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{{Quantum book backlink|Measurement and information}}

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>

[[File:Quantum_projective_measurement_collapse_yellow.jpg|thumb|400px|Projective measurement: a quantum state collapses onto one of the orthogonal eigenstates of the observable, yielding a definite outcome.]]

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

<math display="block">
A = \sum_i a_i P_i,
</math>

where <math>P_i</math> are orthogonal projection operators satisfying

<math display="block">
P_i P_j = \delta_{ij} P_i, \quad \sum_i P_i = I.
</math>

A projective measurement yields outcome <math>a_i</math> with probability

<math display="block">
p(i|\rho) = \operatorname{tr}(P_i \rho),
</math>

where <math>\rho</math> is the state of the system.<ref name=Nielsen/>

After the measurement, the state collapses to

<math display="block">
\rho_i = \frac{P_i \rho P_i}{\operatorname{tr}(P_i \rho)}.
</math>

This process is known as the '''projection postulate'''.<ref name=Neumann/>

== 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]].<ref name=Nielsen/>

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.<ref name=Nielsen/>
* 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 ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Projective measurement|1}}

Physics:Quantum Projective measurement - Revision history

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