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{{Quantum book backlink|Measurement and information}}

A '''quantum measurement''' is the process by which a [[Physics:Quantum Exactly solvable quantum systems|quantum system]] is probed so as to yield a result, such as a position, momentum, spin, or energy value. Unlike in classical physics, the predictions of [[Physics:Quantum mechanics
quantum mechanics]] are generally [[probability|probabilistic]]: the theory does not usually specify a single certain outcome, but gives the probabilities for different possible outcomes. These probabilities are calculated by combining the system’s [[Physics:Quantum Stationary states|quantum state]] with a mathematical description of the measurement, using the [[Physics:Quantum Postulates|Born rule]].<ref name="Peres1995">{{cite book|last=Peres|first=Asher|title=Quantum Theory: Concepts and Methods|publisher=Kluwer Academic Publishers|year=1995|isbn=0-7923-2549-4}}</ref><ref name="Holevo2001">{{cite book|last=Holevo|first=Alexander S.|author-link=Alexander Holevo|title=Statistical Structure of Quantum Theory|publisher=Springer|series=Lecture Notes in Physics|year=2001|isbn=3-540-42082-7}}</ref>

For example, an [[Physics:electron|electron]] can be described by a quantum state that assigns a [[Physics:Probability amplitude|probability amplitude]] to each point in space. Applying the Born rule gives the probabilities of finding the electron in one region or another if its position is measured. The same state can also be used to predict the outcomes of a momentum measurement, but the [[Physics:Quantum Uncertainty principle|uncertainty principle]] implies that position and momentum cannot both be predicted with arbitrary precision at the same time.<ref name="LandauLifshitz1977">{{cite book|last1=Landau|first1=L. D.|author-link1=Lev Landau|last2=Lifshitz|first2=E. M.|author-link2=Evgeny Lifshitz|title=Quantum Mechanics: Non-Relativistic Theory|edition=3rd|volume=3|publisher=Pergamon Press|year=1977|isbn=978-0-08-020940-1}}</ref>

A central feature of quantum measurement is that the act of measurement generally changes the quantum state of the system. In traditional formulations, this is described by the [[Physics:Quantum Wavefunction|collapse of the wavefunction]], or more precisely by the [[Physics:Quantum mechanics measurements|Lüders rule]] in the case of projective measurements.<ref>{{cite journal|last=Lüders|first=Gerhart|author-link=Gerhart Lüders|title=Über die Zustandsänderung durch den Messprozeß|journal=[[Annalen der Physik]]|volume=443|issue=5–8|year=1950|pages=322–328|doi=10.1002/andp.19504430510|bibcode=1950AnP...443..322L}}</ref> More generally, quantum measurements can be represented using [[Physics:Quantum POVM|positive-operator-valued measure]]s (POVMs) and [[Physics:Quantum operation|Kraus operator]]s.<ref name="Wilde2017">{{cite book|last=Wilde|first=Mark M.|author-link=Mark Wilde|title=Quantum Information Theory|publisher=Cambridge University Press|edition=2nd|year=2017|isbn=978-1-107-17616-4|doi=10.1017/9781316809976.001|arxiv=1106.1445}}</ref>

On the conceptual side, quantum measurement has long been at the center of debates about the meaning of quantum mechanics. These debates are closely linked to the [[Physics:Quantum Measurement problem|measurement problem]] and the many [[Physics:Quantum Interpretations of quantum mechanics|interpretations of quantum mechanics]].<ref>{{cite journal|last=Mermin|first=N. David|author-link=N. David Mermin|title=Commentary: Quantum mechanics: Fixing the shifty split|journal=[[Physics Today]]|volume=65|issue=7|year=2012|pages=8–10|doi=10.1063/PT.3.1618|bibcode=2012PhT....65g...8M}}</ref>

<div style="float:right; margin:0 0 12px 12px; background:#fff8cc; border:1px solid #e0d890; padding:8px; max-width:420px;">
[[File:Stern-Gerlach_measurement_yellow_bg.png|400px]]
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Quantum measurement of spin via the Stern–Gerlach apparatus: the system is projected onto discrete eigenstates corresponding to measurement outcomes.
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</div>

== Mathematical formalism ==

=== Observables and Hilbert space ===
In the standard mathematical formulation of quantum mechanics, every physical system is associated with a [[Hilbert space]], and each possible state of the system corresponds to a vector or, more generally, a [[density operator]] on that space.<ref name="Holevo2001" /> Physical quantities such as position, momentum, energy and angular momentum are represented by [[self-adjoint operator]]s, traditionally called '''observables'''.<ref name="Peres1995" />

In finite-dimensional cases, such as the quantum theory of [[Spin (physics)|spin]], the mathematics is comparatively simple. In infinite-dimensional Hilbert spaces, which arise for continuous observables like position and momentum, additional tools from [[functional analysis]] and [[spectral theory]] are needed.<ref name="Peres1995" />

=== Projective measurement ===
In the von Neumann formulation, a measurement is associated with an orthonormal basis of eigenvectors of an observable. If the system is described by a density operator <math>\rho</math>, then the probability of obtaining the outcome corresponding to projection operator <math>\Pi_i</math> is given by the [[Born rule]]:
:<math>P(x_i)=\operatorname{tr}(\Pi_i\rho).</math>

The expectation value of an observable <math>A</math> in the state <math>\rho</math> is
:<math>\langle A\rangle=\operatorname{tr}(A\rho).</math>

A density operator of rank 1 is called a '''pure state'''; all others are '''mixed states'''. A pure state gives certainty for at least one measurement outcome, while mixed states represent statistical uncertainty or entanglement with other systems.<ref name="Holevo2001" /><ref>{{cite journal|last=Kirkpatrick|first=K. A.|title=The Schrödinger-HJW Theorem|journal=[[Foundations of Physics Letters]]|volume=19|issue=1|year=2006|pages=95–102|doi=10.1007/s10702-006-1852-1|arxiv=quant-ph/0305068|bibcode=2006FoPhL..19...95K|s2cid=15995449}}</ref>

A fundamental result related to this formalism is [[Gleason's theorem]], which shows that probability assignments satisfying natural consistency conditions must arise from the Born rule applied to some density operator.<ref>{{cite journal|last=Gleason|first=Andrew M.|author-link=Andrew M. Gleason|title=Measures on the closed subspaces of a Hilbert space|journal=[[Indiana University Mathematics Journal]]|volume=6|issue=4|year=1957|pages=885–893|doi=10.1512/iumj.1957.6.56050|doi-access=free}}</ref><ref>{{cite journal|last=Busch|first=Paul|author-link=Paul Busch (physicist)|title=Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem|journal=[[Physical Review Letters]]|volume=91|issue=12|year=2003|article-number=120403|doi=10.1103/PhysRevLett.91.120403|pmid=14525351|arxiv=quant-ph/9909073|bibcode=2003PhRvL..91l0403B|s2cid=2168715}}</ref>

=== Generalized measurement (POVM) ===
The most general description of a quantum measurement uses a [[positive-operator-valued measure]] (POVM). In a finite-dimensional Hilbert space, a POVM is a set of positive semidefinite operators <math>\{F_i\}</math> satisfying
:<math>\sum_{i=1}^n F_i=I.</math>

If the system is in state <math>\rho</math>, the probability of outcome <math>i</math> is
:<math>\mathrm{Prob}(i)=\operatorname{tr}(\rho F_i).</math>

For a pure state <math>|\psi\rangle</math>, this becomes
:<math>\mathrm{Prob}(i)=\langle\psi|F_i|\psi\rangle.</math>

POVMs generalize the older concept of [[projection-valued measure]]s and are indispensable in [[quantum information science]], where they describe realistic and optimized measurement procedures.<ref name="NielsenChuang">{{cite book|last1=Nielsen|first1=Michael A.|author-link1=Michael Nielsen|last2=Chuang|first2=Isaac L.|author-link2=Isaac Chuang|title=Quantum Computation and Quantum Information|publisher=Cambridge University Press|year=2000|edition=1st|isbn=978-0-521-63503-5}}</ref><ref>{{cite journal|last1=Peres|first1=Asher|author-link1=Asher Peres|last2=Terno|first2=Daniel R.|title=Quantum information and relativity theory|journal=[[Reviews of Modern Physics]]|volume=76|issue=1|year=2004|pages=93–123|doi=10.1103/RevModPhys.76.93|arxiv=quant-ph/0212023|bibcode=2004RvMP...76...93P|s2cid=7481797}}</ref>

=== State change due to measurement ===
A measurement usually alters the state of the system being measured. In the general formalism, each POVM element can be written as
:<math>E_i=A_i^\dagger A_i,</math>
where the operators <math>A_i</math> are [[Kraus operator]]s. If outcome <math>i</math> is obtained, then the post-measurement state is
:<math>\rho\to\rho'=\frac{A_i\rho A_i^\dagger}{\operatorname{tr}(\rho E_i)}.</math>

An important special case is the [[Lüders rule]]. For a projective measurement with projection operators <math>\Pi_i</math>, the state becomes
:<math>\rho\to\rho'=\frac{\Pi_i\rho\Pi_i}{\operatorname{tr}(\rho\Pi_i)}.</math>

For a pure state and rank-1 projectors, the measurement updates the state to the eigenstate corresponding to the observed outcome. This process has historically been called the '''collapse of the wavefunction'''.<ref>{{cite journal|last1=Hellwig|first1=K.-E.|last2=Kraus|first2=K.|author-link2=Karl Kraus (physicist)|title=Pure operations and measurements|journal=[[Communications in Mathematical Physics]]|volume=11|issue=3|year=1969|pages=214–220|doi=10.1007/BF01645807|s2cid=123659396}}</ref><ref>{{cite book|last=Kraus|first=Karl|author-link=Karl Kraus (physicist)|title=States, effects, and operations: fundamental notions of quantum theory|publisher=Springer-Verlag|year=1983|isbn=978-3-5401-2732-1}}</ref><ref>{{cite book|last1=Busch|first1=Paul|author-link1=Paul Busch (physicist)|last2=Lahti|first2=Pekka|title=Lüders Rule|work=Compendium of Quantum Physics|publisher=Springer|year=2009|pages=356–358|doi=10.1007/978-3-540-70626-7_110}}</ref>

If the measurement result is not recorded, then summing over all possible outcomes gives a [[quantum channel]]:
:<math>\rho\to\sum_i A_i\rho A_i^\dagger.</math>

== Examples ==

=== Qubit measurements ===
The simplest finite-dimensional quantum system is a [[qubit]], whose Hilbert space has dimension 2. A pure qubit state can be written as
:<math>|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,</math>
where <math>|\alpha|^2+|\beta|^2=1</math>.

A measurement in the computational basis <math>(|0\rangle,|1\rangle)</math> yields <math>|0\rangle</math> with probability <math>|\alpha|^2</math> and <math>|1\rangle</math> with probability <math>|\beta|^2</math>.<ref name="Wilde2017" />

More generally, an arbitrary qubit state can be represented by a point in the [[Bloch sphere|Bloch ball]]:
:<math>\rho=\tfrac{1}{2}(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z),</math>
where <math>\sigma_x</math>, <math>\sigma_y</math> and <math>\sigma_z</math> are the [[Pauli matrices]]. Measurements of these Pauli observables correspond to measurements along different axes of the Bloch sphere.<ref name="Wilde2017" />

<div style="background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:12px 0;">
[[File:Bloch sphere representation of optimal POVM and states for unambiguous quantum state discrimination.svg|thumb|400px|[[Bloch sphere]] representation of qubit states and a POVM used for unambiguous quantum state discrimination.<ref>{{cite journal|last1=Peres|first1=Asher|author-link1=Asher Peres|last2=Terno|first2=Daniel R.|title=Optimal distinction between non-orthogonal quantum states|journal=[[Journal of Physics A: Mathematical and General]]|volume=31|issue=34|year=1998|pages=7105–7111|doi=10.1088/0305-4470/31/34/013|arxiv=quant-ph/9804031|bibcode=1998JPhA...31.7105P|s2cid=18961213}}</ref>]]
</div>

=== Bell-basis measurement ===
For two qubits, an important projective measurement is the measurement in the [[Bell state|Bell basis]], consisting 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>

Bell-basis measurements are central to [[quantum teleportation]], [[entanglement swapping]] and many protocols in [[quantum information science]].<ref>{{cite book|last1=Rieffel|first1=Eleanor G.|author-link1=Eleanor Rieffel|last2=Polak|first2=Wolfgang H.|title=Quantum Computing: A Gentle Introduction|publisher=MIT Press|year=2011|isbn=978-0-262-01506-6}}</ref>

=== Continuous-variable measurements ===
A standard example involving continuous observables is the [[quantum harmonic oscillator]], defined by the Hamiltonian
:<math>H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2.</math>

Its energy eigenstates satisfy
:<math>H|n\rangle=E_n|n\rangle,</math>
with eigenvalues
:<math>E_n=\hbar\omega\left(n+\tfrac{1}{2}\right).</math>

An energy measurement therefore yields a discrete spectrum, while a position measurement yields a continuous set of possible outcomes described by a [[probability density function]].<ref>{{cite book|last=Weinberg|first=Steven|author-link=Steven Weinberg|title=Lectures on quantum mechanics|publisher=Cambridge University Press|year=2015|edition=2nd|isbn=978-1-107-11166-0}}</ref>

== History ==

=== Early quantum theory ===
Before the modern formulation of quantum mechanics, the so-called [[old quantum theory]] provided a collection of partial rules and semi-classical models developed between 1900 and 1925. Important achievements included [[Max Planck]]'s explanation of [[blackbody radiation]], [[Albert Einstein]]'s account of the [[photoelectric effect]], and [[Niels Bohr]]'s model of the hydrogen atom.<ref>{{cite book|last=Pais|first=Abraham|author-link=Abraham Pais|title=Subtle is the Lord: The Science and the Life of Albert Einstein|publisher=Oxford University Press|year=2005|isbn=978-0-19-280672-7}}</ref><ref>{{cite book|last=ter Haar|first=D.|title=The Old Quantum Theory|publisher=Pergamon Press|year=1967|isbn=978-0-08-012101-7}}</ref>

A landmark experiment in the early history of measurement was the [[Stern–Gerlach experiment]], proposed in 1921 and performed in 1922. Silver atoms were sent through an inhomogeneous magnetic field and deposited on a screen. Instead of producing a continuous distribution, the atoms formed discrete spots, demonstrating the quantization of angular momentum and providing a paradigmatic example of a quantum measurement with distinct outcomes.<ref name="SG1">{{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|year=1922|pages=349–352|doi=10.1007/BF01326983|bibcode=1922ZPhy....9..349G|s2cid=186228677}}</ref><ref>{{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|issue=12|year=2003|pages=53–59|doi=10.1063/1.1650229|bibcode=2003PhT....56l..53F|s2cid=17572089|doi-access=free}}</ref>

<div style="background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:12px 0;">
[[File:Stern-Gerlach experiment.svg|thumb|400px|The [[Stern–Gerlach experiment]] showed that atomic spin measurements yield discrete outcomes rather than a continuous distribution.]]
</div>

=== Uncertainty and hidden variables ===
In the 1920s, the mathematical structure of modern quantum mechanics was established by [[Werner Heisenberg]], [[Max Born]], [[Pascual Jordan]], [[Erwin Schrödinger]] and others. The [[uncertainty principle]] emerged as one of its defining results. In its standard form,
:<math>\sigma_x\sigma_p\ge \frac{\hbar}{2},</math>
meaning that no state can make both position and momentum simultaneously sharp.<ref name="LandauLifshitz1977" />

This raised the question of whether quantum mechanics might be incomplete and whether more fundamental [[hidden variable theory|hidden variables]] could restore deterministic predictions. A major turning point came with [[Bell's theorem]], which showed that broad classes of ''local'' hidden-variable theories are incompatible with the statistical predictions of quantum mechanics.<ref>{{cite journal|last=Bell|first=J. S.|author-link=John Stewart Bell|title=On the Einstein Podolsky Rosen Paradox|journal=[[Physics Physique Физика]]|volume=1|issue=3|year=1964|pages=195–200|doi=10.1103/PhysicsPhysiqueFizika.1.195|doi-access=free|bibcode=1964PhyNY...1..195B}}</ref> Subsequent [[Bell test]] experiments have consistently supported the quantum predictions and ruled out local hidden-variable explanations.<ref>{{cite journal|author=The BIG Bell Test Collaboration|title=Challenging local realism with human choices|journal=[[Nature (journal)|Nature]]|volume=557|issue=7704|year=2018|pages=212–216|doi=10.1038/s41586-018-0085-3|pmid=29743691|arxiv=1805.04431|bibcode=2018Natur.557..212B|s2cid=13665914}}</ref>

=== Decoherence ===
Later work showed that a realistic measuring device must itself be treated as a physical system subject to quantum mechanics. This led to the theory of [[quantum decoherence]], in which interactions with the environment suppress interference effects and make certain states appear effectively classical.<ref>{{cite journal|last=Schlosshauer|first=M.|title=Quantum Decoherence|journal=Physics Reports|volume=831|year=2019|pages=1–57|doi=10.1016/j.physrep.2019.10.001|arxiv=1911.06282|bibcode=2019PhR...831....1S|s2cid=208006050}}</ref> Decoherence helps explain why measurements appear to yield definite outcomes, though it does not by itself settle all aspects of the [[measurement problem]].<ref>{{cite journal|last1=Camilleri|first1=K.|last2=Schlosshauer|first2=M.|title=Niels Bohr as Philosopher of Experiment: Does Decoherence Theory Challenge Bohr's Doctrine of Classical Concepts?|journal=Studies in History and Philosophy of Modern Physics|volume=49|year=2015|pages=73–83|doi=10.1016/j.shpsb.2015.01.005|arxiv=1502.06547|bibcode=2015SHPMP..49...73C|s2cid=27697360}}</ref>

== Quantum information and computation ==
Measurement plays a central role in [[quantum information science]]. The [[von Neumann entropy]]
:<math>S(\rho)=-\operatorname{tr}(\rho\log\rho)</math>
quantifies the uncertainty represented by a quantum state, and reduces to the [[Shannon entropy]] of the eigenvalue distribution of <math>\rho</math>.<ref name="Wilde2017" />

In the [[quantum circuit]] model, computation consists of a sequence of [[quantum gate]]s followed by measurements, usually in the computational basis.<ref>{{cite book|last1=Rieffel|first1=Eleanor G.|author-link1=Eleanor Rieffel|last2=Polak|first2=Wolfgang H.|title=Quantum Computing: A Gentle Introduction|publisher=MIT Press|year=2011|isbn=978-0-262-01506-6}}</ref> In [[measurement-based quantum computation]], measurements are not merely the final readout step but are the essential mechanism by which the computation proceeds.<ref>{{cite journal|last1=Raussendorf|first1=R.|last2=Browne|first2=D. E.|last3=Briegel|first3=H. J.|author-link3=Hans Jürgen Briegel|title=Measurement based Quantum Computation on Cluster States|journal=[[Physical Review A]]|volume=68|issue=2|year=2003|article-number=022312|doi=10.1103/PhysRevA.68.022312|arxiv=quant-ph/0301052|bibcode=2003PhRvA..68b2312R|s2cid=6197709}}</ref>

Measurement theory also underlies [[quantum tomography]], in which a quantum state, channel, or detector is reconstructed from experimental data, and [[quantum metrology]], where quantum effects are used to improve measurement precision.<ref>{{cite journal|last1=Granade|first1=Christopher|last2=Combes|first2=Joshua|last3=Cory|first3=D. G.|title=Practical Bayesian tomography|journal=New Journal of Physics|volume=18|issue=3|year=2016|article-number=033024|doi=10.1088/1367-2630/18/3/033024|arxiv=1509.03770|bibcode=2016NJPh...18c3024G|s2cid=88521187}}</ref><ref>{{cite journal|last1=Braunstein|first1=Samuel L.|last2=Caves|first2=Carlton M.|author-link2=Carlton Caves|title=Statistical distance and the geometry of quantum states|journal=[[Physical Review Letters]]|volume=72|issue=22|year=1994|pages=3439–3443|doi=10.1103/PhysRevLett.72.3439|pmid=10056200|bibcode=1994PhRvL..72.3439B}}</ref>

== Interpretations ==
Quantum measurement remains one of the main philosophical issues in modern physics. Standard textbook formulations distinguish between smooth, deterministic [[unitary]] evolution when a system is isolated and stochastic, discontinuous state change when a measurement occurs.<ref>{{cite book|last=von Neumann|first=John|author-link=John von Neumann|title=Mathematical Foundations of Quantum Mechanics|publisher=Princeton University Press|year=2018|isbn=978-1-40088-992-1}}</ref> Whether this distinction reflects a fundamental feature of nature or merely an effective description is a matter of ongoing debate.

Different [[interpretations of quantum mechanics]] answer this question in different ways. Some treat the wavefunction as a complete physical description, others as a tool for organizing information or beliefs about outcomes.<ref>{{cite journal|last=Peierls|first=Rudolf|author-link=Rudolf Peierls|title=In defence of "measurement"|journal=[[Physics World]]|volume=4|issue=1|year=1991|pages=19–21|doi=10.1088/2058-7058/4/1/19}}</ref><ref>{{cite journal|last=Bell|first=John|author-link=John Stewart Bell|title=Against 'measurement'|journal=[[Physics World]]|volume=3|issue=8|year=1990|pages=33–41|doi=10.1088/2058-7058/3/8/26}}</ref> Despite many proposals, no single interpretation has achieved universal acceptance.<ref>{{cite journal|last1=Schlosshauer|first1=Maximilian|last2=Kofler|first2=Johannes|last3=Zeilinger|first3=Anton|author-link3=Anton Zeilinger|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|year=2013|pages=222–230|doi=10.1016/j.shpsb.2013.04.004|arxiv=1301.1069|bibcode=2013SHPMP..44..222S|s2cid=55537196}}</ref>

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

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

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

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