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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Tsirelson's bound ： ウィキペディア英語版 Tsirelson's bound A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics is non-local, i.e., that quantum mechanical correlations violate Bell inequalities, a natural question to ask is "how non-local can quantum mechanics be?", or, more precisely, by how much can the Bell inequality be violated. The answer is precisely the Tsirelson bound for the particular Bell inequality in question. In general this bound is lower than what would be algebraically possible, and much research has been dedicated to the question of why this is the case. The Tsirelson bounds are named after B. S. Tsirelson, (or Boris Cirel'son, in a different transliteration) the author of the paper〔B. S. Cirel'son, ''(Quantum Generalizations of Bell's Inequality )'', Lett. Math. Phys. 4, 93 (1980). ()〕 in which the first one was derived. ==Tsirelson 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) dichotomic observables $A\_0$, $A\_1$, $B\_0$, $B\_1$ (i.e., two observables for Alice and two for Bob) with outcomes $+1,-1$ such that $()=0$ for all $i,j$, then :$\backslash langle\; A\_0\; B\_0\; \backslash rangle\; +\; \backslash langle\; A\_0\; B\_1\; \backslash rangle\; +\; \backslash langle\; A\_1\; B\_0\; \backslash rangle\; -\; \backslash langle\; A\_1\; B\_1\; \backslash rangle\; \backslash le\; 2\backslash sqrt$ For comparison, in the classical (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of $+1,-1$ is allowed it is 4. The Tsirelson bound is attained already if Alice and Bob each makes measurements on a qubit, the simplest non-trivial quantum system. Lots of proofs have been developed for this bound, but perhaps the most enlightening one is based on the Khalfin-Tsirelson-Landau identity. If we define an observable :$\backslash mathcal\; =\; A\_0\; B\_0\; +\; A\_0\; B\_1\; +\; A\_1\; B\_0\; -\; A\_1\; B\_1$ and $A\_i^2\; =\; B\_j^2\; =\; \backslash mathbb$, i.e., if the outcomes of the observables are associated to projective measurements, then :$\backslash mathcal^2\; =\; 4\; \backslash mathbb\; -\; ()()$ If $()\; =\; 0$ or $()\; =\; 0$, which can be regarded as the classical case, it already follows that $\backslash langle\; \backslash mathcal\; \backslash rangle\; \backslash le\; 2$. In the quantum case, we need only notice that $\backslash |()\backslash |\; \backslash le\; 2\; \backslash |A\_0\backslash |\; \backslash |A\_1\backslash |\; \backslash le\; 2$ and the Tsirelson bound $\backslash langle\; \backslash mathcal\; \backslash rangle\; \backslash le\; 2\backslash sqrt$ follows. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tsirelson's bound」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.