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{{Quantum book backlink|Measurement and information}}

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>
[[File:Bloch sphere.png|thumb|400px|Measurement operators represented on the Bloch sphere. POVMs (red) generalize projective measurements for distinguishing quantum states (blue).]]

== 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]]:
<math display="block">P(m) = \langle \psi | M_m^\dagger M_m | \psi \rangle.</math><ref name="Nielsen"/>

After the measurement, the state changes to:
<math display="block">\frac{M_m |\psi\rangle}{\sqrt{\langle \psi | M_m^\dagger M_m | \psi \rangle}}.</math>

The operators satisfy the completeness relation:
<math display="block">\sum_m M_m^\dagger M_m = I.</math>

== 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.<ref name="Peres">{{cite book |last1=Peres |first1=Asher |title=Quantum Theory: Concepts and Methods |publisher=Kluwer Academic |year=1995}}</ref>
* '''POVMs''' generalize this framework by allowing non-projective measurement elements.<ref name="Nielsen"/>
* '''Kraus operators''' describe the most general state transformations associated with measurement processes.<ref name="Kraus">{{cite book |last1=Kraus |first1=Karl |title=States, Effects, and Operations |publisher=Springer |year=1983}}</ref>

In the POVM formalism, one defines:
<math display="block">E_m = M_m^\dagger M_m,</math>
with:
<math display="block">\sum_m E_m = I.</math>

The probability of outcome <math>m</math> for a general quantum state <math>\rho</math> is:
<math display="block">\mathrm{Prob}(m) = \operatorname{tr}(\rho E_m).</math><ref name="Nielsen"/>

== 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:
<math display="block">E_i = A_i^\dagger A_i.</math>

If outcome <math>i</math> is obtained, the state <math>\rho</math> transforms as:
<math display="block">\rho \to \frac{A_i \rho A_i^\dagger}{\operatorname{tr}(\rho E_i)}.</math><ref name="Kraus"/>

Summing over all possible outcomes gives a [[quantum channel]]:
<math display="block">\rho \to \sum_i A_i \rho A_i^\dagger.</math><ref name="Watrous">{{cite book |last=Watrous |first=John |title=The Theory of Quantum Information |publisher=Cambridge University Press |year=2018}}</ref>

== 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 <math>\{\sigma_i\}</math>, and a measurement is used to determine which state was given.

Using a POVM <math>\{E_i\}</math>, the probability of correctly identifying the state is:
<math display="block">P_{\rm success} = \sum_i p_i \operatorname{tr}(\sigma_i E_i),</math><ref name="Watrous"/>

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]]:
<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>

More generally, optimal measurements can be formulated as optimization problems and solved numerically, for example using [[semidefinite programming]].<ref name="Bae">{{cite journal |last1=Bae |first1=Joonwoo |last2=Kwek |first2=Leong-Chuan |title=Quantum state discrimination and its applications |journal=Journal of Physics A |volume=48 |year=2015}}</ref>

== 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.<ref name="Peres"/>

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

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

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Measurement operators|1}}

Physics:Quantum Measurement operators - Revision history

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