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3.3. Nicht-projektive Zustandsaktualisierung: atomare Instrumente

Im Allgemeinen werden die statistischen Eigenschaften jeder Messung dadurch gekennzeichnet

die Ausgabewahrscheinlichkeitsverteilung 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 Pr\{\text{x}=x\parallel\rho\}} , die Wahrscheinlichkeitsverteilung der Ausgabe 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} der Messung im Eingangszustand 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 \rho }
die Quantenzustandsreduktion 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 \rho\rightarrow\rho_{(X=x)} } , die Zustandsänderung vom Eingangszustand 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 \rho } zum Ausgangszustand 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 \rho\rightarrow\rho_{(X=x)} } abhängig vom Ergebnis 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 \text{X}=x } der Messung.

In von Neumann’s formulation, the statistical properties of any measurement of an observable is uniquely determined by Born’s rule (5) and the projection postulate (6), and they are represented by the map (9), an instrument of von Neumann type. However, von Neumann’s formulation does not reflect the fact that the same observable represented by the Hermitian operator 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{H}} can be measured in many ways.8 Formally, such measurement-schemes are represented by quantum instruments.

In von Neumanns Formulierung werden die statistischen Eigenschaften jeder Messung einer Observablen eindeutig durch die Bornsche Regel (5) und das Projektionspostulat (6) bestimmt, und sie werden durch die Karte (9), ein Instrument vom Typ von Neumann, dargestellt. Die Formulierung von von Neumann spiegelt jedoch nicht die Tatsache wider, dass dieselbe Observable dargestellt durch den hermiteschen Operator 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 \hat{A}} 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/":): {\displaystyle A} kann auf vielfältige Weise gemessen werden.8 Formal werden solche Messschemata durch Quanteninstrumente repräsentiert.

Wir betrachten die einfachsten Quanteninstrumente vom Nicht-von-Neumann-Typ, die als atomare Instrumente bekannt sind. Wir beginnen mit der Erinnerung an den Begriff POVM (Probability Operator Valued Measure); wir beschränken Betrachtungen auf POVMs mit einem diskreten Definitionsbereich 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=\{x_1....,x_N.....\}} . POVM ist eine Karte 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\rightarrow \hat{D}(x)} so dass für jeden , ist ein positiver kontraktiver hermitescher Operator (Effekt genannt) (d. h. 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 \hat{D}(x)^*=\hat{D}(x), 0\leq \langle\psi|\hat{D}(x)\psi\rangle\leq1} oder irgendein 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\in X} , 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 \psi\in\mathcal{H}} ) und die Normalisierungsbedingung

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 \sum_x \hat{D}(x)=I}

hält, wo 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 I} ist der Einheitsoperator. Es wird davon ausgegangen, dass für jede Messung die Wahrscheinlichkeitsverteilung ausgegeben wird

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 Pr\{\text{x}=x||\rho\}} wird von gegeben

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 Pr\{\text{x}=x||\rho\}=Tr [\hat{D}(x)\rho]}

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 (10)}

Wo 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 \hat{D}(x)} ist ein POVM. Bei atomaren Instrumenten wird davon ausgegangen, dass Wirkungen konkret in der Form dargestellt werden

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 \hat{D}(x)=\hat{V}(x)^*\hat{V}(x)}

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 (11)}

Wo 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 {V}(x)} ist ein linearer Operator 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{H}} . Die Normierungsbedingung hat also die Form 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 \sum_x V(x)^*V(x)=I} .9 Die Born-Regel kann ähnlich wie (5) geschrieben werden:

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 Pr\{\text{x}=x||\rho\}=Tr [{V}(x)\rho{V}^*(x)]}

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 (12)}

Es wird angenommen, dass die Zustandstransformation nach der Messung auf der Karte basiert:

*

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 \rho\rightarrow\mathcal{L_A(x)\rho=V(X)\rho V^*(x)}}

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 (13)}

die Quantenzustandsreduktion ist also gegeben durch

*

{\textstyle \rho \rightarrow \rho _{({\text{x}}=x)}={\frac {{\mathcal {L}}_{A}(x)\rho }{Tr[{\mathcal {L}}_{A}(x)\rho ]}}}

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 (14)}

Die Karte 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\mathcal{L_A(x)}} gegeben durch (13) ist ein atomares Quanteninstrument. Wir bemerken, dass die Born-Regel (12) in der Form geschrieben werden kann

*

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 Pr\{\text{x}=x||\rho\}=Tr [\Im_A(x)\rho]}

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 (15)} f

Lassen 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 \hat{A}} sei ein hermitescher Operator 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{H}} . Betrachten Sie eine POVM 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 \hat{D}=\biggl(\hat{D}^A(x)\Biggr)} mit dem Definitionsbereich, der durch das Spektrum von gegeben ist 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 \hat{A}} . Dieses POVM repräsentiert eine Messung von Observable 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 A} wenn die Bornsche Regel gilt:

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 Pr\{\text{A}=x||\rho\}=Tr [\widehat{D}^A(x)\rho]=Tr[\widehat{E}^A(x)\rho]}

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 (16)}

Somit sind im Prinzip Ergebniswahrscheinlichkeiten immer noch in der spektralen Zerlegung von Operatoren kodiert 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 \hat{A}} oder mit anderen Worten Operatoren 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 \biggl(\hat{D}^A(x)\Biggr)} sollten so gewählt werden, dass sie die Wahrscheinlichkeiten erzeugen, die der spektralen Zerlegung der symbolischen Darstellung entsprechen 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 \hat{A}} von Observablen 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 A} , d. h., 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 \biggl(\hat{D}^A(x)\Biggr)} ist eindeutig bestimmt durch 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 \hat{A}} als 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 \hat{D}^A(x)=\hat{E}^A(x)} . Wir können sagen, dass dieser Operator im Gegensatz zum Operator des von Neumann-Schemas nur Informationen über die Wahrscheinlichkeiten von Ergebnissen enthält 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 \hat{A}} codiert nicht die Regel der Zustandsaktualisierung. Bei einem atomaren Instrument Messungen des Observablen $A$ hat die eindeutige Ausgabewahrscheinlichkeitsverteilung nach der Bornschen Regel (16), hat aber viele verschiedene Quantenzustandsreduktionen, abhängig von der Zerlegung des Effekts ${\textstyle {\hat {D}}(x)={\hat {E}}^{A}(x)=V(x)^{*}V(x)}$ Sodass

{\textstyle \rho \rightarrow \rho _{({\text{A}}=x)}={\frac {{V}(x)\rho V(x)^{*}}{Tr[{V}(x)\rho V(x)^{*}]}}}

(17)