Physics:Quantum Projective measurement: Difference between revisions
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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.<ref name=Neumann>{{cite book |last=von Neumann |first=John |title=Mathematical Foundations of Quantum Mechanics |year=1932}}</ref> | 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.<ref name=Neumann>{{cite book |last=von Neumann |first=John |title=Mathematical Foundations of Quantum Mechanics |year=1932}}</ref> | ||
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 measurement]]s (POVMs) and quantum instruments.<ref name=Nielsen>{{cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref> | 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 measurement]]s (POVMs) and quantum instruments.<ref name=Nielsen>{{cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref> | ||
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[[File:Quantum_projective_measurement_collapse_yellow.jpg|thumb|280px|Quantum Projective measurement.]] | |||
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== Definition == | == Definition == | ||
Revision as of 14:01, 17 May 2026
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
where are orthogonal projection operators satisfying
A projective measurement yields outcome with probability
where is the state of the system.[2]
After the measurement, the state collapses to
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.
See also
Table of contents (217 articles)
Index
Full contents
References
Source attribution: Physics:Quantum Projective measurement
