In quantum physics, a quantum instrument is a mathematical description of a quantum measurement, capturing both the classical and quantum outputs. 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

be a countable set denote a collection of trace-non-increasing completely positive maps

is given by

is given by

with

are the Hilbert spaces corresponding to the input and the output systems of the instrument.

Reductions and inductions

Just as a completely positive trace preserving (CPTP) map can always be considered as 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

followed by a unitary conditional on the measurement outcome.. Then one can check that

defines a quantum instrument..

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

immediately induces a 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

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

are any choice of Kraus operators,

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