Tsirelson's bound
Encyclopedia
Tsirelson's bound, also known as Tsirelson's inequality, or in another transliteration, Cirel'son's inequality, is an inequality that imposed an upper limit to quantum mechanical
correlations between distant events. It concerns discussion and experimental determination of whether local hidden variables
are required for, or even compatible with, the representation of experimental results; with particular relevance to the EPR thought experiment
and the CHSH inequality
. It is named for B. S. Tsirelson
, the author of the paper in which it was first derived.
then it follows that:
A suitable choice of inner product (~) in which these operator products are linear, and application to a suitable state vector (s) leads to a corresponding identity of inner product terms:
Further, if the operators F, G, U, and V, as well as any linear combination thereof are self-adjoint
operators for the selected inner product and state vector s, i. e. if
then an inequality is obtained from the above identity by dropping the four last terms:
Finally, if the operators F, G, U, and V are normal for the selected inner product and state vector s, i.e. if
then the inequality reduces to a concise form of Tsirelson's bound:
It is perhaps worth noting that the given elementary derivation is carried out without any explicit requirements or restrictions for the commutator
s
Forms of Tsirelson's bound involving more than four operators can be derived as well.
and given the normalization constraints
In contrast to the product operation (•) used in the elementary derivation above, it must be noted that the product operation (·) here is applied sequentially: any resulting product is required in turn to be an operator which may appear subsequently as a factor in an operator product, and the product operation is required to be associative.
Applied to state vector s, within inner products and with operators self-adjoint and normalized as above, the corresponding identity is obtained as
and using again the above commutators:
The first term of this identity is a real number
; indeed
Consequently the last term of the identity is a real number as well, and therefore
Applying the Cauchy-Bunyakowski-Schwartz inequality to each term of the last expression yields
Further, since
and if
then it follows that
If similarly
then it follows
and therefore the inequality
It has been suggested that Tsirelson's bound is obtained as a consequence of the preceding inequality, since, again in application of the Cauchy-Bunyakowski-Schwartz inequality:
and with the normalization (s~s) = 1,
This resulting expression equals Tsirelson's bound formally.
It must be noted however, that the elementary derivation of Tsirelson's bound above requires considerably weaker assumptions about the operators and the accompanying product operation than the derivation based on Landau's identity shown here. The different strength of assumptions is expressed in the distinction between the symbols for the correspondingly used product operations (• vs. ·).
Similarly, the correctness of related assertions depends on the detailed assumptions made about the operators and their products. For instance,
returning to the identity
it has been suggested that if operator F commutes with operator G, or if operator U commutes with operator V, then the upper limit presented by Tsirelson's bound is lowered from √8 to √4 = 2.
While it is certainly correct that, if all six pairs of operators commute for product operation ·, i. e. if
then
there exist on the other hand examples (see below) of four operators which commute for product operation •, i. e.
for which the equality case of Tsirelson's bound is satisfied:
Signals are to be considered and counted only if A and B detect them trial-by-trial together; i.e. for any one signal which has been detected by A in one particular trial, B must have detected precisely one signal in the same trial, and vice versa.
For any one particular trial it may be consequently distinguished and counted whether
Similarly, it can be distinguished and counted whether
Further, for any one trial j it may be consequently distinguished and counted whether
Summing the counts over all trials j of a given set J of trials, one can evaluate for instance
i.e. the quantum correlation
between the channels or outcomes in which A and B individually detected the signals in the trials of set J; where -1 ≤ P(A↑) (B «)( J ) ≤ 1.
Following Malus
's definition, the correlation values P may be taken as measures of orientation angle φ between the detectors of A and of B, for any particular set of trials:
It is perhaps worth noting that, if numbers φJ and φK were found having different values, then the sets of trials J and K from which those two numbers were obtained were necessarily distinct from each other (though not necessarily disjoint); set J contained trials which were not contained in set K, and (or) set K contained trials which were not contained in set J.
Given experimental data collected in four (not necessarily disjoint) sets of trials J, K, L, and M, for which the measured correlation values were found to satisfy (at least approximately; as can be decided to arbitrary precision, given a sufficiently large number of trials)
or in terms of the corresponding measured orientation angle values, given
then one can find four real number
s, f, g, u, and v, such that
and correspondingly four operators, F, G, U, and V, such that
of course along with
The four operators therefore satisfy the conditions under which Tsirelson's inequality was derived above, and consequently (at least approximately, with arbitrary precision, given a sufficiently large number of trials)
Correspondingly, for any four real numbers f, g, u, and v, holds
or equivalently, for any three real numbers φJ, φK, and φL holds
The equality case of Tsirelson's bound is attained (for instance) for values
, the correlation value obtained from observations collected in the trials of set L,
may be parametrized as
where
The suggested objective local parametrization obtains in particular if
Similarly, the correlation value measured from observations collected in trials of set K,
may be parametrized as
may be parametrized as
may be parametrized as
Tsirelson's bound therefore provides a bound for the correlation values measured in the described experiment if they are expressed in objective-local parametrization:
an experiment is considered with obtained counts and constraints as described above. However, based on a suggestion by J. S. Bell, an additional constraint is imposed: The sets { ι }, { κ }, { λ }, and { μ } which are described and distinguished above are required to be precisely equal to each other;
even in cases in which the sets of trials J, K, L, and M were not all precisely the same set of trials, i. e. specifically
even if the four measured correlation numbers P(A↑) (B «)( J ), P(A↑) (B «)( K ), P(A↑) (B «)( L ), and P(A↑) (B «)( M ) are not all pairwise equal, or correspondingly,
even if the four measured orientation angles φJ, φK, φL, and φM, did not all have pairwise equal value.
Under these stronger assumptions, the CHSH inequality is obtained as
which is stronger than Tsirelson's bound.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
correlations between distant events. It concerns discussion and experimental determination of whether local hidden variables
Local hidden variable theory
In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones....
are required for, or even compatible with, the representation of experimental results; with particular relevance to the EPR thought experiment
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
and the CHSH inequality
CHSH inequality
In physics, the CHSH Bell test is an application of Bell's theorem, intended to distinguish between the entanglement hypothesis of quantum mechanics and local hidden variable theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony and Richard Holt, who described it in a much-cited...
. It is named for B. S. Tsirelson
Boris Tsirelson
Boris Semyonovich Tsirelson is a Soviet-Israeli mathematician and Professor of Mathematics in the Tel Aviv University in Israel.-Biography:Boris Tsirelson was born in Leningrad to a Russian Jewish family...
, the author of the paper in which it was first derived.
Derivation following Tsirelson's elementary proof
Given four operators (F, G, U, and V) together with a product operation (∙) defined for any pair of these four operators, and given that the following four pairs of operators commute:- F ∙ U = U ∙ F, F ∙ V = V ∙ F, G ∙ U = U ∙ G, and G ∙ V = V ∙ G,
then it follows that:
- F ∙ U + F ∙ V + U ∙ G − V ∙ G =
- 1/√2 F ∙ F + 1/√2 G ∙ G + 1/√2 U ∙ U + 1/√2 V ∙ V -
- - (√2 − 1) /8 ((√2 + 1) (F − U) + G − V) ∙ ((√2 + 1) (F − U) + G − V) -
- - (√2 − 1) /8 ((√2 + 1) (F − V) − G − U) ∙ ((√2 + 1) (F − V) − G − U) -
- - (√2 − 1) /8 ((√2 + 1) (G − U) + F + V) ∙ ((√2 + 1) (G − U) + F + V) -
- - (√2 − 1) /8 ((√2 + 1) (G + V) − F − U) ∙ ((√2 + 1) (G + V) − F − U).
A suitable choice of inner product (~) in which these operator products are linear, and application to a suitable state vector (s) leads to a corresponding identity of inner product terms:
- (s ~ (F ∙ U + F ∙ V + G ∙ U − G ∙ V) s) =
- (s ~ (F ∙ U) s) + (s ~ (F ∙ V) s) + (s ~ (G ∙ U) s) − (s ~ (G ∙ V) s) =
- (s ~ (1/√2 F ∙ F + 1/√2 G ∙ G + 1/√2 U ∙ U + 1/√2 V ∙ V -
- - (√2 − 1) /8 ((√2 + 1) (F − U) + G − V) ∙ ((√2 + 1) (F − U) + G − V) -
- - (√2 − 1) /8 ((√2 + 1) (F − V) − G − U) ∙ ((√2 + 1) (F − V) − G − U) -
- - (√2 − 1) /8 ((√2 + 1) (G − U) + F + V) ∙ ((√2 + 1) (G − U) + F + V) -
- - (√2 − 1) /8 ((√2 + 1) (G + V) − F − U) ∙ ((√2 + 1) (G + V) − F − U)) s) =
- 1/√2 (s ~ (F ∙ F) s) + 1/√2 (s ~ (G ∙ G) s) + 1/√2 (s ~ (U ∙ U) s) + 1/√2 (s ~ (V ∙ V) s) -
- - (√2 − 1) /8 (s ~ (((√2 + 1) (F − U) + G − V) ∙ ((√2 + 1) (F − U) + G − V)) s) -
- - (√2 − 1) /8 (s ~ (((√2 + 1) (F − V) − G − U) ∙ ((√2 + 1) (F − V) − G − U)) s) -
- - (√2 − 1) /8 (s ~ (((√2 + 1) (G − U) + F + V) ∙ ((√2 + 1) (G − U) + F + V)) s) -
- - (√2 − 1) /8 (s ~ (((√2 + 1) (G + V) − F − U) ∙ ((√2 + 1) (G + V) − F − U)) s).
Further, if the operators F, G, U, and V, as well as any linear combination thereof are self-adjoint
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
operators for the selected inner product and state vector s, i. e. if
- (s ~ (F ∙ F) s) = (F s ~ F s) >= 0, ...
- (s ~ (((√2 + 1) (F − U) + G − V) ∙ ((√2 + 1) (F − U) + G − V)) s) = (((√2 + 1) (F − U) + G − V) s ~ ((√2 + 1) (F − U) + G − V) s) >= 0, ...
then an inequality is obtained from the above identity by dropping the four last terms:
- (s ~ (F ∙ U) s) + (s ~ (F ∙ V) s) + (s ~ (G ∙ U) s) − (s ~ (G ∙ V) s) =<
- 1/√2 (s ~ (F ∙ F) s) + 1/√2 (s ~ (G ∙ G) s) + 1/√2 (s ~ (U ∙ U) s) + 1/√2 (s ~ (V ∙ V) s).
Finally, if the operators F, G, U, and V are normal for the selected inner product and state vector s, i.e. if
- (F s ~ F s) = 1, (G s ~ G s) = 1, (U s ~ U s) = 1, and (V s ~ V s) = 1,
then the inequality reduces to a concise form of Tsirelson's bound:
- (s ~ (F ∙ U) s) + (s ~ (F ∙ V) s) + (s ~ (G ∙ U) s) − (s ~ (G ∙ V) s) =< 4/√2. = √8.
It is perhaps worth noting that the given elementary derivation is carried out without any explicit requirements or restrictions for the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
s
- F ∙ G − G ∙ F or U ∙ V − V ∙ U.
Forms of Tsirelson's bound involving more than four operators can be derived as well.
The role of Landau's identity in deriving Tsirelson's inequality
An identity involving four operators (F, G, U, and V) and a product operation (·) has been pointed out by L. J. Landau: Given that the following four pairs of operators commute:- F · U = U · F, F · V = V · F, G · U = U · G, and G · V = V · G,
and given the normalization constraints
- F · F = G · G, and U · U = V · V, then Landau's identity holds:
- (F · U + F · V + G · U - G · V) · (F · U + F · V + G · U - G · V) =
- 4 (F · F) · (U · U) - (F · G - G · F) · (U · V - V · U).
In contrast to the product operation (•) used in the elementary derivation above, it must be noted that the product operation (·) here is applied sequentially: any resulting product is required in turn to be an operator which may appear subsequently as a factor in an operator product, and the product operation is required to be associative.
Applied to state vector s, within inner products and with operators self-adjoint and normalized as above, the corresponding identity is obtained as
- (s ~ (F · U + F · V + G · U - G · V) · (F · U + F · V + G · U - G · V) s) = 4 - (s ~ (F · G - G · F) · (U · V - V · U) s),
and using again the above commutators:
- ((F · U + F · V + G · U - G · V) s ~ (F · U + F · V + G · U - G · V) s) = 4 + ((F · G - G · F) s ~ (U · V - V · U) s).
The first term of this identity is a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
; indeed
- ((F · U + F · V + G · U - G · V) s ~ (F · U + F · V + G · U - G · V) s) >= 0.
Consequently the last term of the identity is a real number as well, and therefore
- ((F · G - G · F) s ~ (U · V - V · U) s) =<
- | ((F · G - G · F) s ~ (U · V - V · U) s) | =<
- | ((F · G) s ~ (U · V) s) | + | ((F · G) s ~ (V · U) s) | + | ((G · F) s ~ (U · V) s) | + | ((G · F) s ~ (V · U) s) |.
Applying the Cauchy-Bunyakowski-Schwartz inequality to each term of the last expression yields
- ((F · G - G · F) s ~ (U · V - V · U) s) =<
- | ((F · G) s ~ (U · V) s) | + | ((F · G) s ~ (V · U) s) | + | ((G · F) s ~ (U · V) s) | + | ((G · F) s ~ (V · U) s) | =<
- √ ((F · G) s ~ (F · G) s) √ ((U · V) s ~ (U · V) s) + √ ((F · G) s ~ (F · G) s) √ ((V · U) s ~ (V · U) s) + √ ((G · F) s ~ (G · F) s) √ ((U · V) s ~ (U · V) s) + √ ((G · F) s ~ (G · F) s) √ ((V · U) s ~ (V · U) s).
Further, since
- 0 =< ((F · G) s ~ (F · G) s) = (G s ~ (F · F · G) s) = (s ~ (G · F · F · G) s),
and if
- (s ~ (G · F · F · G) s) = (s ~ (G · G) s),
then it follows that
- √ ((F · G) s ~ (F · G) s) = 1.
If similarly
- (s ~ (F · G · G · F) s) = (s ~ (F · F) s),
- (s ~ (V · U · U · V) s) = (s ~ (V · V) s), as well as
- (s ~ (U · V · V · U) s) = (s ~ (U · U) s),
then it follows
- ((F · G - G · F) s ~ (U · V - V · U) s) =< 4
and therefore the inequality
- ((F · U + F · V + G · U - G · V) s ~ (F · U + F · V + G · U - G · V) s) =< 8.
It has been suggested that Tsirelson's bound is obtained as a consequence of the preceding inequality, since, again in application of the Cauchy-Bunyakowski-Schwartz inequality:
- ((F · U + F · V + G · U - G · V) s ~ s) (s ~ (F · U + F · V + G · U - G · V) s) =< ((F · U + F · V + G · U - G · V) s ~ (F · U + F · V + G · U - G · V) s) (s~s),
- ((F · U + F · V + G · U - G · V) s ~ s) (s ~ (F · U + F · V + G · U - G · V) s) =< 8 (s~s),
and with the normalization (s~s) = 1,
- (s ~ (F · U + F · V + G · U - G · V) s) =< √8.
This resulting expression equals Tsirelson's bound formally.
It must be noted however, that the elementary derivation of Tsirelson's bound above requires considerably weaker assumptions about the operators and the accompanying product operation than the derivation based on Landau's identity shown here. The different strength of assumptions is expressed in the distinction between the symbols for the correspondingly used product operations (• vs. ·).
Similarly, the correctness of related assertions depends on the detailed assumptions made about the operators and their products. For instance,
returning to the identity
- ((F · U + F · V + G · U - G · V) s ~ (F · U + F · V + G · U - G · V) s) = 4 + ((F · G - G · F) s ~ (U · V - V · U) s),
it has been suggested that if operator F commutes with operator G, or if operator U commutes with operator V, then the upper limit presented by Tsirelson's bound is lowered from √8 to √4 = 2.
While it is certainly correct that, if all six pairs of operators commute for product operation ·, i. e. if
- F · G = G · F, U · V = V · U along with
- F · U = U · F, F · V = V · F, G · U = U · G, and G · V = V · G,
then
- (s ~ (F · U + F · V + G · U - G · V) s) =< √4 = 2,
there exist on the other hand examples (see below) of four operators which commute for product operation •, i. e.
- F • G = G • F, U • V = V • U along with
- F • U = U • F, F • V = V • F, G • U = U • G, and G • V = V • G,
for which the equality case of Tsirelson's bound is satisfied:
- (s ~ (F • U) s) + (s ~ (F • V) s) + (s ~ (G • U) s) - (s ~ (G • V) s) = √8.
Application to EPR experiments
The experiments whose results are under certain conditions summarized by the Tsirelson bound or by the CHSH inequality concern measurements obtained by a pair of observers, A and B, who each can detect one signal at a time in one of two distinct own channels or outcomes: for instance A detecting and counting a signal either as (A↑) or (A↓), and B detecting and counting a signal either as (B «), or (B »).Signals are to be considered and counted only if A and B detect them trial-by-trial together; i.e. for any one signal which has been detected by A in one particular trial, B must have detected precisely one signal in the same trial, and vice versa.
For any one particular trial it may be consequently distinguished and counted whether
- A detected a signal as (A↑) and not as (A↓), with corresponding counts nt (A↑) = 1 and nt (A↓) = 0, in this particular trial t, or
- A detected a signal as (A↓) and not as (A↑), with corresponding counts nf (A↑) = 0 and nf (A↓) = 1, in this particular trial f, where trials f and t are evidently distinct.
Similarly, it can be distinguished and counted whether
- B detected a signal as (B «) and not as (B »), with corresponding counts ng (B «) = 1 and ng (B ») = 0, in this particular trial g, or
- B detected a signal as (B ») and not as (B «), with corresponding counts nh (B «) = 0 and nh (B ») = 1, in this particular trial h, where trials g and h are evidently distinct.
Further, for any one trial j it may be consequently distinguished and counted whether
- (A↑), and (B «) were detected together in this particular trial j, or
- (A↑), and (B ») were detected together, or
- (A↓), and (B «) were detected together, or
- (A↓), and (B ») were detected together in this trial.
Summing the counts over all trials j of a given set J of trials, one can evaluate for instance
- P(A↑) (B «)( J ) = { j = first of J Σ last of J}(nj (A↑) - nj (A↓)) (nj (B «) - nj (B »)) / ({ j = first of J Σ last of J} 1),
i.e. the quantum correlation
Quantum correlation
In Bell test experiments the term quantum correlation has come to mean the expectation value of the product of the outcomes on the two sides. In other words, the expected change in physical characteristics as one quantum system passes through an interaction site...
between the channels or outcomes in which A and B individually detected the signals in the trials of set J; where -1 ≤ P(A↑) (B «)( J ) ≤ 1.
Following Malus
Étienne-Louis Malus
- External links :...
's definition, the correlation values P may be taken as measures of orientation angle φ between the detectors of A and of B, for any particular set of trials:
- φJ = arccos ( P(A↑) (B «)( J ) ), φK = arccos ( P(A↑) (B «)( K ) ), and so on.
It is perhaps worth noting that, if numbers φJ and φK were found having different values, then the sets of trials J and K from which those two numbers were obtained were necessarily distinct from each other (though not necessarily disjoint); set J contained trials which were not contained in set K, and (or) set K contained trials which were not contained in set J.
Given experimental data collected in four (not necessarily disjoint) sets of trials J, K, L, and M, for which the measured correlation values were found to satisfy (at least approximately; as can be decided to arbitrary precision, given a sufficiently large number of trials)
- arccos ( P(A↑) (B «)( M ) ) = arccos ( P(A↑) (B «)( J ) ) + arccos ( P(A↑) (B «)( K ) ) + arccos ( P(A↑) (B «)( L ) ),
or in terms of the corresponding measured orientation angle values, given
- φM = φJ + φK + φL,
then one can find four real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, f, g, u, and v, such that
- f - v = φM,
- f - u = φJ,
- u - g = φK,
- g - v = φL,
and correspondingly four operators, F, G, U, and V, such that
- (s ~ (F • U) s) = cos ( f - u ) = cos ( u - f ) = (s ~ (U • F) s),
- (s ~ (F • V) s) = cos ( f - v ) = cos ( v - f ) = (s ~ (V • F) s),
- (s ~ (G • U) s) = cos ( g - u ) = cos ( u - g ) = (s ~ (U • G) s),
- (s ~ (G • V) s) = cos ( g - v ) = cos ( v - g ) = (s ~ (V • G) s),
of course along with
- (s ~ (F • F) s) = cos ( f - f ) = cos ( 0 ) = 1, and so on.
The four operators therefore satisfy the conditions under which Tsirelson's inequality was derived above, and consequently (at least approximately, with arbitrary precision, given a sufficiently large number of trials)
- P(A↑) (B «)( J ) + P(A↑) (B «)( K ) + P(A↑) (B «)( L ) - P(A↑) (B «)( M ) =< √8.
Correspondingly, for any four real numbers f, g, u, and v, holds
- cos ( f - u ) + cos ( u - g ) + cos ( g - v ) - cos ( v - f ) =< √8,
or equivalently, for any three real numbers φJ, φK, and φL holds
- cos ( φJ ) + cos ( φK ) + cos ( φL ) - cos ( φJ + φK + φL ) =< √8.
The equality case of Tsirelson's bound is attained (for instance) for values
- φJ = φK = φL = π/4.
Tsirelson's bound as bound for objective local theories
Given measured correlation values described above as obtained in four (not necessarily disjoint) sets of trials J, K, L, and M, then, following suggestions by J. S. BellJohn Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...
, the correlation value obtained from observations collected in the trials of set L,
- P(A↑) (B «)( L ) = cos ( g - v ) = { l = first of L Σ last of L}(nl (A↑) - nl (A↓)) (nl (B «) - nl (B »)) / ({ l = first of L Σ last of L} 1),
may be parametrized as
- P(A↑) (B «)( L ) = cos ( g - v ) = ∫{ λ } dλ ρ ( λ ) A ( g, λ ) B ( v, λ ),
where
- the A ( g, λ ) and B ( v, λ ) take the value 1 or -1,
- the real numbers g and v are identified as settings of observer A, and of observer B, respectively, in the trials of set J, and
- integration (or summation) is over a set of hidden variables { λ }.
The suggested objective local parametrization obtains in particular if
- the integration (or summation) is over a set of hidden variables { λ } is identified as the summation over the set of trials L,
- each hidden variable value λ of this set is identified (by a one-to-one correspondenceBijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
) as one trial index l - the numbers A ( g, λ ) are identified as the corresponding numbers nl (A↑) - nl (A↓), and
- the numbers B ( v, λ ) are identified as the corresponding numbers nl (B «) - nl (B »).
Similarly, the correlation value measured from observations collected in trials of set K,
- P(A↑) (B «)( K ) = cos ( g - u ),
may be parametrized as
- cos ( g - u ) = ∫{ κ } dκ ρ ( κ ) A ( g, κ ) B ( u, κ ),
- P(A↑) (B «)( J ) = cos ( f - u ),
may be parametrized as
- cos ( f - u ) = ∫{ ι } dι ρ ( ι ) A ( f, ι ) B ( u, ι ), and
- P(A↑) (B «)( M ) = cos ( f - v ),
may be parametrized as
- cos ( f - v ) = ∫{ μ } dμ ρ ( μ ) A ( f, μ ) B ( v, μ ).
Tsirelson's bound therefore provides a bound for the correlation values measured in the described experiment if they are expressed in objective-local parametrization:
- ∫{ ι } dι ρ ( ι ) A ( f, ι ) B ( u, ι ) +
- ∫{ κ } dκ ρ ( κ ) A ( g, κ ) B ( u, κ ) +
- ∫{ λ } dλ ρ ( λ ) A ( g, λ ) B ( v, λ ) -
- ∫{ μ } dμ ρ ( μ ) A ( f, μ ) B ( v, μ ) =< √8.
Comparison with the CHSH inequality
In the derivation of the CHSH inequalityCHSH inequality
In physics, the CHSH Bell test is an application of Bell's theorem, intended to distinguish between the entanglement hypothesis of quantum mechanics and local hidden variable theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony and Richard Holt, who described it in a much-cited...
an experiment is considered with obtained counts and constraints as described above. However, based on a suggestion by J. S. Bell, an additional constraint is imposed: The sets { ι }, { κ }, { λ }, and { μ } which are described and distinguished above are required to be precisely equal to each other;
even in cases in which the sets of trials J, K, L, and M were not all precisely the same set of trials, i. e. specifically
even if the four measured correlation numbers P(A↑) (B «)( J ), P(A↑) (B «)( K ), P(A↑) (B «)( L ), and P(A↑) (B «)( M ) are not all pairwise equal, or correspondingly,
even if the four measured orientation angles φJ, φK, φL, and φM, did not all have pairwise equal value.
Under these stronger assumptions, the CHSH inequality is obtained as
- ∫{ λ } dλ ρ ( λ ) A ( f, λ ) B ( u, λ ) +
- ∫{ λ } dλ ρ ( λ ) A ( g, λ ) B ( u, λ ) +
- ∫{ λ } dλ ρ ( λ ) A ( g, λ ) B ( v, λ ) -
- ∫{ λ } dλ ρ ( λ ) A ( f, λ ) B ( v, λ ) =< 2 < √8,
which is stronger than Tsirelson's bound.