Article
Reference
Bell Inequalities for Arbitrarily High-Dimensional Systems
COLLINS, Daniel Geoffrey, et al.
Abstract
We develop a novel approach to Bell inequalities based on a constraint that the correlations
exhibited by local variable theories must satisfy. This is used to construct a family of Bell
inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly
resistant to noise. In particular, our work gives an analytic description of previous numerical
results and generalizes them to arbitrarily high dimensionality.
COLLINS, Daniel Geoffrey, et al. Bell Inequalities for Arbitrarily High-Dimensional Systems.
Physical review letters, 2002, vol. 88, no. 4
DOI : 10.1103/PhysRevLett.88.040404
Available at:
http://archive-ouverte.unige.ch/unige:36815
Disclaimer: layout of this document may differ from the published version.
1 / 1
VOLUME 88, N
UMBER 4 PHYSICAL REVIEW LETTERS 28J
ANUARY 2002
Bell Inequalities for Arbitrarily High-Dimensional Systems
Daniel Collins,
1,2
Nicolas Gisin,
3
Noah Linden,
4
Serge Massar,
5
and Sandu Popescu
1,2
1
H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
2
BRIMS, Hewlett-Packard Laboratories, Stoke Gifford, Bristol BS12 6QZ, United Kingdom
3
Group of Applied Physics, University of Geneva, 20, rue de l’Ecole-de-Médecine, CH-1211 Geneva 4, Switzerland
4
Department of Mathematics, Bristol University, University Walk, Bristol BS8 1TW, United Kingdom
5
Service de Physique Théorique, Université Libre de Bruxelles, CP 225, Boulevard du Triomphe, B1050 Bruxelles, Belgium
(Received 23 July 2001; published 10 January 2002)
We develop a novel approach to Bell inequalities based on a constraint that the correlations exhibited
by local variable theories must satisfy. This is used to construct a family of Bell inequalities for bipartite
quantum systems of arbitrarily high dimensionality which are strongly resistant to noise. In particular,
our work gives an analytic description of previous numerical results and generalizes them to arbitrarily
high dimensionality.
DOI: 10.1103/PhysRevLett.88.040404 PACS numbers: 03.65.Ud, 03.67.–a
One of the most remarkable aspects of quantum mechan-
ics is its predicted correlations. Indeed, the correlations be-
tween outcomes of measurements performed on systems
composed of several parts in an entangled state have no
classical analog. The most striking aspect of this character-
istic feature of quantum physics is revealed when the parts
are spatially separated: no classical theory based on local
variables can reproduce the quantum correlations. Histori-
cally, this became known as the Einstein-Podolsky-Rosen
paradox and was formulated in terms of measurable quan-
tities by Bell [1] and by Clauser, Horne, Shimony, and
Holt [2] as the nowadays famous inequalities. Other as-
pects of quantum correlation were analyzed in the form
of paradoxes, such as Schrödinger’s cat and the measure-
ment problem. In recent years, these paradoxical aspects
have been overthrown by a more effective approach:
let
us exploit “quantum strangeness” to perform tasks that
are classically impossible has become the new leitmotiv.
From this “conceptual revolution,” the field of quantum in-
formation emerged. Old words became fashionable, such
as “entanglement.” Old questions were revisited, such as
the classifications of quantum correlations.
The variety of known partial results, in particular, about
entanglement measures, makes it today obvious that there
is no one-parameter classification of entanglement. This
Letter concerns classifications related to what is called
“quantum nonlocality,” i.e., the impossibility to reproduce
quantum correlations with theories based on local variables
(often called “local realistic theories”). Specifically we de-
velop a powerful new approach to Bell inequalities which
we then use to write several families of Bell inequalities
for higher-dimensional systems.
Local variable theories cannot exhibit arbitrary correla-
tions. Rather the conditions these correlations must obey
can always be written as inequalities (the Bell inequalities)
which the joint probabilities of outcomes must satisfy. Our
approach to Bell inequalities is based on a logical con-
straint the correlations must satisfy in the case of local
variable theories. In order to introduce this constraint, let
us suppose that one of the parties, Alice, can carry out
two possible measurements,
A
1
or A
2
, and that the other
party, Bob, can carry out two possible measurements, B
1
or B
2
. Each measurement may have d possible outcomes:
A
1
, A
2
, B
1
, B
2
苷 0,...,d 2 1. Without loss of general-
ity a local variable theory can be described by d
4
proba-
bilities c
jklm
共 j, k, l, m 苷 0,...,d 2 1兲 that Alice’s local
variable 共 jk兲 specifies that measurement A
1
gives outcome
j and measurement A
2
gives outcome k and that Bob’s
local variable 共lm兲 specifies that measurement B
1
gives
outcome l and measurement B
2
gives outcome m.(In
this formulation Alice and Bob’s strategy is determinis-
tic since it is completely determined by the value of their
variables jk and lm. Any nondeterministic local theory
can be rephrased in the above way by incorporating the lo-
cal randomness in the probabilities c
jklm
; see, for instance,
[3].) Since the probabilities c
jklm
are positive 共c
jklm
$ 0兲
and sum to one 共
P
jklm
c
jklm
苷 1兲. The joint probabili-
ties take the form P共A
1
苷 j, B
1
苷 l兲 苷
P
km
c
jklm
, and
similarly for P共A
1
苷 j, B
2
苷 m兲, P共A
2
苷 k, B
1
苷 l兲 and
P共A
2
苷 k, B
2
苷 m兲.
Let us consider a particular choice of local variables
jklm (this choice occurs with probability c
jklm
). Since
A
1
苷 j, A
2
苷 k, B
1
苷 l, B
2
苷 m we have
r
0
⬅ B
1
2 A
1
苷 l 2 j ,
s
0
⬅ A
2
2 B
1
苷 k 2 l ,
t
0
⬅ B
2
2 A
2
苷 m 2 k ,
(1)
u
0
⬅ A
1
2 B
2
苷 j 2 m .
We see that the difference, r
0
, between A
1
and B
1
can be
freely chosen by choosing j and l. Similarly the difference,
s
0
, between B
1
and A
2
and the difference, t
0
, between A
2
and B
2
can be freely chosen. But then the difference u
0
between B
2
and A
1
is constrained since we necessarily
have
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r
0
1 s
0
1 t
0
1 u
0
苷 0. (2)
Thus in a local variable theory the relation between three
pairs of operators can be freely chosen, but then the last
relation is constrained.
This constraint plays a central role in our Bell inequali-
ties. Indeed they are written in such a way that their
maximum value can be attained only if this constraint is
frustrated. The simplest such Bell expression is
I ⬅ P共A
1
苷 B
1
兲 1 P共B
1
苷 A
2
1 1兲
1 P共A
2
苷 B
2
兲 1 P共B
2
苷 A
1
兲 , (3)
where we have introduced the probability P共A
a
苷 B
b
1
k兲 that the measurements A
a
and B
b
have outcomes that
differ, modulo d,byk:
P共A
a
苷 B
b
1 k兲⬅
d21
X
j苷0
P共A
a
苷 j, B
b
苷 j 1 k modd兲 .
(4)
Because the difference between A
a
and B
b
is evaluated
modulo d, all the outcomes of A
a
and B
b
are treated on
an equal footing. As we see in Eq. (3) this symmetriza-
tion is the key to reducing Bell inequalities to the logical
constraint that is imposed by local variable theories. In-
deed because of the constraint Eq. (2) any choice of lo-
cal variables jklm can satisfy only three of the relations
appearing in Eq. (3), e.g., A
1
苷 B
1
, B
1
苷 A
2
1 1, etc.
Hence I共local realism兲 # 3. On the other hand, nonlocal
correlations can attain I 苷 4 since they can satisfy all four
relations.
In the case of two-dimensional systems the inequality
I共local variable兲 # 3 is equivalent to the Clauser-Horne-
Shimony-Holt (CHSH) inequality [2]. But the power of
our reformulation is already apparent since this inequal-
ity generalizes the CHSH inequality to arbitrarily large di-
mensions. In fact, the above formulation of the constraint
imposed by local realistic theories allows one to write in
a unified way all previously known Bell inequalities [4].
It can also serve to write completely new Bell inequalities
and this is the subject of the present Letter. Specifically we
have generalized in a nontrivial way [see Eqs. (5) and (6)
below] the Bell expression (3) to d-dimensional systems
(for any d $ 2).
One of the interests of these new Bell expressions is that
they are highly resistant to noise. Indeed Bell inequalities
are sensitive to the presence of noise and above a certain
amount of noise the Bell inequalities will cease to be vio-
lated by a quantum system. However, it has been shown by
numerical optimization [5] that using higher-dimensional
systems can increase the resistance to noise. The measure-
ments that are carried out on the quantum system in order
to obtain an increased violation have been described an-
alytically in [6]. And an analytical proof of the greater
robustness of quantum systems of dimension 3 was given
in [7]. One of the interests of our new Bell inequalities is
that when we apply them to the quantum state and mea-
surement described in [6] for those dimensions 共d # 16兲
for which a numerical optimization was carried out in [6],
we obtain the same resistance to noise as in [6].
The first generalization of the Bell expression Eq. (3) is
I
3
⬅ 1关P共A
1
苷 B
1
兲 1 P共B
1
苷 A
2
1 1兲
1 P共A
2
苷 B
2
兲 1 P共B
2
苷 A
1
兲兴
2 关P共A
1
苷 B
1
2 1兲 1 P共B
1
苷 A
2
兲
1 P共A
2
苷 B
2
2 1兲 1 P共B
2
苷 A
1
2 1兲兴 . (5)
The maximum value of I
3
for nonlocal theories is 4 since
a nonlocal theory could satisfy all four relations that have
a 1 sign in (5). On the other hand, for a local variable
theory I
3
# 2. This should be compared to the constraint
I共local variable兲 # 3 for the expression (3). The origin of
this difference is the 2 signs in (5). Indeed we have seen
when analyzing (3) that only three of the relations with a 1
sign can be satisfied by local realistic theories. But if three
relations with 1 are satisfied in (5), then necessarily one
relation with 2 is also satisfied giving a total of I
3
苷 2.
Alternatively one can satisfy two relations with 1 and two
relations with weight zero (if the dimension is larger than
2), once more giving a total of I
3
苷 2.
For d 苷 2 the inequality I
3
共local variable兲 # 2 is
equivalent to the inequality I共local variable兲 # 3 and
therefore to the CHSH inequality. But for d $ 3 the
inequality based on I
3
is not equivalent to that based on
I. For the quantum measurement described below (when
d $ 3) the inequality based on I
3
(and its generalizations
I
d
given below) is more robust than that based on I.
The Bell expression I
3
can be further generalized when
the dimensionality is greater than 3 by adding extra terms.
The extra terms in I
d
do not change the maximum value at-
tainable by local variable theories 关I
max
d
共local variable兲 苷
2兴, nor do they change the maximum value attainable by
completely nonlocal theories 共I
max
d
苷 4兲. However, these
extra terms allow a better exploitation of the correlations
exhibited by quantum systems.
These new Bell expressions have the form
I
d
⬅
关d兾2兴21
X
k苷0
µ
1 2
2k
d 2 1
∂
兵1关P共A
1
苷 B
1
1 k兲 1 P共B
1
苷 A
2
1 k 1 1兲 1 P共A
2
苷 B
2
1 k兲 1 P共B
2
苷 A
1
1 k兲兴
2 关P共A
1
苷 B
1
2 k 2 1兲 1 P共B
1
苷 A
2
2 k兲 1 P共A
2
苷 B
2
2 k 2 1兲
1 P共B
2
苷 A
1
2 k 2 1兲兴其 . (6)
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As mentioned above the maximum value of I
d
is 4. This
follows immediately from the fact that the maximum
weight of the terms in (6) is 11. And the maximum value
of I
d
for local variable theories is 2. We now prove this
last result.
The proof consists of enumerating all the possible rela-
tions between A
1
, B
1
, A
2
, B
2
allowed by the constraints
(2). This is most easily done by first changing notation.
We do not use the coefficients r
0
, s
0
, t
0
, u
0
defined in (1),
but we use new coefficients r, s, t, u defined by the rela-
tion
A
1
苷 B
1
1 r, B
1
苷 A
2
1 s 1 1,
A
2
苷 B
2
1 t, B
2
苷 A
1
1 u ,
(7)
which obey the constraint
r 1 s 1 t 1 u 1 1 苷 0 modd . (8)
Furthermore we restrict (without loss of generality) r, s,
t, u to lie in the interval
2关d兾2兴 # r, s, t, u # 关共d 2 1兲兾2兴 . (9)
With this notation the value of the Bell inequality for a
given choice of r, s, t, u is
I
d
共r, s, t, u兲 苷 f共r兲 1 f共s兲 1 f共t兲 1 f共u兲 , (10)
where f is given by
f共x兲 苷
(
2
2x
d21
1 1, x $ 0,
2
2x
d21
2
d11
d21
, x , 0.
(11)
We now consider different cases according to the signs of
r, s, t, u.
1. r, s, t, u are all positive. Then (8) and (9) imply that
r 1 s 1 t 1 u 苷 d 2 1. Inserting into (10) and using
(11) one finds I
d
苷 2.
2. Three of the numbers r, s, t, u are positive, one is
strictly negative. Then (8) and (9) imply that either r 1
s 1 t 1 u 苷 d 2 1 or r 1 s 1 t 1 u 苷 21. Inserting
into (10) and using (11) one finds either I
d
苷 22兾共d 2 1兲
or I
d
苷 2.
3. Two of the numbers r, s, t, u are positive, two are
strictly negative. Then (8) and (9) imply that r 1 s 1
t 1 u 苷 21. Inserting into (10) and using (11) one finds
I
d
苷 22兾共d 2 1兲.
4. One of the numbers r, s, t, u is positive, three
are strictly negative. Then (8) and (9) imply that ei-
ther r 1 s 1 t 1 u 苷 21 or r 1 s 1 t 1 u 苷 2d 2
1. Inserting into (10) and using (11) one finds either
I
d
苷 22共d 1 1兲兾共d 2 1兲 or I
d
苷 22兾共d 2 1兲.
5. The numbers r, s, t, u are all strictly negative. Then
(8) and (9) imply that r 1 s 1 t 1 u 苷 2d 2 1. In-
serting into (10) and using (11) one finds I
d
苷 22共d 1
1兲兾共d 2 1兲.
(Note that for small dimensions d not all the possibilities
enumerated above can occur. For instance, for d 苷 2, the
only possible values are I
d
苷 62.) Thus for all possible
choices of r, s, t, u, I
d
共local realism兲 # 2. This concludes
the proof.
Let us now consider the maximum value that can be
attained for the Bell expressions I
d
for quantum measure-
ments on an entangled quantum state. We have carried out
a numerical search for the optimal measurements. It turns
out that the best measurements that we have found numeri-
cally give the same value as the measurements described
in [6]. We do not have a proof that these measurements are
optimal, but our numerical work and the numerical work
that inspired [6] suggests that this is the case.
We therefore first recall the state and the measurement
described in [6]. The quantum state is the maximally en-
tangled state of two d-dimensional systems
c 苷
1
p
d
d21
X
j苷0
jj典
A
≠jj典
B
. (12)
Let the operators A
a
, a 苷 1, 2, measured by Alice and
B
b
, b 苷 1, 2, measured by Bob have the nondegenerate
eigenvectors
jk典
A,a
苷
1
p
d
d21
X
j苷0
exp
µ
i
2p
d
j共k 1a
a
兲
∂
jj典
A
,
jl典
B,b
苷
1
p
d
d21
X
j苷0
exp
µ
i
2p
d
j共2l 1b
b
兲
∂
jj典
B
,
(13)
where a
1
苷 0, a
2
苷 1兾2, b
1
苷 1兾4, and b
2
苷 21兾4.
These measurements can be viewed as being carried out in
two steps: First Alice and Bob give each of the states jj典
A
and jj典
B
a variable phase depending on the measurement
they want to perform; then Alice measures in the Fourier
transform basis and Bob in the inverse Fourier basis. Thus
the joint probabilities are
P
QM
共A
a
苷 k, B
b
苷 l兲 苷
1
d
3
É
d21
X
j苷0
exp
∑
i
2pj
d
共k 2 l 1a
a
1b
b
兲
∏
É
2
苷
1
d
3
sin
2
关p共k 2 l 1a
a
1b
b
兲兴
sin
2
关p共k 2 l 1a
a
1b
b
兲兾d兴
苷
1
2d
3
sin
2
关p共k 2 l 1a
a
1b
b
兲兾d兴
, (14)
where in the last line we have used the values of a
a
and b
b
given above.
Equation (14) shows that these joint probabilities have several symmetries. First of all we have the relation
P
QM
共A
a
苷 k, B
b
苷 l兲 苷 P
QM
共A
a
苷 k 1 c, B
b
苷 l 1 c兲
for all integers c. This symmetry property justifies us considering, as in (4), only the probabilities that A
a
and B
b
differ
by a given constant integer c:
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ANUARY 2002
P
QM
共A
a
苷 B
b
1 c兲 苷
d21
X
j苷0
P
QM
共A
a
苷 j 1 c, B
b
苷 j兲
苷 dP
QM
共A
a
苷 c, B
b
苷 0兲 . (15)
Furthermore we have the relation
P
QM
共A
1
苷 B
1
1 c兲 苷 P
QM
共B
1
苷 A
2
1 c 1 1兲
苷 P
QM
共A
2
苷 B
2
1 c兲
苷 P
QM
共B
2
苷 A
1
1 c兲 . (16)
Using Eqs. (14)–(16) we can rank these probabilities by
decreasing order. Let us denote
q
c
苷 P
QM
共A
1
苷 B
1
1 c兲
苷 1兾兵2d
3
sin
2
关p共c 1 1兾4兲兾d兴其 . (17)
Then we have
q
0
. q
21
. q
1
. q
22
. q
2
. ··· . q
2
关d兾2兴
共.q
关d兾2兴
兲 ,
(18)
where 关x兴 denotes the integer part of x and the last term be-
tween parentheses occurs only for odd dimension d. This
suggests that the quantum probabilities violate the con-
straints imposed by local variable theories. Indeed the
probabilities in (16) are maximized by taking c 苷 0,but
then the four relations that appear in (16) are incompatible
with local realism. In fact, replacing the above probabili-
ties in the expression (3) yields a value I
QM
苷 4dq
0
. 3
for all dimensions d.
However, a stronger violation is obtained if instead of
using the Bell expression I, one uses the Bell expressions
I
d
. In fact, for a d-dimensional quantum system, one can
use all the Bell expressions I
k
for k # d, but the strongest
violation is obtained by using the Bell expression I
d
. This
value, denoted I
d
共QM兲, is given by
I
d
共QM兲 苷 4d
关d兾2兴21
X
k苷0
µ
1 2
2k
d 2 1
∂
共q
k
2 q
2共k11兲
兲 .
(19)
For instance, we find
I
3
共QM兲 苷 4兾共29 1 6
p
3 兲⯝2.872 93 ,
I
4
共QM兲 苷
2
3
共
p
2 1
q
10 2
p
2 兲⯝2.896 24 ,
lim
d!`
I
d
共QM兲 苷
2
p
2
`
X
k苷0
1
共k 1 1兾4兲
2
2
1
共k 1 3兾4兲
2
苷 32 3 Catalan兾p
2
⯝ 2.6981 ,
where Catalan ⯝ 0.9159 is Catalan’s constant.
In the presence of uncolored noise the quantum state
becomes
r 苷 pjc典具cj 1 共1 2 p兲
'
d
2
, (20)
where p is the probability that the state is unaffected by
noise. The value of the Bell inequality for the state r is
I
d
共r兲 苷 pI
d
共QM兲 . (21)
Hence the Bell inequality I
d
is certainly violated if
p .
2
I
d
共QM兲
苷 p
min
d
. (22)
(If there is a quantum measurement giving a value of I
d
greater than that given by Eq. (19), then of course the Bell
inequality would be violated with even more noise. This
remark applies to the various p
min
below.)
As a function of d one finds that p
min
d
is a decreasing
function of d. For instance,
p
min
3
苷 共6
p
3 2 9兲兾2 ⯝ 0.696 15 ,
p
min
4
苷 3兾共
p
2 1
q
10 2
p
2 兲⯝0.690 55 ,
lim
d!`
p
min
d
共d兲 苷 p
2
兾共16 3 Catalan兲⯝0.673 44 .
For d 苷 3 this reproduces the analytical result of [7]. And
combining Eqs. (19) and (22) reproduces the numerical
results of [6] for all dimensions 共2 # d # 16兲 for which
a numerical optimization was carried out.
In summary, our reformulation of Bell inequalities in
terms of a logical constraint local variable theories must
satisfy has provided the basis for constructing a large fam-
ily of Bell inequalities for systems of large dimension. The
numerical work of [5,6] and a numerical search of our
own suggest that these Bell inequalities are optimal in the
same sense that the CHSH inequality is optimal for two-
dimensional systems, namely, both the resistance to noise
and the amount by which the inequality is violated are
maximal. For this reason we hope that the Bell inequali-
ties presented here will have as much interest for physicists
studying entanglement of systems of large dimensional-
ity as the CHSH inequalities have had for bidimensional
systems.
We acknowledge funding by the European Union under
project EQUIP (IST-FET program).
Note added.—While completing this Letter we learned
of a Bell inequality for qutrits [8] that exhibits the same
resistance to noise as that obtained in [5–7].
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