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. It can be equivalently understood as a quantum channel that takes as input a quantum system and has as its output two systems: a classical system containing the outcome of the measurement and a quantum system containing the post-measurement state. Let be a countable set describing the outcomes of a measurement, and let denote a collection of trace-non-increasing completely positive maps, such that the sum of all is trace-preserving, i.e. for all positive operators . Now for describing a quantum measurement by an instrument , the maps are used to model the mapping from an input state to the output state of a measurement conditioned on a classical measurement outcome . Therefore, the probability of measuring a specific outcome on a state is given by The state after a measurement with the specific outcome is given by If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections , then the action of an instrument is given by a quantum channel with Here and are the Hilbert spaces corresponding to the input and the output systems of the instrument. A quantum instrument is an example of a quantum operation in which an "outcome" indicating which operator acted on the state is recorded in a classical register. An expanded development of quantum instruments is given in quantum channel.