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 X be the countable set describing the outcomes of a measurement and { E x } x ∈ X a collection of subnormalized completely positive maps, given in such a way that the sum of all E x is trace preserving, i.e. tr ( ∑ x E x ( ρ ) ) = tr ( ρ ) for all positive operators ρ .
Now for describing a quantum measurement by an instrument I , the maps E x are used to model the mapping from an input state ρ to the outputstate of a measurement conditioned on an classical measurement outcome x . Thereby the probability of measuring a specific outcome x on a state ρ is given by
The state after a measurent with the specific outcome x is given by
If the measurement outcomes are recorded in a classical register, i.e. this can be modelled by a set of orthonormal projections | x ⟩ ⟨ x | ∈ B ( C | x | ) , the action of an instrument I is given by an channel I : B ( H 1 ) → B ( H 2 ) ⊗ B ( C | x | ) with
Here H 1 and H 2 are the Hilbert spaces corresponding to the input and the output quantum system of a measurement.
A quantum instrument is an example of a quantum operation in which an "outcome" x of which operator acted on the state is recorded in a classical register. An expanded development of quantum instruments is given in quantum channel.
