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. Definition. Let formula_1 be a countable set describing the outcomes of a measurement, and let formula_2 denote a collection of trace-non-increasing completely positive maps, such that  the sum of all formula_3 is trace-preserving, i.e. formula_4 for all positive operators formula_5. Now for describing a quantum measurement by an instrument formula_6, the maps formula_3 are used to model the mapping from an input state formula_5 to the output state of a measurement conditioned on a classical measurement outcome formula_9. Therefore, the probability of measuring a specific outcome formula_9 on a state formula_5 is given by formula_12 The state after a measurement with the specific outcome formula_9 is given by formula_14 If the measurement outcomes are recorded in a classical register, whose states are modeled by a  set of orthonormal projections formula_15 , then the action of an instrument formula_6 is given by a quantum channel formula_17 with formula_18 Here formula_19 and formula_20 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" formula_9 indicating which operator acted on the state is recorded in a classical register. An expanded development of quantum instruments is given in quantum channel.