (0.306094 0.603430i)|2i,
F
2
= I |ϕ
22
ihϕ
22
|, |ϕ
22
i = 0.794583|0i + (0.503910 0.071325i)|1i + (0.075809 0.322300i)|2i,
F
3
= I |ϕ
23
ihϕ
23
|, |ϕ
23
i = 0.738612|0i + (0.143632 0.211840i)|1i + (0.594179 + 0.189467i)|2i,
F
4
= |ϕ
24
ihϕ
24
|, |ϕ
24
i = 0.314299|0i + (0.087381 + 0.592536i)|1i + (0.427166 + 0.600009i)|2i,
F
5
= I |ϕ
25
ihϕ
25
|, |ϕ
25
i = 0.745551|0i + (0.060720 + 0.038486i)|1i + (0.610743 0.256863i)|2i
A27:
E
1
= |ϕ
11
ihϕ
11
|, |ϕ
11
i = 0.512740|0i + (0.141298 0.367921i)|1i + (0.118341 0.753500i)|2i,
E
2
= I |ϕ
12
ihϕ
12
|, |ϕ
12
i = 0.429346|0i + (0.490358 + 0.190555i)|1i + (0.588595 0.438697i)|2i,
E
3
= I |ϕ
13
ihϕ
13
|, |ϕ
13
i = 0.649098|0i + (0.034498 + 0.390106i)|1i + (0.648622 + 0.067734i)|2i,
E
4
= |ϕ
14
ihϕ
14
|, |ϕ
14
i = 0.782874|0i + (0.199336 0.104823i)|1i + (0.579621 + 0.020651i)|2i,
E
5
= |ϕ
15
ihϕ
15
|, |ϕ
15
i = 0.504711|0i + (0.266955 0.029362i)|1i + (0.176172 0.801313i)|2i,
F
1
= |ϕ
21
ihϕ
21
|, |ϕ
21
i = 0.477430|0i + (0.243408 + 0.631106i)|1i + (0.024181 + 0.560297i)|2i,
F
2
= |ϕ
22
ihϕ
22
|, |ϕ
22
i = 0.521997|0i + (0.270933 0.132987i)|1i + (0.586914 + 0.540334i)|2i,
F
3
= |ϕ
23
ihϕ
23
|, |ϕ
23
i = 0.631718|0i + (0.176373 + 0.079451i)|1i + (0.678537 + 0.321093i)|2i,
F
4
= |ϕ
24
ihϕ
24
|, |ϕ
24
i = 0.839814|0i + (0.361305 0.101706i)|1i + (0.207777 0.332648i)|2i,
F
5
= |ϕ
25
ihϕ
25
|, |ϕ
25
i = 0.634648|0i + (0.135288 + 0.308277i)|1i + (0.492423 + 0.491328i)|2i
A5:
E
1
= I |ϕ
11
ihϕ
11
|, |ϕ
11
i = 0.079911|0i + (0.347597 0.352563i)|1i + (0.852394 + 0.148034i)|2i,
E
2
= |ϕ
12
ihϕ
12
|, |ϕ
12
i = 0.466812|0i + (0.336458 0.338316i)|1i + (0.063365 0.741896i)|2i,
E
3
= I |ϕ
13
ihϕ
13
|, |ϕ
13
i = 0.700997|0i + (0.090375 + 0.325520i)|1i + (0.625759 0.053829i)|2i,
E
4
= |ϕ
14
ihϕ
14
|, |ϕ
14
i = 0.569742|0i + (0.703808 0.061209i)|1i + (0.405767 0.107957i)|2i,
F
1
= |ϕ
21
ihϕ
21
|, |ϕ
21
i = 0.611974|0i + (0.261472 + 0.553836i)|1i + (0.402289 + 0.297574i)|2i,
F
2
= |ϕ
22
ihϕ
22
|, |ϕ
22
i = 0.743739|0i + (0.644052 0.121119i)|1i + (0.050055 0.121959i)|2i,
F
3
= |ϕ
23
ihϕ
23
|, |ϕ
23
i = 0.327181|0i + (0.492820 + 0.363796i)|1i + (0.442899 + 0.567075i)|2i,
F
4
= I |ϕ
24
ihϕ
24
|, |ϕ
24
i = 0.558366|0i + (0.295353 0.157594i)|1i + (0.593099 + 0.473699i)|2i
Page 7
8
A56:
E
1
= |ϕ
11
ihϕ
11
|, |ϕ
11
i = 0.764669|0i + (0.520735 0.023147i)|1i + (0.314448 0.211429i)|2i,
E
2
= I |ϕ
12
ihϕ
12
|, |ϕ
12
i = 0.523087|0i + (0.660068 + 0.130414i)|1i + (0.115043 + 0.510340i)|2i,
E
3
= I |ϕ
13
ihϕ
13
|, |ϕ
13
i = 0.651881|0i + (0.010176 0.025750i)|1i + (0.599260 + 0.463866i)|2i,
E
4
= I |ϕ
14
ihϕ
14
|, |ϕ
14
i = 0.480244|0i + (0.435821 0.476742i)|1i + (0.370530 + 0.463520i)|2i,
E
5
= I |ϕ
15
ihϕ
15
|, |ϕ
15
i = 0.484893|0i + (0.214118 + 0.403736i)|1i + (0.401826 + 0.628144i)|2i,
F
1
= |ϕ
21
ihϕ
21
|, |ϕ
21
i = 0.704822|0i + (0.050276 0.044858i)|1i + (0.676460 + 0.202702i)|2i,
F
2
= I |ϕ
22
ihϕ
22
|, |ϕ
22
i = 0.279921|0i + (0.406294 + 0.685472i)|1i + (0.534341 + 0.034308i)|2i,
F
3
= I |ϕ
23
ihϕ
23
|, |ϕ
23
i = 0.580814|0i + (0.563163 + 0.064963i)|1i + (0.561359 0.161735i)|2i,
F
4
= I |ϕ
24
ihϕ
24
|, |ϕ
24
i = 0.522791|0i + (0.366663 0.240466i)|1i + (0.161766 0.712921i)|2i,
F
5
= I |ϕ
25
ihϕ
25
|, |ϕ
25
i = 0.575083|0i + (0.352241 + 0.118045i)|1i + (0.170766 0.708598i)|2i
A8:
E
1
= |ϕ
11
ihϕ
11
|, |ϕ
11
i = 0.589845|0i + (0.252414 0.592962i)|1i + (0.067286 + 0.481911i)|2i,
E
2
= |ϕ
12
ihϕ
12
|, |ϕ
12
i = 0.571429|0i + (0.328221 0.214531i)|1i + (0.352103 + 0.629079i)|2i,
E
3
= I |ϕ
13
ihϕ
13
|, |ϕ
13
i = 0.789596|0i + (0.397845 + 0.124284i)|1i + (0.373987 + 0.250887i)|2i,
E
4
= I |ϕ
14
ihϕ
14
|, |ϕ
14
i = 0.588353|0i + (0.068306 0.217513i)|1i + (0.748446 + 0.204184i)|2i,
F
1
= |ϕ
21
ihϕ
21
|, |ϕ
21
i = 0.500028|0i + (0.062398 + 0.498087i)|1i + (0.351826 0.611724i)|2i,
F
2
= |ϕ
22
ihϕ
22
|, |ϕ
22
i = 0.416357|0i + (0.421270 + 0.580072i)|1i + (0.375055 0.414762i)|2i,
F
3
= |ϕ
23
ihϕ
23
|, |ϕ
23
i = 0.555120|0i + (0.275989 0.322007i)|1i + (0.606921 0.378986i)|2i,
F
4
= I |ϕ
24
ihϕ
24
|, |ϕ
24
i = 0.771642|0i + (0.389862 + 0.263652i)|1i + (0.160470 0.396628i)|2i,
F
5
= I |ϕ
25
ihϕ
25
|, |ϕ
25
i = 0.759855|0i + (0.022187 + 0.015402i)|1i + (0.430543 0.486336i)|2i
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9
FIG. 1: Inclusion relation among 89 Bell inequalities, with at most 5 measurements per party, obtained by triangular elimination from facets of CUT
9
. An asterisk (*)
on the right of the serial number ind icates that the inequ ality is relevant to the CHSH inequality for 3 3 isotropic states and that it is a candidate for being relevant
to I
3322
.
Page 9
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    • "The optimal values were calculated by the method of Avis, Imai and Ito [2] using the software SDPA [14] for a set of 89 Bell inequalities [19], which includes two quantum correlation functions: A2 (the CHSH inequality) and A8 (the I 3322 inequality [8]), and the other 87 quantum correlation vectors. The results are reproduced here asTable 1. "
    [Show abstract] [Hide abstract] ABSTRACT: The first part of this paper contains an introduction to Bell inequalities and Tsirelson's theorem for the non-specialist. The next part gives an explicit optimum construction for the "hard" part of Tsirelson's theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson's theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs. Copyright © 2009 The Institute of Electronics, Information and Communication Engineers.
    Full-text · Article · May 2009 · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
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    [Show abstract] [Hide abstract] ABSTRACT: In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. Phys. A: Math. Gen. 38 10971-87) with respect to Bell inequalities. We show that several well known bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329-45) to represent hidden deterministic behaviors, quantum behaviors, and no-signalling behaviors. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the no-signalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviors. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with (m,n) dichotomic observables when m=4,n=4 and when min{m,n}<=3, and give a new general family of correlation inequalities. Comment: 17 pages, 2 figures
    Full-text · Article · May 2006 · Journal of Physics A General Physics
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    [Show abstract] [Hide abstract] ABSTRACT: Bell inequality violation is one of the most widely known manifestations of entanglement in quantum mechanics; indicating that experiments on physically separated quantum mechanical systems cannot be given a local realistic description. However, despite the importance of Bell inequalities, it is not known in general how to determine whether a given entangled state will violate a Bell inequality. This is because one can choose to make many different measurements on a quantum system to test any given Bell inequality and the optimization over measurements is a high-dimensional variational problem. In order to better understand this problem we present algorithms that provide, for a given quantum state, both a lower bound and an upper bound on the maximal expectation value of a Bell operator. Both bounds apply techniques from convex optimization and the methodology for creating upper bounds allows them to be systematically improved. In many cases these bounds determine measurements that would demonstrate violation of the Bell inequality or provide a bound that rules out the possibility of a violation. Examples are given to illustrate how these algorithms can be used to conclude definitively if some quantum states violate a given Bell inequality.
    Full-text · Article · Apr 2007 · Physical Review A
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