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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). | 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=":0" />Thus, quantum instruments mathematically characterize the statistical properties of all the physically realizable quantum measurements. | ||