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3.4. Teoria generale (Davies–Lewis–Ozawa)

Infine, formuliamo la nozione generale di strumento quantistico. Un superoperatore che agisce in 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})} è detto positivo se mappa in se stesso l'insieme degli operatori semidefiniti positivi. Osserviamo che, per ogni 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)}  dato da (13) si può considerare come mappa lineare positiva.

Generalmente qualsiasi mappa 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)} , dove per ogni 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 mappa 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)} è un superoperatore positivo è chiamata strumento quantistico di Davies-Lewis (Davies e Lewis, 1970).[1]

Qui l'indice 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} indica l'osservabile accoppiato a questo strumento. Le probabilità di 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} -risultati sono date dalla regola di Born nella forma (15) e dall'aggiornamento dello stato mediante trasformazione (14). Tuttavia, Yuen (1987)[2] ha sottolineato che la classe degli strumenti Davies-Lewis è troppo generale per escludere strumenti fisicamente non realizzabili. Ozawa (1984)[3] ha introdotto l'importante condizione aggiuntiva per garantire che ogni strumento quantistico sia fisicamente realizzabile. Questa è la condizione di completa positività.

Un superoperatore è detto completamente positivo se la sua estensione naturale 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} al prodotto tensoriale 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})} è ancora un superoperatore positivo su 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})} . Una mappa , dove per ogni 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 x} , la mappa 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)} è un superoperatore completamente positivo è chiamato Davies-Lewis-Ozawa (Davies e Lewis, 1970,[4] Ozawa, 1984[5]) strumento quantistico o semplicemente strumento quantistico. Come vedremo nel paragrafo 4, la completa positività è una condizione sufficiente affinché uno strumento sia fisicamente realizzabile. D'altra parte, la necessità è derivata come segue (Ozawa, 2004).[6]

Ogni osservabile 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} di un sistema 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} è identificato con lo 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} osservabile di un sistema 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'} con qualsiasi sistema 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'} esterno a 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) Quindi, ogni strumento fisicamente realizzabile 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} misurando 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} dovrebbe essere identificato con lo strumento  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 } che misura 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 } tale che 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 } . Ciò implica che 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 } è di nuovo un superoperatore positivo, quindi 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)} è completamente positivo.

Allo stesso modo, qualsiasi strumento fisicamente realizzabile 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)} misurando il sistema 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} dovrebbe avere il suo strumento esteso  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 } che misura il sistema 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'} per qualsiasi sistema esterno 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'} . Questo è soddisfatto solo se  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)} è completamente positivo. Pertanto, la completa positività è una condizione necessaria affinché 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} descrivi uno strumento fisicamente realizzabile.

  1. ↑ Davies E.B., Lewis J.T. An operational approach to quantum probability Comm. Math. Phys., 17 (1970), pp. 239-260 View Record in ScopusGoogle Scholar
  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. Google Scholar
  3. ↑ Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar
  4. ↑ Davies E.B., Lewis J.T. An operational approach to quantum probability Comm. Math. Phys., 17 (1970), pp. 239-260 View Record in ScopusGoogle Scholar
  5. ↑ Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar
  6. ↑ Ozawa M. Uncertainty relations for noise and disturbance in generalized quantum measurements Ann. Phys., NY, 311 (2004), pp. 350-416
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