Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Positive-operator-valued measure

Open this concept in the interactive graph Read the Markdown equivalent

Summary

A weakly countably additive map $E$ from a measurable outcome space to positive operators on a Hilbert space, normalized by $E(\Omega)=I$ .

Record metadata

Canonical name: Positive-operator-valued measure
Concept ID: positive_operator_valued_measure
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Measure
Domains: Measure
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Operator-valued measure

Carrier(s)

measurable outcome space $(\Omega,\Sigma)$
Hilbert space $H$

Data

map $E\colon\Sigma\to\{\text{positive operators on }H\}$

Axioms / constraints

weak countable additivity
$E(\Omega)=I$

Canonically induces

probability measures $p_\rho(B)=\operatorname{Tr}(\rho E(B))$

Notes

A PVM is the special case in which every effect is an orthogonal projection.

Concept sources

Wikipedia — POVM — POVM · encyclopedic reference · source ID wp-positive_operator_valued_measure
nLab — positive-operator-valued measure — positive-operator-valued measure · mathematical reference · source ID nlab-positive_operator_valued_measure
NIST — Quantum Physics Theory — Quantum Physics Theory · NIST research reference · source ID nist-quantum-theory
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Measurable space Positive-operator-valued measure

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: povm_from_positive
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: positive-operator-valued measure

Authored explanation

A POVM adds a normalized, weakly countably additive assignment of a positive operator to each measurable outcome set.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Positive operator — Positive operator · encyclopedic reference · source ID wp-positive_operator
nLab — positive operator — positive operator · mathematical reference · source ID nlab-positive_operator
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum
Wikipedia — POVM — POVM · encyclopedic reference · source ID wp-positive_operator_valued_measure
nLab — positive-operator-valued measure — positive-operator-valued measure · mathematical reference · source ID nlab-positive_operator_valued_measure
NIST — Quantum Physics Theory — Quantum Physics Theory · NIST research reference · source ID nist-quantum-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.