Quantum instrument

In physics, a quantum instrument is a mathematical abstraction of a quantum measurement, capturing both the classical and quantum outputs. It combines the concepts of measurement and quantum operation.

Let ${\displaystyle X}$ be the countable set describing the outcomes of a measurement and ${\displaystyle \{{\mathcal {E}}_{x}\}_{x\in X}}$ a collection of subnormalized completely positive maps, given in such a way that the sum of all ${\displaystyle {\mathcal {E}}_{x}}$ is trace preserving, i.e. ${\textstyle \operatorname {tr} \left(\sum _{x}{\mathcal {E}}_{x}(\rho )\right)=\operatorname {tr} (\rho )}$ for all positive operators ${\displaystyle \rho }$.

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