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]

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Physics:Quantum Measurement operators

Introduction

Relation to other measurement formalisms

State change and Kraus operators

Examples

Physical interpretation

See also

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