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4. Quantum instruments from the scheme of indirect measurements

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 $S$ with a measurement apparatus $M$ . This apparatus consists of a complex physical device interacting with $S$ 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 $A$ for the system $S$. Thus, the indirect measurement scheme involves:

1. the states of the systems $S$ and the apparatus $M$
2. the operator  $U$ representing the interaction-dynamics for the system $S+M$
3. the meter observable $M_{A}$ giving outputs of the pointer of the apparatus $M$.

An indirect measurement model, introduced in Ozawa (1984) as a “(general) measuring process”, is a quadruple

$(H,\sigma ,U,M_{A})$

consisting of a Hilbert space ${\mathcal {H}}$ , a density operator $\sigma \in S({\mathcal {H}})$, a unitary operator  $U$ on the tensor product of the state spaces of  $S$ and$M,U:{\mathcal {H}}\otimes {\mathcal {H}}\rightarrow {\mathcal {H}}\otimes {\mathcal {H}}$ and a Hermitian operator $M_{A}$ on ${\mathcal {H}}$ . By this measurement model, the Hilbert space ${\mathcal {H}}$ describes the states of the apparatus $M$, the unitary operator $U$ describes the time-evolution of the composite system $S+M$, the density operator $\sigma$ describes the initial state of the apparatus $M$ , and the Hermitian operator $M_{A}$ describes the meter observable of the apparatus $M$. Then, the output probability distribution $Pr\{A=x\|\sigma \}$ in the system state $\sigma \in S({\mathcal {H}})$ is given by

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

$(18)$

where $E^{M_{A}}(x)$ is the spectral projection of $M_{A}$ for the eigenvalue $x$.

The change of the state $\sigma$ of the system $S$ caused by the measurement for the outcome  $A=x$ is represented with the aid of the map $\Im _{A}(x)$ in the space of density operators defined as

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

$(19)$

where $Tr_{\mathcal {H}}$ is the partial trace over ${\mathcal {H}}$ . Then, the map  $x\rightarrow \Im _{A}(x)$turn out to be a quantum instrument. Thus, the statistical properties of the measurement realized by any indirect measurement model $(H,\sigma ,U,M_{A})$ is described by a quantum measurement. We remark that conversely any quantum instrument can be represented via the indirect measurement model (Ozawa, 1984). Thus, quantum instruments mathematically characterize the statistical properties of all the physically realizable quantum measurements.