QuantumInfo.Measurements.POVM

Positive Operator-Valued Measures

A Positive Operator-Valued Measures, or POVM, is the most general notion of a quantum "measurement": a collection of positive semidefinite (PSD) operators that sum to the identity. These induce a distribution, `POVM.measure`, of measurement outcomes; and they induce a CPTP map, `POVM.measurement_map`, which changes the state but adds learned information.

Developing this theory is important if one wants to discuss classical information across quantum channels, as POVMs are the route to get back to classical information (a `ProbDistribution` of outcomes).

TODO: They can also evolve under CPTP maps themselves (the Heisenberg picture of quantum evolution), they might commute with each other or not, they might be projective or not.

1 declaration

theorem

$\Lambda.\text{measureForget}$ is a CPTP map with Kraus operators $\sqrt{M_i}$

#measureForget_eq_kraus

Let $\Lambda$ be a Positive Operator-Valued Measure (POVM) with outcomes in a set $X$ acting on a Hilbert space of dimension $d$ . Let $M_i$ denote the positive semidefinite matrix associated with each outcome $i \in X$ . The CPTP map $\Lambda.\text{measureForget}$ , which describes the state of the system after a measurement is performed and the outcome is discarded, is equal to the completely positive trace-preserving map defined by the Kraus operators $K_i = \sqrt{M_i} = M_i^{1/2}$ . That is, for any density matrix $\rho$ , $\Lambda.\text{measureForget}(\rho) = \sum_{i \in X} \sqrt{M_i} \rho \sqrt{M_i}.$

QuantumInfo.Measurements.POVM

Positive Operator-Valued Measures

Λ.measureForget\Lambda.\text{measureForget}Λ.measureForget is a CPTP map with Kraus operators Mi\sqrt{M_i}Mi​​

$\Lambda.\text{measureForget}$ is a CPTP map with Kraus operators $\sqrt{M_i}$