| Tsirelson's Bound |
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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 and Positive 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 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 {Link without Title} : 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) S ~ (U · V) S) + ((F · G) S ~ (V · U) S) + ((G · F) S ~ (U · V) S) + ((G · F) S ~ (V · U) S) | < ''''' |
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