RAPID COMMUNICATIONS
PHYSICAL REVIEW A 72, 050302共R兲 共2005兲
Communication cost of breaking the Bell barrier
Karl Svozil*
Institut für Theoretische Physik, University of Technology Vienna, Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria
共Received 16 December 2004; published 14 November 2005兲
Correlations in an Einstein-Podolsky-Rosen-Bohm experiment can be made stronger than quantum correlations by allowing a single bit of classical communication between the two sides of the experiment.
DOI: 10.1103/PhysRevA.72.050302
PACS number共s兲: 03.67.Hk, 03.65.Ud, 03.65.Ta, 03.67.Mn
From an operational point of view, the nonlocal quantum
correlations giving rise to violations of Bell-type inequalities
amount to the fact that certain joint events at spacelike separated locations occur with greater or smaller frequencies than
can possibly be expected from classical, local realistic models. Two detectors at different locations register pairs of particles or particle properties more frequently or infrequently
as can be explained by the usual classical assumptions such
as value definiteness.
With the rise of quantum algorithms and quantuminformation theory 关1兴, the emphasis shifted to the communication cost and to the quantum communication complexity
related to those quantum correlations. The question of the
expense of obtaining quantum-type correlations from classical systems was stimulated by quantum 关2兴 and classical
关3–6兴 teleportation. In a recent Letter 关7兴, Toner and Bacon,
based on Refs. 关4,8兴, argue that classical systems could
mimic quantum systems by reproducing the cosine law for
correlation functions with the exchange of just one bit of
classical information.
The formal coincidence of the quantum correlation function with classical correlations augmented with the exchange
of a single classical bit might indicate a deep structure in
quantum correlations. One could, for instance, speculate that
two-partite quantum systems may be capable of conferring a
single bit, a property which is reflected by the cosine form of
the expectation function. In what follows it will be argued
that, while this may still be the case for the Toner-Bacon
protocol 关7兴, in general the exchange of a single classical bit
can give rise to stronger than quantum correlations.
Since the systems discussed are entirely planar, whenever
possible, polar angles are used to represent the associated
unit vectors. The same symbols denote polar angles 共without
hat兲 and the associated vectors 共with hat兲. Consider two correlated and spatially separated classical subsystems sharing
common directions ␭i, i = 1 , . . . which are chosen independently of each other and are distributed uniformly. All parameters ␭i are assumed to be identical on each one of the
two subsystems. There are two measurement directions a and
b of two dichotomic observables with values “−1” and “1” at
two spatially separated locations. The measurement direction
a at “Alice’s location” is unknown to an observer “Bob”
*Electronic address: svozil@tuwien.ac.at
1050-2947/2005/72共5兲/050302共3兲/$23.00
measuring b and vice versa. A two-particle correlation function E共␪兲 with ␪ = 兩a − b兩 is defined by averaging the product
of the outcomes O共a兲i, O共b兲i 苸 −1 , 1 in the ith experiment,
N
i.e., E共␪兲 = 共1 / N兲兺i=1
O共a兲iO共b兲i.
The following nonadaptive, memoryless protocols could
give rise to stronger-than-quantum correlations by allowing
the exchange of a single bit per experiment. The protocols
are similar to the one discussed by Toner and Bacon 关7兴, but
require only a single share ␭, and an additional direction
⌬共␦兲, which is obtained by rotating ␭ˆ clockwise around the
origin by an angle ␦ which is a constant shift for all experiments. That is, ⌬共␦兲 = ␭ + ␦. Alice’s observable is given
by ␣ = sgn共â · ␭ˆ 兲 = sgn关cos共a − ␭兲兴. The bit communicated
ˆ 共␦兲兴 = sgn关cos共a
by Alice is given by c共␦兲 = sgn共â · ␭ˆ 兲sgn关â · ⌬
− ␭兲兴sgn cos关a − ⌬共␦兲兴. Bob’s observable is defined by ␤共␦兲
= sgn兵b̂ · 关␭ˆ + c共␦兲⌬ˆ 共␦兲兴其. This protocol becomes Toner and
Bacon’s if ␦ is allowed to vary randomly, with uniform distribution.
The strongest correlations are obtained for ␦ = ␲ / 2, i.e., in
the case where the two directions associated with ␭ and ⌬
= ␭ + ␲ / 2 are orthogonal and the information obtained by
c共␲ / 2兲 is about the location of a within two opposite quadrants. Let H共x兲 stand for the Heaviside step function of x.
The effective shift in the parameter direction ␭ˆ → ␭ˆ ± ␭ˆ ⬜
yields a correlation function of the form
冉
E ␪, ␦ =
冊 冉
冊 冉 冊
冉 冊冉 冊冉
3␲
␲
␲
−␪
=H ␪−
−H
2
4
4
−2 1−
冊
2
3␲
␲
−␪ .
␪ H ␪−
H
␲
4
4
共1兲
To obtain a better understanding for the shift mechanism, in
Fig. 1 a configuration is drawn which, without the shift ␭ˆ
→ ␭ˆ − ␭ˆ ⬜, sgn共b̂ · ␭ˆ 兲 = sgn cos共b − ␭兲 would have contributed
the factor −1. The shift results in a positive contribution
sgn关b̂ · 共␭ˆ − ␭ˆ ⬜兲兴 to the expectation value. This shift mechanism always yields the strongest correlations ±1 as long as
the angle ␪ does not lie between ␲ / 4 and 3␲ / 4.
For general 0 艋 ␦ 艋 ␲ / 2, Fig. 2 depicts numerical evaluations which fit the correlation function
050302-1
©2005 The American Physical Society
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KARL SVOZIL
FIG. 1. Demonstration of the shift mechanism. Concentric
circles represent the measurement directions â and b̂ 共outer circle兲,
as well as parameter directions ␭ˆ and ␭ˆ ⬜ 共inner circle兲 and their
associated projective sign regions. The four measurement regions
spanned by â and b̂ are indicated by “±1,” respectively. Positive and
negative octants spanned by ␭ˆ and ␭ˆ ⬜ are indiated in the inner
circle by “⫾,” respectively. In this configuration, the shift ␭ˆ → ␭ˆ
− ␭ˆ ⬜ pushes ␭ˆ into a positive region.
E共␪,␦兲 =
冦
−1
冉 冊
冉 冊
冉 冊
2
␦
−1+
␪−
␲
2
−2 1−
2
␪
␲
2
␦
1+
␪ − ␲+
␲
2
1
for 0 艋 ␪ 艋
␦
2
␦
for
1
⬍ ␪ 艋 共␲ − ␦兲
2
2
for
1
1
共␲ − ␦兲 ⬍ ␪ 艋 共␲ + ␦兲
2
2
1
␦
for 共␲ + ␦兲 ⬍ ␪ 艋 ␲ −
2
2
for ␲ −
␦
2
⬍ ␪ 艋 ␲.
冧
FIG. 2. 共Color online兲 Classical and stronger-thanquantum correlation functions obtained through the exchange
of a single bit in the memoryless regime for values of ␦
苸 兵0 , ␲ / 10, ␲ / 5 , 3␲ / 10, 2␲ / 5 , ␲ / 2其 between 0 共straight line兲 and
␲ / 2 关cf. Eq 共1兲兴.
formly distributed shares ␭1 and ␭2, and a single communicated bit H关c共␭ˆ 1 , ␭ˆ 2兲兴 with c共␭ˆ 1 , ␭ˆ 2兲 = sgn共â · ␭ˆ 1兲sgn共â · ␭ˆ 2兲
per
measurement
of
= sgn cos共a − ␭1兲sgn cos共a − ␭2兲
ˆ
ˆ
ˆ
sgn共â · ␭1兲sgn关b̂ · 共␭2 − c␭1兲兴. The associated correlation function is
E共␪兲 =
=
1
2␲2
d␭ˆ 1d␭ˆ 2 sgn共â · ␭ˆ 1兲sgn关b̂ · 共␭ˆ 2 − c␭ˆ 1兲兴
d␭ˆ 1 sgn共â · ␭ˆ 1兲
冕
d␭ˆ 2 sgn关b̂ · 共␭ˆ 2 − ␭ˆ 1兲兴.
共3兲
共2兲
Its domains are depicted in Fig. 3. For all nonzero ␦, E共␪ , ␦兲
correlates stronger than quantized systems for some values of
␪. For ␦ = ␲ / 2, the Clauser-Horne-Shimony-Holt 共CHSH兲 inequality 兩E共a , b兲 + E共a , b⬘兲 + E共a⬘ , b兲 − E共a⬘ , b⬘兲兩 艋 2 for a = ␲,
a⬘ = 3␲ / 4, b = 0, b⬘ = ␲ / 4 is violated by 3, a larger value than
the Tsirelson bound for quantum violations 2冑2. This is due
to the fact that, phenomenologically, the strategy allows for
certain joint events to occur with greater or smaller frequencies as can be expected from quantum entangled state measurements. For ␦ = 0, the classical linear correlation function
E共␪兲 = 2␪ / ␲ − 1 is recovered, as can be expected.
The average over all 0 艋 ␦ 艋 ␲ / 2 yields a very similar, but
not identical, behavior as the quantum cosine correlation
function. More precisely, assume two independent, uni-
冕
冕
1
共2␲兲2
FIG. 3. 共Color online兲 Domains of the correlation function
E共␪ , ␦兲. Consecutive sections from left to right cover the domains
1
of Eq. 共2兲 from bottom to top: 共1兲 0 艋 ␪ 艋 ␦ / 2, 共2兲 ␦ / 2 ⬍ ␪ 艋 2 共␲
1
1
1
− ␦兲, 共3兲 2 共␲ − ␦兲 ⬍ ␪ 艋 2 共␲ + ␦兲, 共4兲 2 共␲ + ␦兲 ⬍ ␪ 艋 ␲ − ␦ / 2, 共5兲 ␲
− ␦ / 2 ⬍ ␪ 艋 ␲.
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COMMUNICATION COST OF BREAKING THE BELL BARRIER
The elemination of c at the cost of the prefactor 2 on the
right-hand side of Eq. 共3兲 is achieved by using the symmetries of the outcome under the exchange ␭ˆ 1 ↔ −␭ˆ 1 and
␭ˆ 2 ↔ −␭ˆ 2 as outlined in Ref. 关7兴. Note that, although the correlation function E共␪兲 is nonlocal 共Bob’s output depends on
c, which contains a兲, after some recasting, despite the prefactor of 2 which accounts for nonlocality, it appears to be
perfectly local, since it is the product of two sign functions
containing merely a and 共separately兲 b, respectively. 共The
two parameters ␭1 , ␭2 are common shares.兲
The ␭2 integration can be performed by arranging b̂ along
the positive y axis 共0, 1兲 and by the parametrization ␭ˆ 1
= 共sin t , cos t兲 and ␭ˆ = 共sin ␶ , cos ␶兲. The positive contribu2
tions amount to A+ = 2兰t0d␶ = 2t; thus the negative contributions are A− = 2␲ − A+, and the entire integral is 共A+
− A−兲 / 共2␲兲 = 2t / ␲ − 1 = 2 cos−1共b̂ · ␭ˆ 1兲 / ␲ − 1.
For the ␭1 integration, â is arranged along the positive y
axis 共0, 1兲, b̂ along 共sin r , cos r兲, and ␭ˆ 1 along 共sin ␶ , cos ␶兲.
Since cos−1关cos共x兲兴 = 兩x兩 for −1 艋 x 艋 1, one obtains for the
correlation value 共4 / ␲2兲兰0␲d␶ sgn共cos ␶兲兩␶ − r兩. After evaluating all cases, the ␭ˆ 1 integration, for 0 艋 ␪ = 兩a − b兩 艋 ␲, yields
FIG. 4. The correlation E共␪兲 of Eq. 共4兲 as a function of ␪ = 兩a
− b兩 for the memoryless exchange of a single bit per experiment in
the planar configuration. Note that, although the shape resembles
the quantum cosine law, the function is piecewise quadratic.
which is plotted in Fig. 4. An alternative derivation of Eq. 共4兲
is via 共2 / ␲兲兰0␲/2d␦ E共␪ , ␦兲 with E共␪ , ␦兲 from Eq. 共2兲. The
difference to the cosine law obtained by Toner and Bacon 关7兴
is due to the reduced dimensionality of the problem.
Let us briefly sketch a protocol requiring memory which,
by the exchange of more than one bit, could give rise to
maximal violations 关9兴 of the CHSH inequality. The protocol
is based on locating and communicating information about
Alice’s measurement direction to Bob, who then rotates his
subsystem 共or alternatively his measurement direction兲 so as
to obtain the desired correlation function.
When compared to the protocol discussed by Toner
and Bacon 关7兴 or to the memoryless protocol introduced
above, the adaptive protocols share some similarities. Both
exchange very similar information, as can, for instance,
be seen by a comparison between c2 above and the bit
c = sgn cos共a − ␭1兲sgn cos共a − ␭2兲 exchanged in the Toner and
Bacon protocol. After the exchange of just a few bits, the
adaptive protocols appear to be more efficient from the communication complexity point of view. However, these strategies require memory. Operationally, this presents no problem
for Alice and Bob, but if one insist on nonadaptive, single
particle strategies, these protocols must be excluded.
In summary we have found that, as long as adaptive protocols requiring memory at the receiver side are allowed, the
CHSH inequality can be violated maximally. Furthermore,
we have presented a type of memoryless, nonadaptive protocol giving rise to stronger-than-quantum correlations which
does not yield maximal violations of the CHSH inequality
but rather violates it by 3, as compared to the quantum Tsirelson bound 2冑2. The question still remains open whether
memoryless single bit exchange protocols exist which violate
the CHSH inequality maximally, i.e., by 4. Another open
question is the effect of information exchange between entangled quantum subsystems.
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E共␪兲 =
4
␲2
冋冉
␪2 −
冊 冉 冊冉 冊 册
␲2
␲
− 2H ␪ −
4
2
␪−
␲
2
2
,
共4兲
050302-3