Physics:Quantum Projective measurement: Difference between revisions
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{{Short description|Quantum Collection topic on Quantum Projective measurement}} | |||
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'''Projective measurement''' 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. Let an observable be represented by a self-adjoint operator with spectral decomposition | |||
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== Definition == | == Definition == | ||
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</math> | </math> | ||
where <math>\rho</math> is the state of the system.<ref name=Nielsen/> | where <math>\rho</math> is the state of the system.<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> | ||
After the measurement, the state collapses to | After the measurement, the state collapses to | ||
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</math> | </math> | ||
This process is known as the '''projection postulate'''.<ref name=Neumann/> | This process is known as the '''projection postulate'''.<ref name=Neumann>{{cite book |last=von Neumann |first=John |title=Mathematical Foundations of Quantum Mechanics |year=1932}}</ref> | ||
== Physical interpretation == | == 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 | 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.<ref name=Nielsen/> | ||
However, real physical measurements are often more general and cannot always be described by simple projection operators. | However, real physical measurements are often more general and cannot always be described by simple projection operators. | ||
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Projective measurements are a special case of more general measurement frameworks: | Projective measurements are a special case of more general measurement frameworks: | ||
* A | * A positive operator-valued measurement (POVM) generalizes projective measurements by allowing non-orthogonal measurement operators.<ref name=Nielsen/> | ||
* 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. | ||
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{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}} | {{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}} | ||
= References = | |||
{{reflist|3}} | {{reflist|3}} | ||
{{Author|Harold Foppele}} | {{Author|Harold Foppele}} | ||
{{Sourceattribution|Physics:Quantum Projective measurement|1}} | {{Sourceattribution|Physics:Quantum Projective measurement|1}} | ||
Latest revision as of 00:31, 24 May 2026
Projective measurement 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. Let an observable be represented by a self-adjoint operator with spectral decomposition
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
