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 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.

Reductions and inductions

Just as a completely positive (CPTP) map can always be considered the reduction of unitary evolution on a system with an initially unentangled auxiliary, quantum instruments are the reductions of projective measurement with a conditional unitary, and also reduce to CPTP maps and POVMs when ignore measurement outcomes and state evolution, respectively. In John Smolin's terminology, this is an example of "going to the Church of the Larger Hilbert space".

As a reduction of projective measurement and conditional unitary

Any quantum instrument on a system can be modeled as projective measurement on and (jointly) an uncorrelated auxiliary followed by a unitary conditional on the measurement outcome. Let (with and ) be the normalized initial state of , let (with and ) be a projective measurement on , and let (with ) be unitaries on . Then one can check that

defines a quantum instrument. Furthermore, one can also check that any choice of quantum instrument can be obtained with this construction for some choice of and .

In this sense, a quantum instrument can be thought of as the reduction of a projective measurement combined with a conditional unitary.

Reduction to CPTP map

Any quantum instrument immediately induces a completely positive trace-preserving (CPTP) map, i.e., a quantum channel:

This can be thought of as the overall effect of the measurement on the quantum system if the measurement outcome is thrown away.

Reduction to POVM

Any quantum instrument immediately induces a positive operator-valued measurement (POVM):

where are any choice of Kraus operators for the ,

The Kraus operators are not uniquely determined by the CP maps , but the above definition of the POVM elements is the same for any choice. The POVM can be thought of as the measurement of the quantum system if the information about how the system is affected by the measurement is thrown away.

References


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