Positive Operator Valued Probability Measure

Positive Operator Valued (Probability) Measure , abbreviated as POVM , is a description of the quantum mechanical measurement process in physics . Mathematically speaking, a POVM is a type of probability measure whose values ​​are positive operators rather than positive numbers.

definition

A POVM on a measurement space is a mapping with values ​​in the set of the restricted linear operators of a Hilbert space , the following three conditions are satisfied: ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle \mu \colon {\mathcal {A}}\to B({\mathcal {H}})}$ ${\displaystyle {\mathcal {H}}}$

• The following applies to all (here denotes the identical mapping on the Hilbert space). That is, it is positive and therefore also self-adjoint . ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle 0\leq \mu (A)\leq \operatorname {id} _{\mathcal {H}}}$ ${\displaystyle \operatorname {id} _{\mathcal {H}}}$ ${\displaystyle \mu (A)}$
• ${\displaystyle \mu (\Omega )=\operatorname {id} _{\mathcal {H}}}$.
• For every sequence of pairwise disjoint sets holds ${\displaystyle (A_{n}\in {\mathcal {A}})_{n\in \mathbb {N} }}$

${\displaystyle \mu {\bigg (}\biguplus _{n=1}^{\infty }A_{n}{\bigg )}=\sum _{n=1}^{\infty }\mu (A_{n}),}$

where the infinite series converges in the sense of the strong operator topology .

Explanations

The definition of a POVM is analogous to the Kolmogorov axioms of probability theory , where the probability is described by a positive operator instead of a positive real number. POVM generalize the concept of the spectral measure , which occurs in the spectral theory of self-adjoint operators .

Use in quantum mechanics

In quantum mechanics , POVMs appear to describe more general measurements. Here you usually have a discrete set of so-called effects that fulfill the following: ${\displaystyle {E}_{i}}$

• ${\displaystyle 0\leq E_{i}\leq I}$, here is the identity matrix . In particular, they are positive semidefinite . ${\displaystyle I}$ ${\displaystyle E_{i}}$
• ${\displaystyle \sum _{i}E_{i}=I}$

They describe the different measurement results: If the system is in the state , the probability of the measurement result is given by . ${\displaystyle E_{i}}$ ${\displaystyle \rho }$ ${\displaystyle i}$ ${\displaystyle p_{i}=\operatorname {Tr} (\rho {E}_{i})}$

This approach is more general than that of a Von Neumann measurement (so-called projective measurement), with such an approach the projectors are on the eigenvectors of the observed observables. However, each POVM can be viewed as a Von Neumann measurement on an extended system (original system + auxiliary system). ${\displaystyle E_{i}=|\psi _{i}\rangle \langle \psi _{i}|}$

In particular for quantum information theory, POVMs are relevant for the state differentiation of non-orthogonal states or for eavesdropping strategies in quantum cryptography.

literature

• Diane Martinez, Jody Trout: Asymptotic Spectral Measures, Quantum Mechanics, and E-theory . In: Communications in Mathematical Physics . tape 226 , no. 1 , 2002, p. 41-60 , arxiv : math / 0107091 .