Physics:Quantum Projective measurement: Difference between revisions
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* A '''quantum instrument''' provides a full description of a measurement, including both the classical outcome and the post-measurement quantum state. | * A '''quantum instrument''' provides a full description of a measurement, including both the classical outcome and the post-measurement quantum state. | ||
Revision as of 08:47, 20 May 2026
Projective measurement is a Book I topic in the Quantum Collection. 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. 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. 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. 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.
Definition
Let an observable be represented by a self-adjoint operator with spectral decomposition
where are orthogonal projection operators satisfying
A projective measurement yields outcome with probability
where is the state of the system.[1]
After the measurement, the state collapses to
This process is known as the projection postulate.[2]
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.[1]
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.[1]
- 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.
See also
Table of contents (217 articles)
Index
Full contents
References
Source attribution: Physics:Quantum Projective measurement
