Physics:Quantum Projective measurement

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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.[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]

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

A=∑iaiPi,

where Pi are orthogonal projection operators satisfying

PiPj=δijPi,∑iPi=I.

A projective measurement yields outcome ai with probability

p(i|ρ)=tr⁡(Piρ),

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

After the measurement, the state collapses to

ρi=PiρPitr⁡(Piρ).

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 (185 articles)

Index

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Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ 1.0 1.1 von Neumann, John (1932). Mathematical Foundations of Quantum Mechanics. 
  2. ↑ 2.0 2.1 2.2 2.3 Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. 
Author: Harold Foppele

Source attribution: Physics:Quantum Projective measurement

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