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{{Short description|Quantum Collection topic on Quantum Measurement operators}}
{{Short description|Quantum Collection topic on Quantum Measurement operators}}


{{Quantum book backlink|Measurement and information}}
{{Quantum book backlink|Measurement and information}}
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In [[quantum mechanics]], '''measurement operators''' provide a general mathematical framework for describing the outcomes of a [[Measurement|measurement]] and the associated change of a [[quantum state]]. They unify different types of quantum measurements, including projective measurements and [[Positive operator-valued measure|POVMs]].<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=2000}}</ref>
'''Measurement operators''' is a Book I topic in the Quantum Collection. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. A quantum measurement is described by a set of operators \{M_m\}, each associated with a possible outcome m. If the system is in a state |\psi\rangle, the probability of outcome m is given by the Born rule: Measurement operators provide a unified description of different types of quantum measurements: * Projective measurements correspond to projection operators onto eigenstates of an observable.
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== Introduction ==
== Introduction ==


A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the [[Born rule]]:
A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the Born rule:
<math display="block">P(m) = \langle \psi | M_m^\dagger M_m | \psi \rangle.</math><ref name="Nielsen"/>
<math display="block">P(m) = \langle \psi | M_m^\dagger M_m | \psi \rangle.</math><ref name="Nielsen"/>


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== State change and Kraus operators ==
== State change and Kraus operators ==


Measurement not only yields probabilities but also changes the quantum state. This transformation can be described using [[Kraus operator]]s <math>A_i</math>, such that:
Measurement not only yields probabilities but also changes the quantum state. This transformation can be described using Kraus operatorss <math>A_i</math>, such that:
<math display="block">E_i = A_i^\dagger A_i.</math>
<math display="block">E_i = A_i^\dagger A_i.</math>


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where <math>p_i</math> is the prior probability of state <math>\sigma_i</math>.
where <math>p_i</math> is the prior probability of state <math>\sigma_i</math>.


For two states, the optimal measurement is given by the [[Helstrom measurement]]:
For two states, the optimal measurement is given by the Helstrom measurement:
<math display="block">P_{\rm success} = \tfrac12 + \tfrac12 \| p_0 \sigma_0 - p_1 \sigma_1 \|_1.</math><ref name="Helstrom">{{cite book|first=Carl W.|last=Helstrom|title=Quantum Detection and Estimation Theory|publisher=Academic Press|year=1976}}</ref>
<math display="block">P_{\rm success} = \tfrac12 + \tfrac12 \| p_0 \sigma_0 - p_1 \sigma_1 \|_1.</math><ref name="Helstrom">{{cite book|first=Carl W.|last=Helstrom|title=Quantum Detection and Estimation Theory|publisher=Academic Press|year=1976}}</ref>


Revision as of 08:14, 20 May 2026



← Back to Measurement and information Book I

Measurement operators is a Book I topic in the Quantum Collection. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. A quantum measurement is described by a set of operators \{M_m\}, each associated with a possible outcome m. If the system is in a state |\psi\rangle, the probability of outcome m is given by the Born rule: Measurement operators provide a unified description of different types of quantum measurements: * Projective measurements correspond to projection operators onto eigenstates of an observable.

Quantum Measurement operators.

Introduction

A quantum measurement is described by a set of operators {Mm}, each associated with a possible outcome m. If the system is in a state |ψ⟩, the probability of outcome m is given by the Born rule: P(m)=⟨ψ|Mm†Mm|ψ⟩.[1]

After the measurement, the state changes to: Mm|ψ⟩⟨ψ|Mm†Mm|ψ⟩.

The operators satisfy the completeness relation: ∑mMm†Mm=I.

Relation to other measurement formalisms

Measurement operators provide a unified description of different types of quantum measurements:

  • Projective measurements correspond to projection operators onto eigenstates of an observable.[2]
  • POVMs generalize this framework by allowing non-projective measurement elements.[1]
  • Kraus operators describe the most general state transformations associated with measurement processes.[3]

In the POVM formalism, one defines: Em=Mm†Mm, with: ∑mEm=I.

The probability of outcome m for a general quantum state ρ is: Prob(m)=tr⁡(ρEm).[1]

State change and Kraus operators

Measurement not only yields probabilities but also changes the quantum state. This transformation can be described using Kraus operatorss Ai, such that: Ei=Ai†Ai.

If outcome i is obtained, the state ρ transforms as: ρ→AiρAi†tr⁡(ρEi).[3]

Summing over all possible outcomes gives a quantum channel: ρ→∑iAiρAi†.[4]

Examples

Measurement operators play a central role in quantum-information tasks such as quantum state discrimination. In this setting, a system is prepared in one of several possible states {σi}, and a measurement is used to determine which state was given.

Using a POVM {Ei}, the probability of correctly identifying the state is: Psuccess=∑ipitr⁡(σiEi),[4]

where pi is the prior probability of state σi.

For two states, the optimal measurement is given by the Helstrom measurement: Psuccess=12+12‖p0σ0−p1σ1‖1.[5]

More generally, optimal measurements can be formulated as optimization problems and solved numerically, for example using semidefinite programming.[6]

Physical interpretation

Measurement operators encode both the probabilities of outcomes and the transformation of the quantum state. Unlike classical measurements, quantum measurements generally disturb the system, reflecting the non-commutative structure of quantum observables.[2]

Description

Measurement operators is a matter-scale concept used to organize how quantum theory describes atoms, particles, fields, condensed matter, plasma, or spacetime-related systems. In the Quantum Collection it is placed by scale so the reader can move from materials and molecules down to subatomic degrees of freedom.

Quantum context

At this scale, the relevant behavior is controlled by quantized states, interactions, conservation laws, and the way excitations or particles are observed. The concept is normally linked to measurable properties such as energy, momentum, charge, spin, spectra, scattering rates, or collective modes.

Role in the collection

This page provides a compact reference point for related pages in Book II. It should be read together with nearby matter-scale topics and the corresponding foundations in quantum mechanics.[7]

Interpretation

For measurement operators, the quantum description is useful because it separates the allowed states, interactions, and measurable quantities from the classical picture. The same concept may appear differently in spectroscopy, scattering, condensed matter, field theory, or cosmology.

Typical measurements involve spectra, decay products, transition rates, transport behavior, correlation functions, or detector signatures. These observations provide the empirical link between the page topic and the wider Quantum Collection.

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 1.2 Cite error: Invalid <ref> tag; no text was provided for refs named Nielsen
  2. ↑ 2.0 2.1 Peres, Asher (1995). Quantum Theory: Concepts and Methods. Kluwer Academic. 
  3. ↑ 3.0 3.1 Kraus, Karl (1983). States, Effects, and Operations. Springer. 
  4. ↑ 4.0 4.1 Watrous, John (2018). The Theory of Quantum Information. Cambridge University Press. 
  5. ↑ Helstrom, Carl W. (1976). Quantum Detection and Estimation Theory. Academic Press. 
  6. ↑ Bae, Joonwoo; Kwek, Leong-Chuan (2015). "Quantum state discrimination and its applications". Journal of Physics A 48. 
  7. ↑ "Quantum mechanics". https://en.wikipedia.org/wiki/Quantum_mechanics. 


Author: Harold Foppele


Source attribution: Physics:Quantum Measurement operators

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