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{{Short description|Quantum measurement, probability, and state change in quantum mechanics}}
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Measurement in quantum physics is the manipulation or testing of a physical system to  generate a numerical result. Quantum theory's  foundational feature is that the outcome it predicts is probabilistic. 

In quantum mechanics, calculating the likelihood of an outcome involves linking the mathematical description of a system, its quantum state, with a representation of the measurement . This procedure is formalized by the Born rule. An electron: its quantum state assigns a complex value, or a probability amplitude. The Born rule converts these amplitudes into the chances of detecting the electron in a particular location during a measurement. Quantum theory does not provide certainty, it only provides probabilities. The same quantum state can also be used to predict other properties, such as momentum, if that observable is measured instead. The uncertainty principle makes that precise knowledge of one quantity, like position, comes at the expense of unpredictability in its complementary quantity, like momentum. The fact that experiments violate Bell inequalities shows that this probabilistic behavior cannot be explained by local hidden variables. Randomness is a fundamental feature.When a measurement is performed on a quantum system, observing it alters the state that characterizes it. This behavior is the heart of quantum mechanics, intertwining mathematical structure with conceptual difference. The theoretical framework used to predict possible measurement outcomes and describe how quantum states evolve was developed throughout the 20th century, drawing on the methods of linear algebra and functional analysis. Quantum theory has demonstrated experimental accuracy and a range of applications. 

At a more philosophical level, discussions continue concerning the interpretation of what measurement means within the theory. Competing  interpretations of quantum mechanics offer different resolutions to the so-called measurement problem, the question of how and when quantum possibilities give definite outcomes.
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[[File:Quantum_mechanics_measurements_yellow.jpg|thumb|280px|Quantum measurement process: a system in superposition is measured, yielding probabilistic outcomes and collapsing to a definite eigenstate.]]
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== Common experimental measuring devices ==
{| class="wikitable"
!Observable
!Typical Measurement Device
|-
|'''Position'''
|Photodetectors, CCD cameras, scanning tunneling microscopes (STM)
|-
|'''Momentum'''
|Time-of-flight detectors, Doppler measurement setups
|-
|'''Spin'''
|Stern–Gerlach apparatus, spin-resolving detectors
|-
|'''Energy / Spectra'''
|Spectrometers, calorimeters, photomultiplier tubes
|-
|'''Photon number / field observables'''
|Photodiodes, homodyne detectors, superconducting qubits readout
|}

== Axiomatic formalism ==
=== "Observables" represented by self-adjoint operators ===
For more coverage of this topic see 
*Obervable (quantum mechanics)
*Canonical quantization
*Dirac–von Neumann axioms
In quantum mechanics, physical systems are described by a Hilbert space, where elements correspond to the system’s quantum states. The mathematical framework developed by John von Neumann, learns that a measurement is represented by a self-adjoint operator acting on this space, known as an "observable".<ref name=":30">{{Cite book|title=Statistical Structure of Quantum Theory|last=Holevo|first=Alexander S.|publisher=Springer|year=2001|isbn=3-540-42082-7|series=Lecture Notes in Physics|oclc=318268606|author-link=W:en:Alexander Holevo}}</ref> Operators correspond to the measurable quantities from classical physics such as position, momentum, energy, and angular momentum.

The dimensionality of a system’s Hilbert space depends of the system being modeled. For example, the Hilbert space describing a particle with a continuous degree of freedom, like position along a line, is infinite-dimensional and consists of square-integrable functions. In contrast, systems characterized by discrete properties, such as spin, are associated with finite-dimensional Hilbert spaces. Because the mathematics of the finite-dimensional case is considerably simpler, it is often mentioned in pedagogical treatments.

Introductory presentations of quantum mechanics often omit the advanced mathematical subtleties that arise for continuous observables and infinite-dimensional spaces—issues such as bounded versus unbounded operators, convergence of sequences, or unusual spectra like Cantor sets. These complications are rigorously addressed through spectral theory, but detailed discussions are usually reserved for advanced texts. In quantum mechanics, each physical system is associated with a Hilbert space, whose elements represent possible states. In the framework codified by John von Neumann, a measurement is represented by a self-adjoint operator on this space, called an "observable".<ref name=":3">{{Cite book|title=Statistical Structure of Quantum Theory|last=Holevo|first=Alexander S.|publisher=Springer|year=2001|isbn=3-540-42082-7|series=Lecture Notes in Physics|oclc=318268606|author-link=W:en:Alexander Holevo}}</ref> Observables correspond to familiar classical quantities such as position, momentum, energy, and angular momentum. The dimension of the Hilbert space may be infinite, as for the space of square-integrable functions describing a continuous degree of freedom, or finite, as for spin degrees of freedom. Many treatments focus on the finite-dimensional case, where the mathematics is simpler. Introductory texts often gloss over technical issues arising for continuous-valued observables and infinite-dimensional Hilbert spaces, including convergence, bounded versus unbounded operators, and exotic sets of eigenvalues.<ref name=":1" /><ref>{{cite web|url=https://terrytao.wordpress.com/2014/08/12/avila-bhargava-hairer-mirzakhani/|title=Avila, Bhargava, Hairer, Mirzakhani|last=Tao|first=Terry|author-link=W:Terence Tao|date=12 August 2014|website=What's New|access-date=9 February 2020}}</ref> While these issues can be rigorously treated using spectral theory;<ref name=":1" />, this article will generally avoid them.

=== Measurement based on state projection ===
See also Projection-valued measure

The eigenvectors of a von Neumann observable form an orthonormal basis for the Hilbert space, and each possible outcome of that measurement corresponds to one of the vectors comprising the basis. A density operator is a positive-semidefinite operator on the Hilbert space whose trace is equal to 1.<ref name=":3" /><ref name=":1" /> For each measurement that can be defined, the probability distribution over the outcomes of that measurement can be computed from the density operator. The procedure for doing so is the Born rule, which states that
:<math>P(x_i) = \operatorname{tr}(\Pi_i \rho),</math>
where <math>\rho</math> is the density operator, and <math>\Pi_i</math> is the projection operator onto the basis vector corresponding to the measurement outcome <math>x_i</math>. The average of the eigenvalues of a von&nbsp;Neumann observable, weighted by the Born rule probabilities, is the expectation value of that observable. For an observable <math>A</math>, the expectation value given a quantum state <math>\rho</math> is
:<math> \langle A \rangle = \operatorname{tr} (A\rho).</math>

A density operator that is a rank-1 projection is known as a ''pure'' quantum state, and all quantum states that are not pure are designated ''mixed''. Pure states are also known as ''wavefunctions''. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system (i.e., <math>P(x) = 1</math> for some outcome <math>x</math>). Any mixed state can be written as a convex combination of pure states, though not in a unique way.<ref>{{Cite journal|last=Kirkpatrick|first=K. A.|date=February 2006|title=The Schrödinger-HJW Theorem|journal=Foundations of Physics Letters|volume=19|issue=1|pages=95–102|arxiv=quant-ph/0305068|bibcode=2006FoPhL..19...95K|doi=10.1007/s10702-006-1852-1|issn=0894-9875|s2cid=15995449}}</ref> The state space of a quantum system is the set of all states, pure and mixed, that can be assigned to it.

The Born rule associates a probability with each unit vector in the Hilbert space, in such a way that these probabilities sum to 1 for any set of unit vectors comprising an orthonormal basis. Moreover, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like a choice of basis for that vector to be embedded in. Gleason's theorem establishes the converse: all assignments of probabilities to unit vectors (or, equivalently, to the operators that project onto them) that satisfy these conditions take the form of applying the Born rule to some density operator.<ref>{{cite journal|last=Gleason|first=Andrew M.|author-link=W:en:Andrew M. Gleason|year=1957|title=Measures on the closed subspaces of a Hilbert space|journal=Indiana University Mathematics Journal|volume=6|issue=4|pages=885–893|doi=10.1512/iumj.1957.6.56050|mr=0096113|doi-access=free}}</ref><ref>{{Cite journal|last=Busch|first=Paul|author-link=W:en:Paul Busch (physicist)|date=2003|title=Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem|journal=Physical Review Letters|volume=91|issue=12|arxiv=quant-ph/9909073|bibcode=2003PhRvL..91l0403B|doi=10.1103/PhysRevLett.91.120403|pmid=14525351|article-number=120403|s2cid=2168715}}</ref><ref>{{Cite journal|last1=Caves|first1=Carlton M.|author-link=W:en:Carlton M. Caves|last2=Fuchs|first2=Christopher A.|last3=Manne|first3=Kiran K.|last4=Renes|first4=Joseph M.|date=2004|title=Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements|journal=Foundations of Physics|volume=34|issue=2|pages=193–209|arxiv=quant-ph/0306179|bibcode=2004FoPh...34..193C|doi=10.1023/B:FOOP.0000019581.00318.a5|s2cid=18132256}}</ref>

=== Positive Operator-Valued Measure (POVM) ===
See PQVM<br>

A generalized measurement, or positive operator-valued measure (POVM), extends the standard projective measurement framework in quantum mechanics.

In a projective measurement, each outcome corresponds to a projection operator <math>P_i</math> satisfying
<math>P_i P_j = \delta_{ij} P_i</math> and <math>\sum_i P_i = I</math>.
For a system in state <math>\rho</math>, the probability of obtaining outcome <math>i</math> is
<math>p(i) = \mathrm{Tr}(P_i \rho)</math>,
and after the measurement, the state collapses to
<math>\rho' = \frac{P_i \rho P_i}{p(i)}</math>.

In contrast, a POVM consists of a set of positive semi-definite operators <math>{ E_i }</math> that satisfy
<math>E_i \ge 0</math> and <math>\sum_i E_i = I</math>.
The probability of obtaining outcome <math>i</math> is given by
<math>p(i) = \mathrm{Tr}(E_i \rho)</math>.
The post-measurement state depends on the specific measurement implementation and is not uniquely determined by the POVM elements.

POVMs provide a more general description of measurements, encompassing cases such as noisy or coarse-grained detectors and indirect measurements involving ancillary systems. Every projective measurement is a special case of a POVM, but not all POVMs correspond to projective measurements on the original system alone. According to Naimark’s theorem, any POVM can be realized as a projective measurement on an extended Hilbert space that includes an auxiliary system.
In functional analysis and quantum measurement theory, a positive-operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVMs) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement|projection-valued measure described by PVMs. In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see Schrödinger–HJW theorem); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics, and can also be used in quantum field theory.<ref>{{cite journal|last1=Peres|first1=Asher|last2=Terno|first2=Daniel R.|year=2004|title=Quantum information and relativity theory|journal=[[Wikipedia:Reviews of Modern Physics]|Reviews of Modern Physics]]|volume=76|pages=93–123|arxiv=quant-ph/0212023|bibcode=2004RvMP...76...93P|doi=10.1103/RevModPhys.76.93|author-link1=Asher Peres|number=1|s2cid=7481797}}</ref> They are extensively used in the field of quantum information.

In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite, matrices <math>\{F_i\} </math> on a Hilbert space <math> \mathcal{H} </math> that sum to the identity matrix,<ref name="mike_ike">{{Cite book|title=Quantum Computation and Quantum Information|title-link=Quantum Computation and Quantum Information|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac L.|publisher=[[Wikipedia:en:Cambridge University Press]|Cambridge University Press]]|year=2000|isbn=978-0-521-63503-5|edition=1st|location=Cambridge|oclc=634735192|author-link1=W:en:Michael Nielsen|author-link2=Isaac Chuang}}</ref>{{rp|90}}

:<math>\sum_{i=1}^n F_i = \operatorname{I}.</math>

In quantum mechanics, the POVM element <math>F_i</math> is associated with the measurement outcome <math>i</math>, such that the probability of obtaining it when making a measurement on the quantum state <math>\rho</math> is given by

:<math>\text{Prob}(i) = \operatorname{tr}(\rho F_i) </math>,

where <math>\operatorname{tr}</math> is the trace operator. When the quantum state being measured is a pure state <math>|\psi\rangle</math> this formula reduces to

:<math>\text{Prob}(i) = \operatorname{tr}(|\psi\rangle\langle\psi| F_i) = \langle\psi|F_i|\psi\rangle</math>.

===State change due to measurement===
*Main resource Quantum operation

A measurement upon a quantum system will generally bring about a change of the quantum state of that system. Writing a POVM does not provide the complete information necessary to describe this state-change process.<ref name=":2" /> To remedy this, further information is specified by decomposing each POVM element into a product:
:<math>E_i =  A^\dagger_{i} A_{i}.</math>
The Kraus operators <math>A_{i}</math>, named for Karl Kraus, provide a specification of the state-change process.{{efn|Hellwig and Kraus<ref>{{Cite journal|last1=Hellwig|first1=K. -E.|last2=Kraus|first2=K.|author-link2=W:en:Karl Kraus (physicist)|date=September 1969|title=Pure operations and measurements|url=https://projecteuclid.org/download/pdf_1/euclid.cmp/1103841220|journal=Communications in Mathematical Physics|language=en|volume=11|issue=3|pages=214–220|doi=10.1007/BF01645807|s2cid=123659396|issn=0010-3616}}</ref><ref>{{Cite book
| publisher = Springer-Verlag
| isbn = 978-3-5401-2732-1
| oclc = 925001331
| last = Kraus
| first = Karl
| author-link = W:en:Karl Kraus (physicist)
| title = States, effects, and operations: fundamental notions of quantum theory
| series = Lectures in mathematical physics at the University of Texas at Austin
| volume = 190
| date = 1983
| url = https://books.google.com/books?id=fRBBAQAAIAAJ
}}</ref> originally introduced operators with two indices, <math>A_{ij}</math>, such that <math>\textstyle \sum_j A_{ij} A^\dagger_{ij} = E_i</math>. The extra index does not affect the computation of the measurement outcome probability, but it does play a role in the state-update rule, with the post-measurement state being now proportional to <math>\textstyle \sum_j A^\dagger_{ij} \rho A_{ij}</math>. This can be regarded as representing <math>\textstyle E_i</math> as a coarse-graining together of multiple outcomes of a more fine-grained POVM.<ref>{{Cite journal|last1=Barnum|first1=Howard|last2=Nielsen|first2=M. A.|author-link2=w:en:Michael Nielsen|last3=Schumacher|first3=Benjamin|author-link3=W:en:Benjamin Schumacher|date=1 June 1998|title=Information transmission through a noisy quantum channel|journal=Physical Review A|language=en|volume=57|issue=6|pages=4153–4175|arxiv=quant-ph/9702049|doi=10.1103/PhysRevA.57.4153|bibcode=1998PhRvA..57.4153B|s2cid=13717391|issn=1050-2947}}</ref><ref>{{Cite journal|last1=Fuchs|first1=Christopher A.|last2=Jacobs|first2=Kurt|date=16 May 2001|title=Information-tradeoff relations for finite-strength quantum measurements|journal=Physical Review A|language=en|volume=63|issue=6|article-number=062305|arxiv=quant-ph/0009101|bibcode=2001PhRvA..63f2305F|doi=10.1103/PhysRevA.63.062305|s2cid=119476175|issn=1050-2947}}</ref><ref>{{Cite journal|last=Poulin|first=David|date=7 February 2005|title=Macroscopic observables|journal=Physical Review A|language=en|volume=71|issue=2|article-number=022102|arxiv=quant-ph/0403212|bibcode=2005PhRvA..71b2102P|doi=10.1103/PhysRevA.71.022102|s2cid=119364450|issn=1050-2947}}</ref> Kraus operators with two indices also occur in generalized models of system-environment interaction.<ref name="mike_ike"/>{{rp|364}}}} They are not necessarily self-adjoint, but the products <math>A^\dagger_{i} A_{i}</math> are. If upon performing the measurement the outcome <math>E_i</math> is obtained, then the initial state <math>\rho</math> is updated to
:<math>\rho \to \rho' =  \frac{A_{i} \rho A^\dagger_{i}}{\mathrm{Prob}(i)} =  \frac{A_{i} \rho A^\dagger_{i}}{\operatorname{tr} (\rho E_i)}.</math>
An important special case is the Lüders rule, named for Gerhart Lüders.<ref>{{cite journal|last=Lüders|first=Gerhart|author-link=W:en:Gerhart Lüders|year=1950|title=Über die Zustandsänderung durch den Messprozeß|journal=Annalen der Physik|volume=443|issue=5–8|page=322|bibcode=1950AnP...443..322L|doi=10.1002/andp.19504430510}} Translated by K. A. Kirkpatrick as {{Cite journal|last=Lüders|first=Gerhart|author-link=W:en:Gerhart Lüders|date=3 April 2006|title=Concerning the state-change due to the measurement process|journal=Annalen der Physik|volume=15|issue=9|pages=663&ndash;670|arxiv=quant-ph/0403007|bibcode=2006AnP...518..663L|doi=10.1002/andp.200610207|s2cid=119103479}}</ref><ref name="Busch2009">{{Citation|last1=Busch|first1=Paul|author-link=W:Paul Busch (physicist) |title=Lüders Rule|date=2009|work=Compendium of Quantum Physics|pages=356–358|editor-last=Greenberger|editor-first=Daniel|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-70626-7_110|isbn=978-3-540-70622-9|last2=Lahti|first2=Pekka|editor2-last=Hentschel|editor2-first=Klaus|editor3-last=Weinert|editor3-first=Friedel}}</ref> If the POVM is itself a PVM, then the Kraus operators can be taken to be the projectors onto the eigenspaces of the von&nbsp;Neumann observable:
:<math>\rho \to \rho'  = \frac{\Pi_i \rho \Pi_i}{\operatorname{tr} (\rho \Pi_i)}.</math>
If the initial state <math>\rho</math> is pure, and the projectors <math>\Pi_i</math> have rank 1, they can be written as projectors onto the vectors <math>|\psi\rangle</math> and <math>|i\rangle</math>, respectively. The formula simplifies thus to
:<math>\rho = |\psi\rangle\langle\psi| \to \rho' = \frac{|i\rangle\langle i | \psi\rangle\langle\psi | i \rangle\langle i|}{|\langle i |\psi \rangle|^2} = |i\rangle\langle i|.</math>
Lüders rule has historically been known as the "reduction of the wave packet" or the "collapse of the wavefunction".<ref name="Busch2009"/><ref>{{cite book |last=Jammer |first=Max |author-link=W:Max Jammer |chapter=A Consideration of the Philosophical Implications of the New Physics |date=1979 |title=The Structure and Development of Science |series=Boston Studies in the Philosophy of Science |volume=59 |pages=41–61 |editor-last=Radnitzky |editor-first=Gerard |chapter-url=http://link.springer.com/10.1007/978-94-009-9459-1_3 |access-date=2024-03-26 |place=Dordrecht |publisher=Springer Netherlands |doi=10.1007/978-94-009-9459-1_3 |isbn=978-90-277-0995-0 |editor2-last=Andersson |editor2-first=Gunnar}}</ref><ref>{{cite book|first=Osvaldo |last=Pessoa |chapter=The Measurement Problem |title=The Oxford Handbook of the History of Quantum Interpretations |pages=281–302 |editor-first=Olival |editor-last=Freire |publisher=Oxford University Press |year=2022 |isbn=978-0-191-88008-7 |doi=10.1093/oxfordhb/9780198844495.013.0012|doi-broken-date=1 July 2025 }}</ref> The pure state <math>|i\rangle</math> implies a probability-one prediction for any von&nbsp;Neumann observable that has <math>|i\rangle</math> as an eigenvector. Introductory texts on quantum theory often express this by saying that if a quantum measurement is repeated in quick succession, the same outcome will occur both times. This is an oversimplification, since the physical implementation of a quantum measurement may involve a process like the absorption of a photon; after the measurement, the photon does not exist to be measured again.<ref name="mike_ike"/>{{rp|91}}

We can define a linear, trace-preserving, completely positive map, by summing over all the possible post-measurement states of a POVM without the normalisation:
:<math>\rho \to \sum_i A_i \rho A^\dagger_i.</math>
It is an example of a quantum channel,<ref name=":2" />{{Rp|150}} and can be interpreted as expressing how a quantum state changes if a measurement is performed but the result of that measurement is lost.<ref name=":2" />{{Rp|159}}

===Examples===
[[Image:Bloch sphere representation of optimal POVM and states for unambiguous quantum state discrimination.svg|thumb|right|Bloch sphere representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination<ref>{{Cite journal|last1=Peres|first1=Asher|author-link=W:Asher Peres|last2=Terno|first2=Daniel R.|date=1998|title=Optimal distinction between non-orthogonal quantum states|journal=Journal of Physics A: Mathematical and General|language=en|volume=31|issue=34|pages=7105–7111|doi=10.1088/0305-4470/31/34/013|issn=0305-4470 |arxiv=quant-ph/9804031|bibcode=1998JPhA...31.7105P|s2cid=18961213}}</ref>
 on the states <math>|\psi\rangle=|0\rangle</math> and <math>|\varphi\rangle=(|0\rangle+|1\rangle)/\sqrt2</math>. Note that on the Bloch sphere orthogonal states are antiparallel.]]
The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. A pure state for a qubit can be written as a linear combination of two orthogonal basis states <math>|0 \rangle </math> and <math>|1 \rangle </math> with complex coefficients:
: <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle  </math>
A measurement in the <math>(|0\rangle, |1\rangle)</math> basis will yield outcome <math>|0 \rangle </math> with probability <math>| \alpha |^2</math> and outcome  <math>|1 \rangle </math> with probability <math>| \beta |^2</math>, so by normalization,
: <math>| \alpha |^2 + | \beta |^2 = 1.</math>

An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for <math>2 \times 2</math> self-adjoint matrices:<ref name=":2" />{{Rp|126}}

:<math>\rho = \tfrac{1}{2}\left(I + r_x \sigma_x + r_y \sigma_y + r_z \sigma_z\right),</math>
where the real numbers <math>(r_x, r_y, r_z)</math> are the coordinates of a point within the unit ball and
:<math>
  \sigma_x =
    \begin{pmatrix}
      0&1\\
      1&0
    \end{pmatrix}, \quad
  \sigma_y =
    \begin{pmatrix}
      0&-i\\
      i&0
    \end{pmatrix}, \quad
  \sigma_z =
    \begin{pmatrix}
      1&0\\
      0&-1
    \end{pmatrix} .</math>
POVM elements can be represented likewise, though the trace of a POVM element is not fixed to equal 1. The Pauli matrices are traceless and orthogonal to one another with respect to the Hilbert–Schmidt inner product, and so the coordinates <math>(r_x, r_y, r_z)</math> of the state <math>\rho</math> are the expectation values of the three von Neumann measurements defined by the Pauli matrices.<ref name=":2" />{{Rp|126}} If such a measurement is applied to a qubit, then by the Lüders rule, the state will update to the eigenvector of that Pauli matrix corresponding to the measurement outcome. The eigenvectors of <math>\sigma_z</math> are the basis states <math>|0\rangle</math> and <math>|1\rangle</math>, and a measurement of <math>\sigma_z</math> is often called a measurement in the "computational basis."<ref name=":2">{{Cite book|title=Quantum Information Theory|last=Wilde|first=Mark M.|publisher=Cambridge University Press|year=2017|isbn=978-1-107-17616-4|edition=2nd|arxiv=1106.1445|doi=10.1017/9781316809976.001|oclc=973404322|author-link=W:en:Mark Wilde|s2cid=2515538}}</ref>{{Rp|76}} After a measurement in the computational basis, the outcome of a <math>\sigma_x</math> or <math>\sigma_y</math> measurement is maximally uncertain.

A pair of qubits together form a system whose Hilbert space is 4-dimensional. One significant von Neumann measurement on this system is that defined by the Bell basis,<ref name=":0" />{{Rp|36}} a set of four maximally entangled states:
:<math>\begin{align}
|\Phi^+\rangle &= \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B) \\
|\Phi^-\rangle &= \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B) \\
|\Psi^+\rangle &= \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B) \\
|\Psi^-\rangle &= \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B)
\end{align}</math>

[[Image:Aufenthaltswahrscheinlichkeit harmonischer Oszillator.png|thumb|Probability density <math>P_n(x)</math> for the outcome of a position measurement given the energy eigenstate <math>|n\rangle</math> of a 1D harmonic oscillator]]A common and useful example of quantum mechanics applied to a continuous degree of freedom is the quantum harmonic oscillator.<ref>{{Cite book|last=Weinberg|first=Steven|title=Lectures on quantum mechanics|publisher=Cambridge University Press|year=2015|isbn=978-1-107-11166-0|edition=Second|location=Cambridge, United Kingdom|oclc=910664598|author-link=W:Steven Weinberg}}</ref>{{Rp|24}} This system is defined by the Hamiltonian
:<math>{H} = \frac{{p}^2}{2m} + \frac{1}{2}m\omega^2 {x}^2,</math>
where <math>{H}</math>, the momentum operator <math>{p}</math> and the position operator <math>{x}</math> are self-adjoint operators on the Hilbert space of square-integrable functions on the real line. The energy eigenstates solve the time-independent Schrödinger equation:
:<math>{H} |n\rangle = E_n |n\rangle.</math>
These eigenvalues can be shown to be given by
:<math>E_n = \hbar\omega\left(n + \tfrac{1}{2}\right),</math>

and these values give the possible numerical outcomes of an energy measurement upon the oscillator. The set of possible outcomes of a ''position'' measurement on a harmonic oscillator is continuous, and so predictions are stated in terms of a probability density function <math>P(x)</math> that gives the probability of the measurement outcome lying in the infinitesimal interval from <math>x</math> to <math>x + dx</math>.

==History of the measurement concept==
===The "old quantum theory"===
The old quantum theory is a collection of results from the years 1900–1925<ref>{{cite book |title=Subtle is the Lord: The Science and the Life of Albert Einstein |edition=illustrated |first1=Abraham |last1=Pais |author-link=W:Abraham Pais |publisher=Oxford University Press |year=2005 |isbn=978-0-19-280672-7 |page=28 |url=https://books.google.com/books?id=0QYTDAAAQBAJ}}</ref> which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics.<ref>{{cite book|last = ter Haar|first =D.|title =The Old Quantum Theory|url = https://archive.org/details/oldquantumtheory0000haar|url-access = registration|publisher=Pergamon Press|year=1967|pages = [https://archive.org/details/oldquantumtheory0000haar/page/206 206]|isbn = 978-0-08-012101-7}}</ref> The theory is now understood as a semi-classical approximation<ref>{{cite web|title=Semi-classical approximation |website=Encyclopedia of Mathematics |url=https://www.encyclopediaofmath.org/index.php?title=Semi-classical_approximation |access-date=1 February 2020}}</ref> to modern quantum mechanics.<ref>{{cite book|last1=Sakurai |first1=J. J. |author-link1=J. J. Sakurai |last2=Napolitano |first2=J. |title=Modern Quantum Mechanics|publisher=Pearson|year=2014|isbn=978-1-292-02410-3|chapter=Quantum Dynamics |oclc=929609283}}</ref> Notable results from this period include Max Planck's calculation of the blackbody radiation spectrum, Albert Einstein's explanation of the photoelectric effect, Einstein and Peter Debye's work on the specific heat of solids, Niels Bohr and |Hendrika van Leeuwen's proof that classical physics cannot account for magnetism, Bohr's model of the hydrogen atom and ArnoWld Sommerfeld's extension of the Bohr model to include relativistic effects.

[[File:Quantum_book1_mechanics_measurements_yellow.png|300px|thumb|Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result.]]
The Stern–Gerlach experiment, proposed in 1921 and implemented in 1922,<ref name=SG>{{cite journal
 |last1=Gerlach |first1=W.
 |last2=Stern |first2=O.
 |title=Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld
 |journal=Zeitschrift für Physik
 |volume=9 |issue=1
 |pages=349–352
 |year=1922
 |doi=10.1007/BF01326983
|bibcode = 1922ZPhy....9..349G |s2cid=186228677
 }}</ref><ref>{{cite journal
 |last1=Gerlach |first1=W.
 |last2=Stern |first2=O.
 |title=Das magnetische Moment des Silberatoms
 |journal=Zeitschrift für Physik
 |volume=9 |issue=1
 |pages=353–355
 |year=1922
 |doi=10.1007/BF01326984
|bibcode = 1922ZPhy....9..353G |s2cid=126109346
 }}</ref><ref>{{cite journal
 |last1=Gerlach |first1=W.
 |last2=Stern |first2=O.
 |title=Der experimentelle Nachweis des magnetischen Moments des Silberatoms
 |journal=Zeitschrift für Physik
 |volume=8 |pages=110–111
 |year=1922
 |issue=1
 |doi=10.1007/BF01329580
|bibcode = 1922ZPhy....8..110G |s2cid=122648402
 |url=https://zenodo.org/record/1525119
 }}</ref> became a prototypical example of a quantum measurement having a discrete set of possible outcomes. In the original experiment, silver atoms were sent through a spatially varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment are deflected, due to the magnetic field gradient, from a straight path. The screen reveals discrete points of accumulation, rather than a continuous distribution, owing to the particles' quantized spin.<ref>{{cite book|first1=Allan |last1=Franklin |author-link1=Allan Franklin |first2=Slobodan |last2=Perovic |chapter= Experiment in Physics, Appendix 5 |title=The Stanford Encyclopedia of Philosophy |edition= Winter 2016 |editor=Edward N. Zalta |chapter-url=https://plato.stanford.edu/archives/win2016/entries/physics-experiment/app5.html |access-date=14 August 2018}}</ref><ref name="FH2003">{{cite journal
 |last1=Friedrich |first1=B. |last2=Herschbach |first2=D.
 |title=Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics
 |journal=Physics Today
 |volume=56 |page=53
 |year=2003
 |doi=10.1063/1.1650229
 |issue=12
|bibcode = 2003PhT....56l..53F |s2cid=17572089 |doi-access=free}}</ref><ref>{{Cite journal|last1=Zhu|first1=Guangtian|last2=Singh|first2=Chandralekha|author-link2=Chandralekha Singh|date=May 2011|title=Improving students' understanding of quantum mechanics via the Stern–Gerlach experiment|url=http://aapt.scitation.org/doi/10.1119/1.3546093|journal=American Journal of Physics|language=en|volume=79|issue=5|pages=499–507|doi=10.1119/1.3546093|arxiv=1602.06367 |bibcode=2011AmJPh..79..499Z |s2cid=55077698 |issn=0002-9505}}</ref>

===Transition to the "new" quantum theory===
A 1925 paper by Werner Heisenberg, known in English as "Quantum theoretical re-interpretation of kinematic and mechanical relations", marked a pivotal moment in the maturation of quantum physics.<ref name="sources-intro">{{cite encyclopedia |first=B. L. |last=van der Waerden |author-link=Bartel Leendert van der Waerden |title=Introduction, Part II |encyclopedia=Sources of Quantum Mechanics |publisher=Dover |year=1968 |isbn=0-486-61881-1}}</ref> Heisenberg sought to develop a theory of atomic phenomena that relied only on "observable" quantities. At the time, and in contrast with the later standard presentation of quantum mechanics, Heisenberg did not regard the position of an electron bound within an atom as "observable". Instead, his principal quantities of interest were the frequencies of light emitted or absorbed by atoms.<ref name="sources-intro"/>

The uncertainty principle dates to this period. It is frequently attributed to Heisenberg, who introduced the concept in analyzing a thought experiment where one attempts to measure an electron's position and momentum simultaneously. However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant. The precise mathematical statement of the position-momentum uncertainty principle is due to Earle Hesse Kennard, Wolfgang Pauli, and Hermann Weyl, and its generalization to arbitrary pairs of noncommuting observables is due to Howard P. Robertson and Erwin Schrödinger.<ref>{{Cite journal|last1=Busch|first1=Paul|author-link1=Paul Busch (physicist) |last2=Lahti|first2=Pekka|last3=Werner|first3=Reinhard F.|date=17 October 2013|title=Proof of Heisenberg's Error-Disturbance Relation|journal=Physical Review Letters|language=en|volume=111|issue=16|article-number=160405|doi=10.1103/PhysRevLett.111.160405|pmid=24182239|arxiv=1306.1565|issn=0031-9007|bibcode=2013PhRvL.111p0405B|s2cid=24507489}}</ref><ref>{{Cite journal|last=Appleby|first=David Marcus|date=6 May 2016|title=Quantum Errors and Disturbances: Response to Busch, Lahti and Werner|journal=Entropy|language=en|volume=18|issue=5|page=174|doi=10.3390/e18050174|arxiv=1602.09002|bibcode=2016Entrp..18..174A|doi-access=free}}</ref>

Writing <math>{x}</math> and <math>{p}</math> for the self-adjoint operators representing position and momentum respectively, a standard deviation of position can be defined as
:<math>\sigma_x=\sqrt{\langle {x}^2 \rangle-\langle {x}\rangle^2},</math>
and likewise for the momentum:
:<math>\sigma_p=\sqrt{\langle {p}^2 \rangle-\langle {p}\rangle^2}.</math>
The Kennard–Pauli–Weyl uncertainty relation is
:<math>\sigma_x \sigma_p \geq \frac{\hbar}{2}.</math>
This inequality means that no preparation of a quantum particle can imply simultaneously precise predictions for a measurement of position and for a measurement of momentum.<ref name="L&L">{{cite book
 |first1=L.D. |last1=Landau |author-link1=Lev Landau
 |first2=E.M. |last2=Lifschitz |author-link2=Evgeny Lifshitz
 |year=1977
 |title=Quantum Mechanics: Non-Relativistic Theory
 |edition=3rd |volume=3
 |publisher=Pergamon Press
 |isbn=978-0-08-020940-1
 |oclc=2284121
 |url=https://archive.org/details/QuantumMechanics_104
}}</ref> The Robertson inequality generalizes this to the case of an arbitrary pair of self-adjoint operators <math>A</math> and <math>B</math>. The commutator of these two operators is
:<math>[A,B]=AB-BA,</math>
and this provides the lower bound on the product of standard deviations:
:<math>\sigma_A \sigma_B \geq \left| \frac{1}{2i}\langle[A,B]\rangle \right| = \frac{1}{2}\left|\langle[A,B]\rangle \right|.</math>
Substituting in the canonical commutation relation <math>[{x},{p}] = i\hbar</math>, an expression first postulated by Max Born in 1925,<ref>{{Cite journal | last1 = Born | first1 = M. | author-link1 =W:Max Born | last2 = Jordan | first2 = P. | author-link2 =W:Pascual Jordan | doi = 10.1007/BF01328531 | title = Zur Quantenmechanik | journal = Zeitschrift für Physik | volume = 34 | pages = 858–888 | year = 1925 | issue = 1 |bibcode = 1925ZPhy...34..858B | s2cid = 186114542 }}</ref> recovers the Kennard–Pauli–Weyl statement of the uncertainty principle.

===From uncertainty to no-hidden-variables===
The existence of the uncertainty principle naturally raises the question of whether quantum mechanics can be understood as an approximation to a more exact theory. Do there exist "hidden variables", more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory can provide? A collection of results, most significantly Bell's theorem, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics.

John Stewart Bell published the theore|John Stewart Bellm now known by his name in 1964, investigating more deeply a thought experiment originally proposed in 1935 by Einstein, Boris Podolsky and Nathan Rosen.<ref>{{cite journal | last1 = Bell | first1 = J. S. | author-link =w:John Stewart Bell | year = 1964 | title = On the Einstein Podolsky Rosen Paradox | url = https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf | journal = Physics Physique Физика | volume = 1 | issue = 3| pages = 195–200 | doi = 10.1103/PhysicsPhysiqueFizika.1.195 | doi-access = free }}</ref><ref name="EPR">{{cite journal | title = Can Quantum-Mechanical Description of Physical Reality be Considered Complete? | date = 15 May 1935 | first1 = A | last1 = Einstein |author-link1=Albert Einstein |first2=B |last2=Podolsky |author-link2=Boris Podolsky |first3=N |last3=Rosen |author-link3=Nathan Rosen | journal = Physical Review | volume = 47 | issue = 10 | pages = 777–780 |bibcode = 1935PhRv...47..777E |doi = 10.1103/PhysRev.47.777 |doi-access = free }}</ref> According to Bell's theorem, if nature actually operates in accord with any theory of ''local'' hidden variables, then the results of a Bell test will be constrained in a particular, quantifiable way. If a Bell test is performed in a laboratory and the results are ''not'' thus constrained, then they are inconsistent with the hypothesis that local hidden variables exist. Such results would support the position that there is no way to explain the phenomena of quantum mechanics in terms of a more fundamental description of nature that is more in line with the rules of classical physics. Many types of Bell test have been performed in physics laboratories, often with the goal of ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. This is known as "closing loopholes in Bell tests".  To date, Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.<ref name="NAT-20180509">{{cite journal |author=The BIG Bell Test Collaboration |title=Challenging local realism with human choices |date=9 May 2018 |journal=Nature |volume=557 |issue=7704 |pages=212–216 |doi=10.1038/s41586-018-0085-3 |pmid=29743691 |bibcode=2018Natur.557..212B |arxiv=1805.04431 |s2cid=13665914 }}</ref><ref>{{Cite web|url=https://www.quantamagazine.org/20170207-bell-test-quantum-loophole/|title=Experiment Reaffirms Quantum Weirdness|last=Wolchover|first=Natalie|author-link=W:Natalie Wolchover|date=7 February 2017|work=Quanta Magazine|language=en-US|access-date=8 February 2020}}</ref>

===Quantum systems as measuring devices===
The Robertson–Schrödinger uncertainty principle establishes that when two observables do not commute, there is a tradeoff in predictability between them. The Wigner–Araki–Yanase theorem demonstrates another consequence of non-commutativity: the presence of a conservation law limits the accuracy with which observables that fail to commute with the conserved quantity can be measured.<ref>See, for example:
*{{Citation|last=Wigner|first=E. P.|title=Philosophical Reflections and Syntheses|chapter=Die Messung quantenmechanischer Operatoren|date=1995|pages=147–154|editor-last=Mehra|editor-first=Jagdish|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-642-78374-6_10|isbn=978-3-540-63372-3|author-link=W:Eugene Wigner|editor-link=w:Jagdish Mehra}}
*{{Cite journal|last1=Araki|first1=Huzihiro|author-link=W:Huzihiro Araki|last2=Yanase|first2=Mutsuo M.|date=15 October 1960|title=Measurement of Quantum Mechanical Operators|journal=Physical Review|language=en|volume=120|issue=2|pages=622–626|doi=10.1103/PhysRev.120.622|bibcode=1960PhRv..120..622A|issn=0031-899X}}
*{{Cite journal|last=Yanase|first=Mutsuo M.|date=15 July 1961|title=Optimal Measuring Apparatus|journal=Physical Review|language=en|volume=123|issue=2|pages=666–668|doi=10.1103/PhysRev.123.666|bibcode=1961PhRv..123..666Y|issn=0031-899X}}
*{{Cite journal|last1=Ahmadi|first1=Mehdi|last2=Jennings|first2=David|last3=Rudolph|first3=Terry|date=28 January 2013|title=The Wigner–Araki–Yanase theorem and the quantum resource theory of asymmetry|journal=New Journal of Physics|language=en|volume=15|issue=1|article-number=013057|doi=10.1088/1367-2630/15/1/013057|arxiv=1209.0921|bibcode=2013NJPh...15a3057A|issn=1367-2630|doi-access=free}}</ref> Further investigation in this line led to the formulation of the Wigner–Yanase skew information.<ref>{{cite journal|doi=10.1103/PhysRevLett.91.180403 |first=Shenlong |last=Luo |title=Wigner–Yanase Skew Information and Uncertainty Relations |journal=Physical Review Letters |year=2003 |volume=91 |number=18 |article-number=180403|pmid=14611271 |bibcode=2003PhRvL..91r0403L }}</ref>

Historically, experiments in quantum physics have often been described in semiclassical terms. For example, the spin of an atom in a Stern–Gerlach experiment might be treated as a quantum degree of freedom, while the atom is regarded as moving through a magnetic field described by the classical theory of Maxwell's equations.<ref name=":1" />{{Rp|24}} But the devices used to build the experimental apparatus are themselves physical systems, and so quantum mechanics should be applicable to them as well. Beginning in the 1950s, Léon Rosenfeld, Carl Friedrich von Weizsäcker and others tried to develop consistency conditions that expressed when a quantum-mechanical system could be treated as a measuring apparatus.<ref name="camilleri2015" /> One proposal for a criterion regarding when a system used as part of a measuring device can be modeled semiclassically relies on the Wigner function, a quasiprobability distribution that can be treated as a probability distribution on phase space in those cases where it is everywhere non-negative.<ref name=":1" />{{Rp|375}}

===Decoherence===
A quantum state for an imperfectly isolated system will generally evolve to be entangled with the quantum state for the environment. Consequently, even if the system's initial state is pure, the state at a later time, found by taking the partial trace of the joint system-environment state, will be mixed. This phenomenon of entanglement produced by system-environment interactions tends to obscure the more exotic features of quantum mechanics that the system could in principle manifest. Quantum decoherence, as this effect is known, was first studied in detail during the 1970s.<ref name="schlosshauer2019">{{cite journal|first=M. |last=Schlosshauer |title=Quantum Decoherence |journal=Physics Reports |volume=831 |year=2019 |pages=1–57 |arxiv=1911.06282 |doi=10.1016/j.physrep.2019.10.001 |bibcode=2019PhR...831....1S|s2cid=208006050 }}</ref> (Earlier investigations into how classical physics might be obtained as a limit of quantum mechanics had explored the subject of imperfectly isolated systems, but the role of entanglement was not fully appreciated.<ref name="camilleri2015">{{cite journal|first1=K. |last1=Camilleri |first2=M. |last2=Schlosshauer |title=Niels Bohr as Philosopher of Experiment: Does Decoherence Theory Challenge Bohr's Doctrine of Classical Concepts? |arxiv=1502.06547 |journal=Studies in History and Philosophy of Modern Physics |volume=49 |pages=73–83 |year=2015 |doi=10.1016/j.shpsb.2015.01.005|bibcode=2015SHPMP..49...73C |s2cid=27697360 }}</ref>) A significant portion of the effort involved in quantum computing research is to avoid the deleterious effects of decoherence.<ref>{{Cite journal|last1=DiVincenzo|first1=David|author-link=W:David DiVincenzo|last2=Terhal|first2=Barbara|author-link2=W:Barbara Terhal|date=March 1998|title=Decoherence: the obstacle to quantum computation|journal=Physics World|volume=11|issue=3|pages=53–58|doi=10.1088/2058-7058/11/3/32|issn=0953-8585}}</ref><ref name=":0" />{{Rp|239}}

To illustrate, let <math>\rho_S</math> denote the initial state of the system, <math>\rho_E</math> the initial state of the environment and <math>H</math> the Hamiltonian specifying the system-environment interaction. The density operator <math>\rho_E</math> can be diagonalized and written as a linear combination of the projectors onto its eigenvectors:
:<math>\rho_E = \sum_i p_i |\psi_i\rangle\langle \psi_i|.</math>
Expressing time evolution for a duration <math>t</math> by the unitary operator <math>U = e^{-iHt/\hbar}</math>, the state for the system after this evolution is
:<math>\rho_S' = {\rm tr}_E U \left[\rho_S \otimes \left(\sum_i p_i |\psi_i\rangle\langle \psi_i|\right)\right] U^\dagger,</math>
which evaluates to
:<math>\rho_S' = \sum_{ij} \sqrt{p_i} \langle \psi_j | U | \psi_i \rangle \rho_S \sqrt{p_i}\langle \psi_i | U^\dagger | \psi_j \rangle.</math>
The quantities surrounding <math>\rho_S</math> can be identified as Kraus operators, and so this defines a quantum channel.<ref name="schlosshauer2019"/>

Specifying a form of interaction between system and environment can establish a set of "pointer states," states for the system that are (approximately) stable, apart from overall phase factors, with respect to environmental fluctuations. A set of pointer states defines a preferred orthonormal basis for the system's Hilbert space.<ref name=":1">{{Cite book|title=Quantum Theory: Concepts and Methods|last=Peres|first=Asher|publisher=Kluwer Academic Publishers|year=1995|isbn=0-7923-2549-4|author-link=W:en:Asher Peres}}</ref>{{Rp|423}}

==Quantum information and computation==
Quantum information science studies how information science and its application as technology depend on quantum-mechanical phenomena. Understanding measurement in quantum physics is important for this field in many ways, some of which are briefly surveyed here.

===Measurement, entropy, and distinguishability===
The von Neumann entropy is a measure of the statistical uncertainty represented by a quantum state. For a density matrix <math>\rho</math>, the von Neumann entropy is
:<math>S(\rho) = -{\rm tr}(\rho \log \rho);</math>
writing <math>\rho</math> in terms of its basis of eigenvectors,
:<math>\rho = \sum_i \lambda_i |i\rangle\langle i|,</math>
the von Neumann entropy is
:<math>S(\rho) = -\sum_i \lambda_i \log \lambda_i.</math>

This is the Shannon entropy of the set of eigenvalues interpreted as a probability distribution, and so the von Neumann entropy is the Shannon entropy of the random variable defined by measuring in the eigenbasis of <math>\rho</math>. Consequently, the von Neumann entropy vanishes when <math>\rho</math> is pure.<ref name=":2" />{{Rp|320}} The von Neumann entropy of <math>\rho</math> can equivalently be characterized as the minimum Shannon entropy for a measurement given the quantum state <math>\rho</math>, with the minimization over all POVMs with rank-1 elements.<ref name=":2" />{{Rp|323}}

Many other quantities used in quantum information theory also find motivation and justification in terms of measurements. For example, the trace distance between quantum states is equal to the largest ''difference in probability'' that those two quantum states can imply for a measurement outcome:<ref name=":2" />{{Rp|254}}
:<math>\frac{1}{2}||\rho-\sigma|| = \max_{0\leq E \leq I} [{\rm tr}(E \rho) - {\rm tr}(E \sigma)].</math>
Similarly, the fidelity of two quantum states, defined by
:<math>F(\rho, \sigma) = \left(\operatorname{Tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}\right)^2,</math>
expresses the probability that one state will pass a test for identifying a successful preparation of the other. The trace distance provides bounds on the fidelity via the Fuchs–van de Graaf inequalities:<ref name=":2" />{{Rp|274}}
:<math>1 - \sqrt{F(\rho,\sigma)} \leq \frac{1}{2}||\rho-\sigma|| \leq \sqrt{1 - F(\rho,\sigma)}.</math>

===Quantum circuits===
[[Image:Qcircuit measure-arrow.svg|150px|thumb|Circuit representation of measurement. The single line on the left-hand side stands for a qubit, while the two lines on the right-hand side represent a classical bit.]]
Quantum circuits are a model for quantum computation in which a computation is a sequence of quantum gates followed by measurements.<ref name=":0">{{Cite book|title-link= Quantum Computing: A Gentle Introduction |title=Quantum Computing: A Gentle Introduction|last1=Rieffel|first1=Eleanor G.|last2=Polak|first2=Wolfgang H.|date=4 March 2011|publisher=MIT Press|isbn=978-0-262-01506-6|language=en|author-link=W:Eleanor Rieffel}}</ref>{{Rp|93}} The gates are reversible transformations on a quantum mechanical analog of an ''n''-bit register. This analogous structure is referred to as an ''n''-qubit register. Measurements, drawn on a circuit diagram as stylized pointer dials, indicate where and how a result is obtained from the quantum computer after the steps of the computation are executed. Without loss of generality, one can work with the standard circuit model, in which the set of gates are single-qubit unitary transformations and controlled NOT gates on pairs of qubits, and all measurements are in the computational basis.<ref name=":0" />{{Rp|93}}<ref>{{Cite journal|last=Terhal|first=Barbara M.|author-link=W:Barbara Terhal|date=7 April 2015|title=Quantum error correction for quantum memories|journal=Reviews of Modern Physics|language=en|volume=87|issue=2|pages=307–346|arxiv=1302.3428|bibcode=2013arXiv1302.3428T|doi=10.1103/RevModPhys.87.307|s2cid=118646257|issn=0034-6861}}</ref>

===Measurement-based quantum computation===
Measurement-based quantum computation (MBQC) is a model of quantum computing in which the answer to a question is, informally speaking, created in the act of measuring the physical system that serves as the computer.<ref name=":0" />{{Rp|317}}<ref>{{cite journal |first1=R. |last1=Raussendorf |first2=D. E. |last2=Browne |first3=H. J. |last3=Briegel |author-link3=W:Hans Jürgen Briegel | title=Measurement based Quantum Computation on Cluster States| journal=Physical Review A| year=2003| volume=68 | issue=2 | article-number=022312 |arxiv=quant-ph/0301052|doi=10.1103/PhysRevA.68.022312|bibcode = 2003PhRvA..68b2312R |s2cid=6197709 }}</ref><ref>{{Cite journal|last1=Childs|first1=Andrew M.|author-link=W:Andrew Childs|last2=Leung|first2=Debbie W.|author-link2=Debbie Leung|last3=Nielsen|first3=Michael A.|author-link3=Michael Nielsen|date=17 March 2005|title=Unified derivations of measurement-based schemes for quantum computation|journal=Physical Review A|language=en|volume=71|issue=3|article-number=032318|arxiv=quant-ph/0404132|doi=10.1103/PhysRevA.71.032318|bibcode=2005PhRvA..71c2318C|s2cid=27097365|issn=1050-2947}}</ref>

===Quantum tomography===
Quantum state tomography is a process by which, given a set of data representing the results of quantum measurements, a quantum state consistent with those measurement results is computed.<ref name="granade2016">{{Cite journal|last1=Granade|first1=Christopher|last2=Combes|first2=Joshua|last3=Cory|first3=D. G.|date=1 January 2016|title=Practical Bayesian tomography|journal=New Journal of Physics|language=en|volume=18|issue=3|article-number=033024|arxiv=1509.03770|doi=10.1088/1367-2630/18/3/033024|issn=1367-2630|bibcode=2016NJPh...18c3024G|s2cid=88521187}}</ref> It is named by analogy with tomography, the reconstruction of three-dimensional images from slices taken through them, as in a CT scan. Tomography of quantum states can be extended to tomography of quantum channels<ref name="granade2016"/> and even of measurements.<ref>{{Cite journal|last1=Lundeen|first1=J. S.|last2=Feito|first2=A.|last3=Coldenstrodt-Ronge|first3=H.|last4=Pregnell|first4=K. L.|last5=Silberhorn|first5=Ch|last6=Ralph|first6=T. C.|last7=Eisert|first7=J.|last8=Plenio|first8=M. B.|last9=Walmsley|first9=I. A.|date=2009|title=Tomography of quantum detectors|journal=Nature Physics|language=en|volume=5|issue=1|pages=27–30|doi=10.1038/nphys1133|arxiv=0807.2444|bibcode=2009NatPh...5...27L|s2cid=119247440 |issn=1745-2481}}</ref>

===Quantum metrology===
Quantum metrology is the use of quantum physics to aid the measurement of quantities that, generally, had meaning in classical physics, such as exploiting quantum effects to increase the precision with which a length can be measured.<ref name="BraunstenCaves1994">{{cite journal | last1=Braunstein | first1=Samuel L. | last2=Caves | first2=Carlton M. |author-link2=W:Carlton Caves | title=Statistical distance and the geometry of quantum states | journal=Physical Review Letters |volume=72 | issue=22 | date=30 May 1994 | doi=10.1103/physrevlett.72.3439 | pmid=10056200 | pages=3439–3443 | bibcode=1994PhRvL..72.3439B}}</ref> A celebrated example is the introduction of squeezed light into the LIGO experiment, which increased its sensitivity to gravitational waves.<ref>{{Cite web|url=https://www.universetoday.com/144272/ligo-will-squeeze-light-to-overcome-the-quantum-noise-of-empty-space/|title=LIGO Will Squeeze Light To Overcome The Quantum Noise Of Empty Space|last=Koberlein|first=Brian|date=5 December 2019|website=Universe Today|language=en-US|access-date=2 February 2020}}</ref><ref>{{Cite journal|last=Ball|first=Philip|author-link=W:Philip Ball |date=5 December 2019|title=Focus: Squeezing More from Gravitational-Wave Detectors|journal=Physics|language=en|volume=12|page=139 |doi=10.1103/Physics.12.139|s2cid=216538409 }}</ref>

==Laboratory implementations==
The range of physical procedures to which the mathematics of quantum measurement can be applied is very broad.<ref name="Peierls"/> In the early years of the subject, laboratory procedures involved the recording of spectral lines, the darkening of photographic film, the observation of scintillations, finding tracks in cloud chambers, and hearing clicks from Geiger counters.{{efn|The glass plates used in the Stern–Gerlach experiment did not darken properly until Stern breathed on them, accidentally exposing them to sulfur from his cheap cigars.<ref name="FH2003"/><ref name="Barad"/>}} Language from this era persists, such as the description of measurement outcomes in the abstract as "detector clicks".<ref>{{Cite journal|last=Englert|first=Berthold-Georg|author-link=W:Berthold-Georg Englert|date=22 November 2013|title=On quantum theory|journal=The European Physical Journal D|language=en|volume=67|issue=11|article-number=238|arxiv=1308.5290|doi=10.1140/epjd/e2013-40486-5|bibcode=2013EPJD...67..238E|s2cid=119293245|issn=1434-6079}}</ref>

The double-slit experiment is a prototypical illustration of quantum interference, typically described using electrons or photons. The first interference experiment to be carried out in a regime where both wave-like and particle-like aspects of photon behavior are significant was G. I. Taylor's test in 1909. Taylor used screens of smoked glass to attenuate the light passing through his apparatus, to the extent that, in modern language, only one photon would be illuminating the interferometer slits at a time. He recorded the interference patterns on photographic plates; for the dimmest light, the exposure time required was roughly three months.<ref>{{cite journal
| first1=G. I. |last1=Taylor | author-link1=W:Geoffrey Ingram Taylor
| title=Interference Fringes with Feeble Light
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| volume=15
| page=114
| date=1909
| url=https://archive.org/details/proceedingsofcam15190810camb/page/114/mode/2up | access-date=7 December 2024}}</ref><ref>{{cite web|url=https://skullsinthestars.com/2018/08/25/taylor-sees-the-feeble-light-1909/ |title=Taylor sees the (feeble) light (1909) |last=Gbur |first=Greg |author-link=W:Greg Gbur |website=Skulls in the Stars |date=25 August 2018 |access-date=24 October 2020}}</ref> In 1974, the Italian physicists Pier Giorgio Merli, Gian Franco Missiroli, and Giulio Pozzi implemented the double-slit experiment using single electrons and a television tube (CRT).<ref>{{cite journal |last1= Merli |first1= P G |last2= Missiroli |first2= G F |last3= Pozzi |first3= G |year= 1976 |title= On the statistical aspect of electron interference phenomena |journal= American Journal of Physics |volume= 44 |issue= 3|pages= 306–307 |doi= 10.1119/1.10184 |bibcode = 1976AmJPh..44..306M}}</ref> A quarter-century later, a team at the University of Vienna performed an interference experiment with buckyballs, in which the buckyballs that passed through the interferometer were ionized by a laser, and the ions then induced the emission of electrons, emissions which were in turn amplified and detected by an electron multiplier.<ref>{{Cite journal |doi = 10.1038/44348|title = Wave–particle duality of C60 molecules|journal = Nature|volume = 401|issue = 6754|pages = 680–682|year = 1999|last1 = Arndt|first1 = Markus|last2 = Nairz|first2 = Olaf|last3 = Vos-Andreae|first3 = Julian|last4 = Keller|first4 = Claudia|last5 = Van Der Zouw|first5 = Gerbrand|last6 = Zeilinger|first6 = Anton|bibcode = 1999Natur.401..680A|pmid = 18494170|s2cid = 4424892}}</ref>

Modern quantum optics experiments can employ single-photon detectors. For example, in the "BIG Bell test" of 2018, several of the laboratory setups used single-photon avalanche diodes. Another laboratory setup used superconducting qubits.<ref name="NAT-20180509"/> The standard method for performing measurements upon superconducting qubits is to couple a qubit with a resonator in such a way that the characteristic frequency of the resonator shifts according to the state for the qubit, and detecting this shift by observing how the resonator reacts to a probe signal.<ref>{{Cite journal|last1=Krantz|first1=Philip|last2=Bengtsson|first2=Andreas|last3=Simoen|first3=Michaël|last4=Gustavsson|first4=Simon|last5=Shumeiko|first5=Vitaly|last6=Oliver|first6=W. D.|last7=Wilson|first7=C. M.|last8=Delsing|first8=Per|last9=Bylander|first9=Jonas|date=9 May 2016|title=Single-shot read-out of a superconducting qubit using a Josephson parametric oscillator|journal=Nature Communications|language=en|volume=7|issue=1|article-number=11417|doi=10.1038/ncomms11417|pmid=27156732|pmc=4865746|arxiv=1508.02886|bibcode=2016NatCo...711417K|issn=2041-1723|doi-access=free}}</ref>

==Interpretations of quantum mechanics==
[[Image:Niels Bohr Albert Einstein4 by Ehrenfest cr.jpg|thumb|Niels Bohr and Albert Einstein, pictured here at Paul Ehrenfest's home in Leiden (December 1925), had a long-running collegial dispute about what quantum mechanics implied for the nature of reality.]]
Despite the consensus among scientists that quantum physics is in practice a successful theory, disagreements persist on a more philosophical level. Many debates in the area known as quantum foundations concern the role of measurement in quantum mechanics. Recurring questions include which interpretation of probability theory is best suited for the probabilities calculated from the Born rule; and whether the apparent randomness of quantum measurement outcomes is fundamental, or a consequence of a deeper deterministic process.<ref name="snapshot">{{Cite journal| arxiv=1301.1069 | title=A Snapshot of Foundational Attitudes Toward Quantum Mechanics | journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics | volume=44 | issue=3 | pages=222–230 | date=6 January 2013 | last1=Schlosshauer | first1=Maximilian | last2=Kofler | first2=Johannes | last3=Zeilinger | first3=Anton | author-link3=Anton Zeilinger | doi=10.1016/j.shpsb.2013.04.004 | bibcode=2013SHPMP..44..222S | s2cid=55537196 }}</ref><ref name="map">{{Cite book|title=What is Quantum Information? |last=Cabello |first=Adán |publisher=Cambridge University Press |year=2017 |isbn=978-1-107-14211-4 |editor-last=Lombardi |editor-first=Olimpia|editor-link=W:Olimpia Lombardi |pages=138&ndash;143 |chapter=Interpretations of quantum theory: A map of madness |arxiv=1509.04711 |editor2-last=Fortin |editor2-first=Sebastian |editor3-last=Holik |editor3-first=Federico |editor4-last=López |editor4-first=Cristian |bibcode=2015arXiv150904711C |doi=10.1017/9781316494233.009|s2cid=118419619 }}</ref><ref>{{Cite journal|last1=Schaffer|first1=Kathryn|last2=Barreto Lemos|first2=Gabriela|date=24 May 2019|title=Obliterating Thingness: An Introduction to the "What" and the "So What" of Quantum Physics|journal=Foundations of Science|volume=26 |pages=7–26 |language=en|arxiv=1908.07936|doi=10.1007/s10699-019-09608-5|s2cid=182656563|issn=1233-1821}}</ref> Worldviews that present answers to questions like these are known as "interpretations" of quantum mechanics; as the physicist N. David Mermin once quipped, "New interpretations appear every year. None ever disappear."<ref name="mermin2012">{{Cite journal|last=Mermin|first=N. David|author-link=W:N. David Mermin |date=1 July 2012|title=Commentary: Quantum mechanics: Fixing the shifty split|journal=Physics Today|volume=65|issue=7|pages=8–10|doi=10.1063/PT.3.1618|issn=0031-9228|bibcode=2012PhT....65g...8M|doi-access=free}}</ref>

A central concern within quantum foundations is the "quantum measurement problem," though how this problem is delimited, and whether it should be counted as one question or multiple separate issues, are contested topics.<ref name="Barad">{{Cite book|title=Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning|last=Barad|first=Karen|publisher=Duke University Press|year=2007|isbn=978-0-8223-3917-5|language=en|oclc=1055296186|author-link=W:Karen Barad}}</ref><ref>{{cite encyclopedia|first1=Jeffrey |last1=Bub |author-link1=Jeffrey Bub |first2=Itamar |last2=Pitowsky |title=Two dogmas about quantum mechanics |arxiv=0712.4258 |encyclopedia=Many Worlds? |year=2010 |pages=433–459 |publisher= |isbn=978-0-19-956056-1 |oclc=696602007}}</ref> Of primary interest is the seeming disparity between apparently distinct types of time evolution. Von Neumann declared that quantum mechanics contains "two fundamentally different types" of quantum-state change.<ref>{{cite book |title=Mathematical Foundations of Quantum Mechanics. New Edition |first=John |last=von Neumann |author-link=W:John von Neumann |translator=Robert T. Beyer |editor-first=Nicholas A. |editor-last=Wheeler |publisher=Princeton University Press |date=2018 |isbn=9-781-40088-992-1 |oclc=1021172445}}</ref>{{rp|§V.1}} First, there are those changes involving a measurement process, and second, there is unitary time evolution in the absence of measurement. The former is stochastic and discontinuous, writes von Neumann, and the latter deterministic and continuous. This dichotomy has set the tone for much later debate.<ref>{{Citation|last=Wigner|first=E. P.|title=Philosophical Reflections and Syntheses|chapter=Review of the Quantum-Mechanical Measurement Problem|date=1995|pages=225–244|editor-last=Mehra|editor-first=Jagdish|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-642-78374-6_19|isbn=978-3-540-63372-3|author-link=W:Eugene Wigner|editor-link=W:Jagdish Mehra}}</ref><ref name="stanford3">{{Cite book|last=Faye|first=Jan|title=Stanford Encyclopedia of Philosophy|publisher=Metaphysics Research Lab, Stanford University|year=2019|editor-last=Zalta|editor-first=Edward N.|chapter=Copenhagen Interpretation of Quantum Mechanics|author-link=W:Jan Faye|chapter-url=https://plato.stanford.edu/entries/qm-copenhagen/}}</ref> Some interpretations of quantum mechanics find the reliance upon two different types of time evolution distasteful and regard the ambiguity of when to invoke one or the other as a deficiency of the way quantum theory was historically presented.<ref name="Bell-against-measurement">{{Cite journal|last=Bell|first=John|author-link=W:John Stewart Bell |date=1990|title=Against 'measurement'|journal=Physics World |language=en|volume=3|issue=8|pages=33–41|doi=10.1088/2058-7058/3/8/26|issn=2058-7058}}</ref> To bolster these interpretations, their proponents have worked to derive ways of regarding "measurement" as a secondary concept and deducing the seemingly stochastic effect of measurement processes as approximations to more fundamental deterministic dynamics. However, consensus has not been achieved among proponents of the correct way to implement this program, and in particular how to justify the use of the Born rule to calculate probabilities.<ref>{{cite encyclopedia|first=Adrian |last=Kent |author-link=W:Adrian Kent |title=One world versus many: the inadequacy of Everettian accounts of evolution, probability, and scientific confirmation |arxiv=0905.0624 |encyclopedia=Many Worlds? |year=2010 |pages=307–354 |publisher=Oxford University Press |isbn=978-0-19-956056-1 |oclc=696602007}}</ref><ref name="stanford1">{{Cite book|chapter-url=https://plato.stanford.edu/entries/qm-everett/|title=Stanford Encyclopedia of Philosophy|last=Barrett|first=Jeffrey|publisher=Metaphysics Research Lab, Stanford University|year=2018|editor-last=Zalta|editor-first=Edward N.|chapter=Everett's Relative-State Formulation of Quantum Mechanics}}</ref> Other interpretations regard quantum states as statistical information about quantum systems, thus asserting that abrupt and discontinuous changes of quantum states are not problematic, simply reflecting updates of the available information.<ref name="Peierls">{{Cite journal|last=Peierls|first=Rudolf|author-link=W:Rudolf Peierls |date=1991|title=In defence of "measurement"|journal=Physics World |language=en|volume=4|issue=1|pages=19–21|doi=10.1088/2058-7058/4/1/19|issn=2058-7058}}</ref><ref name="stanford2">{{Cite book|chapter-url=https://plato.stanford.edu/entries/quantum-bayesian/|title=Stanford Encyclopedia of Philosophy|last=Healey|first=Richard|publisher=Metaphysics Research Lab, Stanford University|year=2016|editor-last=Zalta|editor-first=Edward N.|chapter=Quantum-Bayesian and Pragmatist Views of Quantum Theory}}</ref> Of this line of thought, Bell asked, "''Whose'' information? Information about ''what''?"<ref name="Bell-against-measurement" /> Answers to these questions vary among proponents of the informationally-oriented interpretations.<ref name="map" /><ref name="stanford2" />
==Look at==
{{Div col|colwidth=20em}}
* Einstein's thought experiments
* Holevo's theorem
* Quantum error correction
* Quantum limit
* Quantum logic
* Quantum Zeno effect
* Schrödinger's cat
* SIC-POVM
{{div col end}}

==Further reading==
{{Wikiquote}}
* {{cite book|editor-first1=John A. |editor-last1=Wheeler |editor-link1=John Archibald Wheeler |editor-first2=Wojciech H. |editor-last2=Zurek |editor-link2=Wojciech H. Zurek  |title=Quantum Theory and Measurement| publisher=Princeton University Press| year=1983| isbn= 978-0-691-08316-2}}
* {{cite book|author-link1=Vladimir Braginsky |first1=Vladimir B. |last1=Braginsky |first2=Farid Ya. |last2=Khalili| title=Quantum Measurement| publisher=Cambridge University Press |year=1992| isbn= 978-0-521-41928-4}}
* {{cite book|title= The Quantum Challenge: Modern Research On The Foundations Of Quantum Mechanics |first1=George S. |last1=Greenstein  |first2=Arthur G. |last2=Zajonc  |edition=2nd|year=2006|isbn= 978-0-7637-2470-2}}
* {{cite book|first1=Orly |last1=Alter |author-link2=Yoshihisa Yamamoto (scientist) |first2=Yoshihisa |last2=Yamamoto |title=Quantum Measurement of a Single System|location=New York |publisher=Wiley|year=2001|doi=10.1002/9783527617128|isbn=978-0-471-28308-9 }}
* {{cite book |author1=Andrew N. Jordan |author2=Irfan A. Siddiqi |title=Quantum Measurement: Theory and Practice |publisher=Cambridge University Press |year=2024 |isbn=978-1-009-10006-9}}

== Notes ==
{{Notelist|30em}}
==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
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=References=
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{{Author|Harold Foppele}}
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