Tsirelson's bound

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Tsirelson's bound

Tsirelson's bound, also known as Tsirelson's inequality, or in another transliteration, Cirel'son's inequality, arises in quantum mechanics, in 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.

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 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 commutators

: "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; 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 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:

: "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 numbers, "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. Bell  , 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 correspondence) 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 inequality an experiment is considered with obtained counts and constraints as described above. However, based on a suggestion by J. S. Bell  , an additional contraint 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.

References

:  B. S. Cirel'son, "Quantum Generalizations of Bell's Inequality", Lett. Math. Phys. 4, 93 (1980).:  L. J. Landau, "Experimental Tests of General Quantum Theories", Lett. Math. Phys. 14, 33 (1987).:  J. S. Bell, "On the Einstein-Podolski-Rosen paradox", Physics 1, 195 (1964).

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