A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics violates
Bell inequalities (i.e., it cannot be described by a
local hidden-variable theory
In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the principle of locality. These models attempt to account for the probabilistic features of quantum mechanics via the mechanism ...
), a natural question to ask is how large can the violation be. The answer is precisely the Tsirelson bound for the particular Bell inequality in question. In general, this bound is lower than the bound that would be obtained if more general theories, only constrained by "no-signalling" (i.e., that they do not permit communication faster than light), were considered, and much research has been dedicated to the question of why this is the case.
The Tsirelson bounds are named after
Boris S. Tsirelson (or Cirel'son, in a different
transliteration
Transliteration is a type of conversion of a text from one script to another that involves swapping letters (thus '' trans-'' + '' liter-'') in predictable ways, such as Greek → and → the digraph , Cyrillic → , Armenian → or L ...
), the author of the article in which the first one was derived.
Bound for the CHSH inequality
The first Tsirelson bound was derived as an upper bound on the correlations measured in the
CHSH inequality. It states that if we have four (
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...
) dichotomic observables
,
,
,
(i.e., two observables for
Alice
Alice may refer to:
* Alice (name), most often a feminine given name, but also used as a surname
Literature
* Alice (''Alice's Adventures in Wonderland''), a character in books by Lewis Carroll
* ''Alice'' series, children's and teen books by ...
and two for
Bob) with outcomes
such that
for all
, then
:
For comparison, in the classical case (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of
is allowed, it is 4. The Tsirelson bound is attained already if Alice and Bob each make measurements on a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical syste ...
, the simplest non-trivial quantum system.
Several proofs of this bound exist, but perhaps the most enlightening one is based on the Khalfin–Tsirelson–Landau identity. If we define an observable
:
and
, i.e., if the observables' outcomes are
, then
:
If
or
, which can be regarded as the classical case, it already follows that
. In the quantum case, we need only notice that
, and the Tsirelson bound
follows.
Other Bell inequalities
Tsirelson also showed that for any bipartite full-correlation Bell inequality with ''m'' inputs for Alice and ''n'' inputs for Bob, the ratio between the Tsirelson bound and the local bound is at most
where
and
is the
Grothendieck constant of order ''d''. Note that since
, this bound implies the above result about the CHSH inequality.
In general, obtaining a Tsirelson bound for a given Bell inequality is a hard problem that has to be solved on a case-by-case basis. It is not even known to be decidable. The best known computational method for upperbounding it is a convergent hierarchy of
semidefinite programs, the NPA hierarchy, that in general does not halt.
The exact values are known for a few more Bell inequalities:
For the Braunstein–Caves inequalities we have that
:
For the WWŻB inequalities the Tsirelson bound is
:
For the
inequality the Tsirelson bound is not known exactly, but concrete realisations give a lower bound of , and the NPA hierarchy gives an upper bound of . It is conjectured that only infinite-dimensional quantum states can reach the Tsirelson bound.
Derivation from physical principles
Significant research has been dedicated to finding a physical principle that explains why quantum correlations go only up to the Tsirelson bound and nothing more. Three such principles have been found: no-advantage for non-local computation,
information causality and macroscopic locality. That is to say, if one could achieve a CHSH correlation exceeding Tsirelson's bound, all such principles would be violated.
Tsirelson's bound also follows if the Bell experiment admits a strongly positive quantal measure.
Tsirelson's problem
There are two different ways of defining the Tsirelson bound of a Bell expression. One by demanding that the measurements are in a tensor product structure, and another by demanding only that they commute. Tsirelson's problem is the question of whether these two definitions are equivalent. More formally, let
:
be a Bell expression, where
is the probability of obtaining outcomes
with the settings
. The tensor product Tsirelson bound is then the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
of the value attained in this Bell expression by making measurements
and
on a quantum state
:
:
The commuting Tsirelson bound is the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
of the value attained in this Bell expression by making measurements
and
such that
on a quantum state
:
:
Since tensor product algebras in particular commute,
. In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras, so only for infinite dimensions it is possible that
. Tsirelson's problem is the question of whether for all Bell expressions
.
This question was first considered by
Boris Tsirelson in 1993, where he asserted without proof that
. Upon being asked for a proof by
Antonio Acín in 2006, he realized that the one he had in mind didn't work, and issued the question as an open problem. Together with
Miguel Navascués and
Stefano Pironio, Antonio Acín had developed an hierarchy of semidefinite programs, the NPA hierarchy, that converged to the commuting Tsirelson bound
from above,
and wanted to know whether it also converged to the tensor product Tsirelson bound
, the most physically relevant one.
Since one can produce a converging sequencing of approximations to
from below by considering finite-dimensional states and observables, if
, then this procedure can be combined with the NPA hierarchy to produce a halting algorithm to compute the Tsirelson bound, making it a
computable number
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers, computable reals, or recursive reals ...
(note that in isolation neither procedure halts in general). Conversely, if
is not computable, then
. In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed to have proven that
is not computable, thus solving Tsirelson's problem in the negative. Tsirelson's problem has been shown to be equivalent to
Connes' embedding problem, so the same proof also implies that the Connes embedding problem is false.
See also
*
Quantum nonlocality
*
Bell's theorem
Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measuremen ...
*
EPR paradox
EPR may refer to:
Science and technology
* EPR (nuclear reactor), European Pressurised-Water Reactor
* EPR paradox (Einstein–Podolsky–Rosen paradox), in physics
* Earth potential rise, in electrical engineering
* East Pacific Rise, a mid-ocea ...
*
CHSH inequality
*
Quantum pseudo-telepathy
Quantum pseudo-telepathy describes the use of quantum entanglement to eliminate the need for classical communications.
A nonlocal game is said to display quantum pseudo-telepathy if players who can use entanglement can win it with certainty whi ...
References
{{DEFAULTSORT:Tsirelson's Bound
Quantum measurement