A projective measurement (also called a von Neumann measurement) is a fundamental type of measurement in quantum mechanics in which the state of a system is projected onto an eigenstate of an observable.^[1]

Projective measurements represent the simplest and most idealized form of quantum measurement and form the basis for more general frameworks such as positive operator-valued measurements (POVMs) and quantum instruments.^[2]

Definition

Let an observable be represented by a self-adjoint operator with spectral decomposition

$A = \sum_{i} a_{i} P_{i},$

where $P_{i}$ are orthogonal projection operators satisfying

$P_{i} P_{j} = δ_{i j} P_{i}, \sum_{i} P_{i} = I .$

A projective measurement yields outcome $a_{i}$ with probability

$p (i | ρ) = tr (P_{i} ρ),$

where $ρ$ is the state of the system.^[2]

After the measurement, the state collapses to

$ρ_{i} = \frac{P_{i} ρ P_{i}}{tr (P_{i} ρ)} .$

This process is known as the projection postulate.^[1]

Physical interpretation

Projective measurements correspond to ideal measurements in which the system is sharply projected onto an eigenstate of the observable. They are often associated with textbook examples such as spin measurements using a Stern–Gerlach experiment.^[2]

However, real physical measurements are often more general and cannot always be described by simple projection operators.

Relation to POVMs and quantum instruments

Projective measurements are a special case of more general measurement frameworks:

A positive operator-valued measurement (POVM) generalizes projective measurements by allowing non-orthogonal measurement operators.^[2]
A quantum instrument provides a full description of a measurement, including both the classical outcome and the post-measurement quantum state.

In this broader framework, projective measurements correspond to the case where the measurement operators are orthogonal projections and the post-measurement state follows directly from the projection.

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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 Projective measurement

Definition

Physical interpretation

Relation to POVMs and quantum instruments

See also

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