POVMs

Positive operator-valued measures describe the most general quantum measurement, going beyond projective observables.

The general measurement

A projective measurement uses orthogonal projectors, one per outcome. But the most general measurement allowed by quantum mechanics is a POVM: a set of positive operators {E_k}, called effects, satisfying sum_k E_k = I. Outcome k occurs with probability Tr(E_k rho). The effects need not be projectors, need not be orthogonal, and their number can exceed the Hilbert-space dimension.

What POVMs add

POVMs let a d-dimensional system have more than d distinct outcomes, allow measurements that are not repeatable, and describe measurements optimized for a task rather than tied to an observable's eigenbasis. They arise naturally when a projective measurement is performed on a system coupled to an ancilla: Naimark's theorem says every POVM is a projective measurement on an enlarged space. So POVMs are not exotic; they are ordinary measurements viewed from the system alone.

import numpy as np
# unambiguous discrimination style trine POVM on a qubit
kets=[np.array([1,0]),np.array([-0.5,np.sqrt(3)/2]),np.array([-0.5,-np.sqrt(3)/2])]
E=[(2/3)*np.outer(k,k) for k in kets]
print(np.round(sum(E),3))  # identity -> valid POVM

POVM vs measurement operators

A POVM gives outcome probabilities but not the post-measurement state. To describe the state after measurement one needs measurement operators M_k with E_k = M_k-dagger M_k; the state updates to M_k rho M_k-dagger over its probability. Many different M_k can produce the same E_k, so the same statistics can leave the system in different post-measurement conditions. This finer object is a generalized measurement or quantum instrument.

Uses

POVMs are essential for optimal state discrimination, where non-orthogonal states must be told apart as well as physics allows, and for unambiguous discrimination that never errs but sometimes returns inconclusive. Symmetric informationally complete POVMs enable full state tomography with a single measurement setting. In quantum key distribution and metrology, tailoring the POVM to the task extracts more information than any fixed projective observable could.