In quantum physics, a quantum instrument is a mathematical description of a quantum measurement, capturing both the classical and quantum outputs.
[1] 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.
be a countable set describing the outcomes of a quantum measurement, and let
denote a collection of trace-non-increasing completely positive maps, such that the sum of all
= tr ( ρ )
Now for describing a measurement by an instrument
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 that a specific measurement outcome
ρ ) = tr (
The state after a measurement with the specific outcome
If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections
are the Hilbert spaces corresponding to the input and the output systems of the instrument.
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.
[4] In John Smolin's terminology, this is an example of "going to the Church of the Larger Hilbert space".
Any quantum instrument on a system
can be modeled as a projective measurement on
and (jointly) an uncorrelated auxiliary
followed by a unitary conditional on the measurement outcome.
) be the normalized initial state of
Then one can check that defines a quantum instrument.
[4] Furthermore, one can also check that any choice of quantum instrument
can be obtained with this construction for some choice of
[4] In this sense, a quantum instrument can be thought of as the reduction of a projective measurement combined with a conditional unitary.
immediately induces a CPTP map, i.e., a quantum channel:[4] This can be thought of as the overall effect of the measurement on the quantum system if the measurement outcome is thrown away.
immediately induces a positive operator-valued measurement (POVM): where
are any choice of Kraus operators for
are not uniquely determined by the CP maps
, but the above definition of the POVM elements
[4] 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.