POVM

Positive Operator Valued (probability ) measure, abbreviated as POVM is a description of the quantum mechanical measurement process in physics. Mathematically is a POVM a kind of probability measure whose values ​​are positive operators instead of positive numbers.

Definition

A POVM to a measurement space is a mapping with values ​​in the set of bounded linear operators of a Hilbert space, satisfies the following three conditions:

• For all ( here denotes the identical picture on the Hilbert space ). That is, a positive and, therefore, self-adjoint.
• .
• Pairwise disjoint sets For each result

Notes

The definition of a POVM is in analogy with the Kolmogorov axioms of probability theory, the probability is described by a positive operator instead by a positive real number. POVM generalize the concept of Spektralmaßes that occurs in the spectral theory of self-adjoint operators.

Use in quantum mechanics

In quantum mechanics POVM occur for the description of general measurements. Here one usually has a discrete set of so-called effects that satisfy the following:

• , Here is the identity matrix. In particular, the positive semi-definite.

Describe the various measurement results, when the system is in the state, the probability of the measurement result is given by.

This approach is more general than that of a von Neumann - measurement (so-called projective measurement), in such a projectors are the eigenvectors of the measured observables. However, one can interpret each POVM as a von Neumann measurement on an extended system ( the original system auxiliary system ).

In particular, for quantum information theory are relevant POVM in the state discrimination nonorthogonal states or eavesdropping strategies in quantum cryptography.