Store:QLMfr08

Go to top

3.4. Théorie générale (Davies-Lewis-Ozawa)

Enfin, nous formulons la notion générale d'instrument quantique. Un super-opérateur agissant dans Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}(\mathcal{H})} appelé positif s'il mappe l'ensemble des opérateurs semi-définis positifs sur lui-même. Nous remarquons que, pour chaque Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,\Im_A(x)} donné par (13) peut être considéré comme une application positive linéaire.

Généralement, toute carte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\rightarrow\Im_A(x)} , où pour chaque Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , la carte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im_A(x)} est un superopérateur positif est appelée instrument quantique de Davies-Lewis (Davies et Lewis, 1970).^[1]

Ici, l'indice 0Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A} désigne l'observable couplé à cet instrument. Les probabilités de résultats Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A} sont données par la règle de Born sous la forme (15) et la mise à jour d'état par transformation (14). Cependant, Yuen (1987)^[2] a souligné que la classe des instruments Davies-Lewis est trop générale pour exclure les instruments physiquement non réalisables. Ozawa (1984)^[3] a introduit la condition supplémentaire importante pour s'assurer que chaque instrument quantique est physiquement réalisable. C'est la condition de la positivité complète.

Un superopérateur est dit complètement positif si son extension naturelle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \jmath\otimes I} au produit tensoriel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})=\mathcal{L}(\mathcal{H}\otimes\mathcal{H})} est à nouveau un superopérateur positif sur Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})} . Une carte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\rightarrow\Im_A(x)} , où pour chaque ${\textstyle x}$ , la carte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im_A(x)} est un superopérateur complètement positif est appelée Davies–Lewis–Ozawa (Davies et Lewis, 1970,^[1] Ozawa, 1984^[3]) instrument quantique ou simplement instrument quantique. Comme nous le verrons dans la section 4, la positivité complète est une condition suffisante pour qu'un instrument soit physiquement réalisable. D'autre part, la nécessité est dérivée comme suit (Ozawa, 2004).^[4]

Chaque Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A} observable d'un système Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S} est identifié avec le Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A\otimes I} observable d'un système Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S+S'} avec tout système Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S'} externe à Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S} .(10)

Ensuite, tout instrument physiquement réalisable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im_A} mesurant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A} doit être identifié avec l'instrument Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Im_A{_\otimes}_I } mesurant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A{\otimes}I } tel que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Im_A{_\otimes}_I(x)=\Im_A(x)\otimes I } . Cela implique que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Im_A(x)\otimes I } est à nouveau un super-opérateur positif, de sorte que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im_A(x)} est complètement positif.

De même, tout système de mesure d'instrument Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im_A(x)} physiquement réalisable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S} devrait avoir son instrument étendu Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Im_A(x)\otimes I } système de mesure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S+S'} pour tout système externe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S'} . Ceci n'est rempli que si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im_A(x)} est complètement positif. Ainsi, la positivité complète est une condition nécessaire pour que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im_A} décrive un instrument physiquement réalisable.

↑ ^1.0 ^1.1 Davies E.B., Lewis J.T. An operational approach to quantum probability. Comm. Math. Phys., 17 (1970), pp. 239-260
↑ Yuen, H. P., 1987. Characterization and realization of general quantum measurements. M. Namiki and others (ed.) Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, pp. 360–363.
↑ ^3.0 ^3.1 Ozawa M. Quantum measuring processes for continuous observables J. Math. Phys., 25 (1984), pp. 79-87
↑ Ozawa M. Uncertainty relations for noise and disturbance in generalized quantum measurements Ann. Phys., NY, 311 (2004), pp. 350-416

[:0-1] 1.0 ^1.1 Davies E.B., Lewis J.T. An operational approach to quantum probability. Comm. Math. Phys., 17 (1970), pp. 239-260

[2] Yuen, H. P., 1987. Characterization and realization of general quantum measurements. M. Namiki and others (ed.) Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, pp. 360–363.

[:1-3] 3.0 ^3.1 Ozawa M. Quantum measuring processes for continuous observables J. Math. Phys., 25 (1984), pp. 79-87

[4] Ozawa M. Uncertainty relations for noise and disturbance in generalized quantum measurements Ann. Phys., NY, 311 (2004), pp. 350-416

[1]

[2]

[3]

[4]