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4. Instrumentos cuánticos del esquema de medidas indirectas.

El modelo básico para la construcción de instrumentos cuánticos se basa en el esquema de medidas indirectas. Este esquema formaliza la siguiente situación: los resultados de la medición se generan a través de la interacción de 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/":): {\displaystyle S} con un aparato de medida 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 M} .Este aparato consiste en un dispositivo físico complejo que interactúa con 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 S} y un puntero que muestra el resultado de la medición, digamos girar hacia arriba o hacia abajo. Un observador solo puede ver las salidas del puntero y asocia estas salidas con los valores del 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} para el 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/":): {\displaystyle S} .Así, el esquema de medición indirecta implica:

los estados del sistemas 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 S} y el aparato 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 M}
El operador 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 U} representando la dinámica de interacción para el 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/":): {\displaystyle S+M}
el metro 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 M_A} dando salidas del puntero del aparato 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 M} .

Un modelo de medición indirecta, introducido en Ozawa (1984)^[1] como un "proceso de medición (general)", es un cuádruple

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 (H,\sigma,U,M_A)}

que consta de un espacio de Hilbert 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 \mathcal{H}} ,un operador de densidad 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 \sigma\in S(\mathcal{H})} , Un operador unitario 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 U} sobre el producto tensorial de los espacios de estado de 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 S} y 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 M,U:\mathcal{H}\otimes\mathcal{H}\rightarrow \mathcal{H}\otimes\mathcal{H}} y un operador hermitiano 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 M_A} on 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 \mathcal{H}} . Por este modelo de medida, el espacio de Hilbert 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 \mathcal{H}} describe los estados del aparato 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 M} , El operador unitario 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 U} describe la evolución temporal del sistema compuesto 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 S+M} , El operador de densidad 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 \sigma} describe el estado inicial del aparato 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 M} , y el operador hermitiano 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 M_A} describe el metro observable del aparato 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 M} . Entonces, la distribución de probabilidad de salida. 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 Pr\{A=x\|\sigma\}} en el estado del 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/":): {\displaystyle \sigma\in S(\mathcal{H})} es dado por

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 Pr\{A=x\|\rho\}=Tr[\Bigl(I\otimes E^M{^{_A}(x)\Bigr)}U(\rho \otimes\sigma)U^*] }

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 (18)}

dónde 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 E^{M_{A}}(x)} es la proyección espectral de 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 M_A} para el valor propio 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} .

El cambio de estado 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 \sigma} del 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/":): {\displaystyle S} causado por la medición para el resultado $A=x$ se representa con la ayuda del mapa 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)} en el espacio de operadores de densidad definidos como

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 \mathcal{P}_A(x)\rho= Tr_\mathcal{H}[\Bigl(I\otimes E^M{^{_A}(x)\Bigr)}U(\rho \otimes\sigma)U^*]}

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 (19)}

dónde 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 Tr_\mathcal{H}} es la traza parcial sobre 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 \mathcal{H}} . Entonces, el mapa 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)} resultar ser un instrumento cuántico. Así, las propiedades estadísticas de la medida realizada por cualquier modelo de medida indirecta 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 (H,\sigma,U,M_A)} se describe mediante una medida cuántica. Resaltamos que, a la inversa, cualquier instrumento cuántico puede representarse mediante el modelo de medición indirecta (Ozawa, 1984).^[1]Así, los instrumentos cuánticos caracterizan matemáticamente las propiedades estadísticas de todas las medidas cuánticas físicamente realizables.

↑ ^1.0 ^1.1 Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar

[:Ozawa_M_(1984)-1] 1.0 ^1.1 Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar

[1]

@@ Line 1: / Line 1: @@
-==4. Quantum instruments from the scheme of indirect measurements==
+==4. Instrumentos cuánticos del esquema de medidas indirectas.==
-The basic model for construction of quantum instruments is based on the scheme of indirect measurements. This scheme formalizes the following situation: measurement’s outputs are generated via interaction of a system <math>S</math> with a measurement apparatus <math>M</math> . This apparatus consists of a complex physical device interacting with <math>S</math> and a pointer that shows the result of measurement, say spin up or spin down. An observer can see only outputs of the pointer and he associates these outputs with the values of the observable <math>A</math> for the system <math>S</math>. Thus, the indirect measurement scheme involves:
+El modelo básico para la construcción de instrumentos cuánticos se basa en el esquema de medidas indirectas. Este esquema formaliza la siguiente situación: los resultados de la medición se generan a través de la interacción de un sistema <math>S</math> con un aparato de medida <math>M</math> .Este aparato consiste en un dispositivo físico complejo que interactúa con <math>S</math> y un puntero que muestra el resultado de la medición, digamos girar hacia arriba o hacia abajo. Un observador solo puede ver las salidas del puntero y asocia estas salidas con los valores del observable.<math>A</math> para el sistema <math>S</math>.Así, el esquema de medición indirecta implica:
-# the states of the systems <math>S</math> and the apparatus <math>M</math>
+# los estados del sistemas <math>S</math> y el aparato <math>M</math>
-# the operator  <math>U</math> representing the interaction-dynamics for the system <math>S+M</math>
+# El operador <math>U</math> representando la dinámica de interacción para el sistema <math>S+M</math>
-# the meter observable <math>M_A</math> giving outputs of the pointer of the apparatus <math>M</math>.
+# el metro observable <math>M_A</math> dando salidas del puntero del aparato <math>M</math>.
-An ''indirect measurement model'', introduced in Ozawa (1984)<ref name=":Ozawa M (1984)">Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar</ref> as a “(general) measuring process”, is a quadruple
+Un modelo de medición indirecta, introducido en Ozawa (1984)<ref name=":Ozawa M (1984)">Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar</ref> como un "proceso de medición (general)", es un cuádruple
 <math>(H,\sigma,U,M_A)</math>
-consisting of a Hilbert space <math>\mathcal{H}</math> , a density operator <math>\sigma\in S(\mathcal{H})</math>, a unitary operator  <math>U</math> on the tensor product of the state spaces of  <math>S</math> and<math>M,U:\mathcal{H}\otimes\mathcal{H}\rightarrow \mathcal{H}\otimes\mathcal{H}</math> and a Hermitian operator <math>M_A</math> on <math>\mathcal{H}</math> . By this measurement model, the Hilbert space <math>\mathcal{H}</math> describes the states of the apparatus <math>M</math>, the unitary operator <math>U</math> describes the time-evolution of the composite system <math>S+M</math>, the density operator <math>\sigma</math> describes the initial state of the apparatus <math>M</math> , and the Hermitian operator <math>M_A</math> describes the meter observable of the apparatus <math>M</math>. Then, the output probability distribution <math>Pr\{A=x\|\sigma\}</math> in the system state <math>\sigma\in S(\mathcal{H})</math> is given by
+que consta de un espacio de Hilbert <math>\mathcal{H}</math> ,un operador de densidad <math>\sigma\in S(\mathcal{H})</math>, Un operador unitario  <math>U</math>sobre el producto tensorial de los espacios de estado de  <math>S</math> y <math>M,U:\mathcal{H}\otimes\mathcal{H}\rightarrow \mathcal{H}\otimes\mathcal{H}</math>y un operador hermitiano <math>M_A</math> on <math>\mathcal{H}</math> . Por este modelo de medida, el espacio de Hilbert <math>\mathcal{H}</math>describe los estados del aparato <math>M</math>, El operador unitario <math>U</math> describe la evolución temporal del sistema compuesto <math>S+M</math>, El operador de densidad <math>\sigma</math> describe el estado inicial del aparato <math>M</math> , y el operador hermitiano <math>M_A</math> describe el metro observable del aparato <math>M</math>. Entonces, la distribución de probabilidad de salida.  <math>Pr\{A=x\|\sigma\}</math>en el estado del sistema <math>\sigma\in S(\mathcal{H})</math> es dado por
 {| width="80%" |
@@ Line 20: / Line 20: @@
 |}
-where <math>E^{M_{A}}(x)</math> is the spectral projection of <math>M_A</math> for the eigenvalue <math>x</math>.
+dónde <math>E^{M_{A}}(x)</math> es la proyección espectral de <math>M_A</math> para el valor propio <math>x</math>.
-The change of the state <math>\sigma</math> of the system <math>S</math> caused by the measurement for the outcome  <math>A=x</math> is represented with the aid of the map <math>\Im_A(x)</math> in the space of density operators defined as
+El cambio de estado <math>\sigma</math> del sistema <math>S</math> causado por la medición para el resultado <math>A=x</math>  se representa con la ayuda del mapa <math>\Im_A(x)</math>en el espacio de operadores de densidad definidos como
 {| width="80%" |
@@ Line 32: / Line 32: @@
 |}
-where <math>Tr_\mathcal{H}</math> is the partial trace over <math>\mathcal{H}</math> . Then, the map  <math>x\rightarrow\Im_A(x)</math>turn out to be a quantum instrument. Thus, the statistical properties of the measurement realized by any indirect measurement model <math>(H,\sigma,U,M_A)</math> is described by a quantum measurement. We remark that conversely any quantum instrument can be represented via the indirect measurement model (Ozawa, 1984).<ref name=":Ozawa M (1984)">Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar</ref>Thus, quantum instruments mathematically characterize the statistical properties of all the physically realizable quantum measurements.
+dónde <math>Tr_\mathcal{H}</math>es la traza parcial sobre <math>\mathcal{H}</math> . Entonces, el mapa <math>x\rightarrow\Im_A(x)</math>  resultar ser un instrumento cuántico. Así, las propiedades estadísticas de la medida realizada por cualquier modelo de medida indirecta <math>(H,\sigma,U,M_A)</math>se describe mediante una medida cuántica. Resaltamos que, a la inversa, cualquier instrumento cuántico puede representarse mediante el modelo de medición indirecta (Ozawa, 1984).<ref name=":Ozawa M (1984)">Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar</ref>Así, los instrumentos cuánticos caracterizan matemáticamente las propiedades estadísticas de todas las medidas cuánticas físicamente realizables.