In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs.^[1]

Introduction

A quantum measurement is described by a set of operators ${M_{m}}$ , each associated with a possible outcome $m$ . If the system is in a state $| ψ ⟩$ , the probability of outcome $m$ is given by the Born rule: $P (m) = ⟨ ψ | M_{m}^{†} M_{m} | ψ ⟩ .$ ^[1]

After the measurement, the state changes to: $\frac{M_{m} | ψ ⟩}{\sqrt{⟨ ψ | M_{m}^{†} M_{m} | ψ ⟩}} .$

The operators satisfy the completeness relation: $\sum_{m} M_{m}^{†} M_{m} = I .$

Relation to other measurement formalisms

Measurement operators provide a unified description of different types of quantum measurements:

Projective measurements correspond to projection operators onto eigenstates of an observable.^[2]
POVMs generalize this framework by allowing non-projective measurement elements.^[1]
Kraus operators describe the most general state transformations associated with measurement processes.^[3]

In the POVM formalism, one defines: $E_{m} = M_{m}^{†} M_{m},$ with: $\sum_{m} E_{m} = I .$

The probability of outcome $m$ for a general quantum state $ρ$ is: $P r o b (m) = tr (ρ E_{m}) .$ ^[1]

State change and Kraus operators

Measurement not only yields probabilities but also changes the quantum state. This transformation can be described using Kraus operators $A_{i}$ , such that: $E_{i} = A_{i}^{†} A_{i} .$

If outcome $i$ is obtained, the state $ρ$ transforms as: $ρ \to \frac{A_{i} ρ A_{i}^{†}}{tr (ρ E_{i})} .$ ^[3]

Summing over all possible outcomes gives a quantum channel: $ρ \to \sum_{i} A_{i} ρ A_{i}^{†} .$ ^[4]

Examples

Measurement operators play a central role in quantum-information tasks such as quantum state discrimination. In this setting, a system is prepared in one of several possible states ${σ_{i}}$ , and a measurement is used to determine which state was given.

Using a POVM ${E_{i}}$ , the probability of correctly identifying the state is: $P_{s u c c e s s} = \sum_{i} p_{i} tr (σ_{i} E_{i}),$ ^[4]

where $p_{i}$ is the prior probability of state $σ_{i}$ .

For two states, the optimal measurement is given by the Helstrom measurement: $P_{s u c c e s s} = \frac{1}{2} + \frac{1}{2} ‖ p_{0} σ_{0} - p_{1} σ_{1} ‖_{1} .$ ^[5]

More generally, optimal measurements can be formulated as optimization problems and solved numerically, for example using semidefinite programming.^[6]

Physical interpretation

Measurement operators encode both the probabilities of outcomes and the transformation of the quantum state. Unlike classical measurements, quantum measurements generally disturb the system, reflecting the non-commutative structure of quantum observables.^[2]

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Measurement operators

Introduction

Relation to other measurement formalisms

State change and Kraus operators

Examples

Physical interpretation

See also

Table of contents (138 articles)

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