Physics:Quantum mechanics measurements

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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.

Quantum measurement process: a system in superposition is measured, yielding probabilistic outcomes and collapsing to a definite eigenstate.

Common experimental measuring devices

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". 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". 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. While these issues can be rigorously treated using spectral theory;, 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. 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

$P(x_i)=\mathrm{tr}({\Pi }_i\rho ),$

where $\rho$ is the density operator, and ${\Pi }_i$ is the projection operator onto the basis vector corresponding to the measurement outcome $x_i$. The average of the eigenvalues of a von Neumann observable, weighted by the Born rule probabilities, is the expectation value of that observable. For an observable $A$, the expectation value given a quantum state $\rho$ is

$⟨A⟩=\mathrm{tr}(A\rho ).$

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., $P(x)=1$ for some outcome $x$). Any mixed state can be written as a convex combination of pure states, though not in a unique way. 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.

Positive Operator-Valued Measure (POVM)

See PQVM

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 $P_i$ satisfying $P_i{P}_j={\delta }_{ij}{P}_i$ and $\sum _i{P}_i=I$. For a system in state $\rho$, the probability of obtaining outcome $i$ is $p(i)=\mathrm{T}\mathrm{r}(P_i\rho )$, and after the measurement, the state collapses to ${\rho }^{\prime }=\frac{P_i\rho P_i}{p(i)}$.

In contrast, a POVM consists of a set of positive semi-definite operators $E_i$ that satisfy $E_i\ge 0$ and $\sum _i{E}_i=I$. The probability of obtaining outcome $i$ is given by $p(i)=\mathrm{T}\mathrm{r}(E_i\rho )$. 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. 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 $\{F_i\}$ on a Hilbert space $\mathscr{H}$ that sum to the identity matrix,

$\sum _{i=1}^n{F}_i=\mathrm{I}.$

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

$\text{Prob}(i)=\mathrm{tr}(\rho F_i)$,

where $\mathrm{tr}$ is the trace operator. When the quantum state being measured is a pure state $|\psi ⟩$ this formula reduces to

$\text{Prob}(i)=\mathrm{tr}(|\psi ⟩⟨\psi |F_i)=⟨\psi |F_i|\psi ⟩$.

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. To remedy this, further information is specified by decomposing each POVM element into a product:

$E_i=A_i^{†}{A}_i.$

The Kraus operators $A_i$, named for Karl Kraus, provide a specification of the state-change process. They are not necessarily self-adjoint, but the products $A_i^{†}{A}_i$ are. If upon performing the measurement the outcome $E_i$ is obtained, then the initial state $\rho$ is updated to

$\rho \to {\rho }^{\prime }=\frac{A_i\rho A_i^{†}}{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}(i)}=\frac{A_i\rho A_i^{†}}{\mathrm{tr}(\rho E_i)}.$

An important special case is the Lüders rule, named for Gerhart Lüders. If the POVM is itself a PVM, then the Kraus operators can be taken to be the projectors onto the eigenspaces of the von Neumann observable:

$\rho \to {\rho }^{\prime }=\frac{{\Pi }_i\rho {\Pi }_i}{\mathrm{tr}(\rho {\Pi }_i)}.$

If the initial state $\rho$ is pure, and the projectors ${\Pi }_i$ have rank 1, they can be written as projectors onto the vectors $|\psi ⟩$ and $|i⟩$, respectively. The formula simplifies thus to

$\rho =|\psi ⟩⟨\psi |\to {\rho }^{\prime }=\frac{|i⟩⟨i|\psi ⟩⟨\psi |i⟩⟨i|}{|⟨i|\psi ⟩{|}^2}=|i⟩⟨i|.$

Lüders rule has historically been known as the "reduction of the wave packet" or the "collapse of the wavefunction". The pure state $|i⟩$ implies a probability-one prediction for any von Neumann observable that has $|i⟩$ 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.

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:

$\rho \to \sum _i{A}_i\rho A_i^{†}.$

It is an example of a quantum channel, and can be interpreted as expressing how a quantum state changes if a measurement is performed but the result of that measurement is lost.

Examples

Bloch sphere representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination on the states $|\psi ⟩=|0⟩$ and $|\phi ⟩=(|0⟩+|1⟩)/\sqrt{2}$. 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 $|0⟩$ and $|1⟩$ with complex coefficients:

$|\psi ⟩=\alpha |0⟩+\beta |1⟩$

A measurement in the $(|0⟩,|1⟩)$ basis will yield outcome $|0⟩$ with probability $|\alpha {|}^2$ and outcome $|1⟩$ with probability $|\beta {|}^2$, so by normalization,

$|\alpha {|}^2+|\beta {|}^2=1.$

An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for $2\times 2$ self-adjoint matrices:

$\rho =\frac{1}{2}(I+r_x{\sigma }_x+r_y{\sigma }_y+r_z{\sigma }_z),$

where the real numbers $(r_x,r_y,r_z)$ are the coordinates of a point within the unit ball and

${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$

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 $(r_x,r_y,r_z)$ of the state $\rho$ are the expectation values of the three von Neumann measurements defined by the Pauli matrices. 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 ${\sigma }_z$ are the basis states $|0⟩$ and $|1⟩$, and a measurement of ${\sigma }_z$ is often called a measurement in the "computational basis." After a measurement in the computational basis, the outcome of a ${\sigma }_x$ or ${\sigma }_y$ 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, a set of four maximally entangled states:

$\begin{array}{cc}|{\Phi }^{+}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |0{⟩}_B+|1{⟩}_A\otimes |1{⟩}_B)\\ |{\Phi }^{-}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |0{⟩}_B-|1{⟩}_A\otimes |1{⟩}_B)\\ |{\Psi }^{+}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |1{⟩}_B+|1{⟩}_A\otimes |0{⟩}_B)\\ |{\Psi }^{-}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |1{⟩}_B-|1{⟩}_A\otimes |0{⟩}_B)\end{array}$

Probability density $P_n(x)$ for the outcome of a position measurement given the energy eigenstate $|n⟩$ 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. This system is defined by the Hamiltonian

$H=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{x}^2,$

where $H$, the momentum operator $p$ and the position operator $x$ 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:

$H|n⟩=E_n|n⟩.$

These eigenvalues can be shown to be given by

$E_n=\hslash \omega (n+\frac{1}{2}),$

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 $P(x)$ that gives the probability of the measurement outcome lying in the infinitesimal interval from $x$ to $x+dx$.

History of the measurement concept

The "old quantum theory"

The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as a semi-classical approximation to modern quantum mechanics. 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.

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, 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.

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. 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.

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.

Writing $x$ and $p$ for the self-adjoint operators representing position and momentum respectively, a standard deviation of position can be defined as

${\sigma }_x=\sqrt{⟨x^2⟩-⟨x{⟩}^2},$

and likewise for the momentum:

${\sigma }_p=\sqrt{⟨p^2⟩-⟨p{⟩}^2}.$

The Kennard–Pauli–Weyl uncertainty relation is

${\sigma }_x{\sigma }_p\ge \frac{\hslash }{2}.$

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. The Robertson inequality generalizes this to the case of an arbitrary pair of self-adjoint operators $A$ and $B$. The commutator of these two operators is

$[A,B]=AB-BA,$

and this provides the lower bound on the product of standard deviations:

${\sigma }_A{\sigma }_B\ge |\frac{1}{2i}⟨[A,B]⟩|=\frac{1}{2}|⟨[A,B]⟩|.$

Substituting in the canonical commutation relation $[x,p]=i\hslash$, an expression first postulated by Max Born in 1925, 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. 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.

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. Further investigation in this line led to the formulation of the Wigner–Yanase skew information.

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. 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. 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.

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. (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.) A significant portion of the effort involved in quantum computing research is to avoid the deleterious effects of decoherence.

To illustrate, let ${\rho }_S$ denote the initial state of the system, ${\rho }_E$ the initial state of the environment and $H$ the Hamiltonian specifying the system-environment interaction. The density operator ${\rho }_E$ can be diagonalized and written as a linear combination of the projectors onto its eigenvectors:

${\rho }_E=\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|.$

Expressing time evolution for a duration $t$ by the unitary operator $U=e^{-iHt/\hslash }$, the state for the system after this evolution is

${{\rho }_S}^{\prime }={\mathrm{t}\mathrm{r}}_{E}U[{\rho }_S\otimes (\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|)]U^{†},$

which evaluates to

${{\rho }_S}^{\prime }=\sum _{ij}\sqrt{p_i}⟨{\psi }_j|U|{\psi }_i⟩{\rho }_S\sqrt{p_i}⟨{\psi }_i|U^{†}|{\psi }_j⟩.$

The quantities surrounding ${\rho }_S$ can be identified as Kraus operators, and so this defines a quantum channel.

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.

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 $\rho$, the von Neumann entropy is

$S(\rho )=-\mathrm{t}\mathrm{r}(\rho \log \rho );$

writing $\rho$ in terms of its basis of eigenvectors,

$\rho =\sum _i{\lambda }_i|i⟩⟨i|,$

the von Neumann entropy is

$S(\rho )=-\sum _i{\lambda }_i\log {\lambda }_i.$

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 $\rho$. Consequently, the von Neumann entropy vanishes when $\rho$ is pure. The von Neumann entropy of $\rho$ can equivalently be characterized as the minimum Shannon entropy for a measurement given the quantum state $\rho$, with the minimization over all POVMs with rank-1 elements.

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:

$\frac{1}{2}||\rho -\sigma ||=\underset{0\le E\le I}{\max }[\mathrm{t}\mathrm{r}(E\rho )-\mathrm{t}\mathrm{r}(E\sigma )].$

Similarly, the fidelity of two quantum states, defined by

$F(\rho ,\sigma )={(\mathrm{Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }})}^2,$

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:

$1-\sqrt{F(\rho ,\sigma )}\le \frac{1}{2}||\rho -\sigma ||\le \sqrt{1-F(\rho ,\sigma )}.$

Quantum circuits

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. 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.

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.

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. 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 and even of measurements.

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. A celebrated example is the introduction of squeezed light into the LIGO experiment, which increased its sensitivity to gravitational waves.

Laboratory implementations

The range of physical procedures to which the mathematics of quantum measurement can be applied is very broad. 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. Language from this era persists, such as the description of measurement outcomes in the abstract as "detector clicks".

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. 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). 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.

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. 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.

Interpretations of quantum mechanics

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. 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."

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. 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. 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. 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. 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. 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. Of this line of thought, Bell asked, "Whose information? Information about what?" Answers to these questions vary among proponents of the informationally-oriented interpretations.

Look at

• Einstein's thought experiments
• Holevo's theorem
• Quantum error correction
• Quantum limit
• Quantum logic
• Quantum Zeno effect
• Schrödinger's cat
• SIC-POVM

Further reading

• Quantum Theory and Measurement. Princeton University Press. 1983. ISBN 978-0-691-08316-2. 
• Braginsky, Vladimir B.; Khalili, Farid Ya. (1992). Quantum Measurement. Cambridge University Press. ISBN 978-0-521-41928-4. 
• Greenstein, George S.; Zajonc, Arthur G. (2006). The Quantum Challenge: Modern Research On The Foundations Of Quantum Mechanics (2nd ed.). ISBN 978-0-7637-2470-2. 
• Alter, Orly; Yamamoto, Yoshihisa (2001). Quantum Measurement of a Single System. New York: Wiley. doi:10.1002/9783527617128. ISBN 978-0-471-28308-9. 
• Andrew N. Jordan; Irfan A. Siddiqi (2024). Quantum Measurement: Theory and Practice. Cambridge University Press. ISBN 978-1-009-10006-9. 

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

9. Physics:Quantum mechanics measurements

10. Physics:Quantum state

11. Physics:Quantum system

12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

19. Physics:Quantum Measurement problem

20. Physics:Quantum Bell's theorem

21. Physics:Quantum Hidden variable theory

22. Physics:Quantum nonlocality

23. Physics:Quantum contextuality

24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

37. Physics:Quantum Runge–Lenz vector

38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (21) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Aharonov-Casher effect

77. Physics:Quantum Scattering theory

78. Physics:Quantum Scattering matrix

79. Physics:Quantum S-matrix

80. Physics:Quantum tunnelling

81. Physics:Quantum speed limit

82. Physics:Quantum revival

83. Physics:Quantum reflection

84. Physics:Quantum oscillations

85. Physics:Quantum jump

86. Physics:Quantum boomerang effect

87. Physics:Quantum chaos

7. Measurement and information (9) Back to index

88. Physics:Quantum Measurement theory

89. Physics:Quantum Measurement operators

90. Physics:Quantum Projective measurement

91. Physics:Quantum POVM

92. Physics:Quantum Weak measurement

93. Physics:Quantum Measurement collapse

94. Physics:Quantum entanglement

95. Physics:Quantum Zeno effect

96. Physics:Quantum limit

8. Quantum information and computing (15) Back to index

97. Physics:Quantum information theory

98. Physics:Quantum Qubit

99. Physics:Quantum Entanglement

100. Physics:Quantum Bell state

101. Physics:Quantum Gates and circuits

102. Physics:Quantum BB84

103. Physics:Quantum No-cloning theorem

104. Physics:Quantum Computing Algorithms in the NISQ Era

105. Physics:Quantum Noisy Qubits

106. Physics:Quantum error correction

107. Physics:Quantum Boson sampling

108. Physics:Quantum random access code

109. Physics:Quantum pseudo-telepathy

110. Physics:Quantum network

111. Physics:Quantum money

9. Quantum optics and experiments (10) Back to index

112. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

113. Physics:Quantum Stern-Gerlach experiment

114. Physics:Quantum Experimental quantum physics

115. Physics:Quantum optics

116. Physics:Quantum optics beam splitter experiments

117. Physics:Quantum Mach-Zehnder interferometer

118. Physics:Quantum Hong-Ou-Mandel effect

119. Physics:Quantum eraser experiment

120. Physics:Quantum delayed-choice quantum eraser

121. Physics:Quantum Ultra fast lasers

122. Template:Quantum optics operators

10. Open quantum systems (15) Back to index

123. Physics:Quantum Open systems

124. Physics:Quantum channel

125. Physics:Quantum Kraus operators

126. Physics:Quantum Amplitude damping

127. Physics:Quantum Phase damping

128. Physics:Quantum Depolarizing channel

129. Physics:Quantum Master equation

130. Physics:Quantum Lindblad equation

131. Physics:Quantum Decoherence

132. Physics:Quantum Dynamical decoupling

133. Physics:Quantum dissipation

134. Physics:Quantum Markov semigroup

135. Physics:Quantum Markovian dynamics

136. Physics:Quantum Non-Markovian dynamics

137. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

138. Physics:Quantum field theory (QFT) basics

139. Physics:Quantum field theory (QFT) core

140. Physics:Quantum Fields and Particles

141. Physics:Quantum Second quantization

142. Physics:Quantum Fock space

143. Physics:Quantum Harmonic Oscillator field modes

144. Physics:Quantum Creation and annihilation operators

145. Physics:Quantum vacuum fluctuations

146. Physics:Quantum Casimir effect

147. Physics:Quantum Propagators in quantum field theory

148. Physics:Quantum Feynman diagrams

149. Physics:Quantum Path integral formulation

150. Physics:Quantum Renormalization in field theory

151. Physics:Quantum Renormalization group

152. Physics:Quantum Field Theory Gauge symmetry

153. Physics:Quantum Spontaneous symmetry breaking

154. Physics:Quantum Non-Abelian gauge theory

155. Physics:Quantum Electrodynamics (QED)

156. Physics:Quantum chromodynamics (QCD)

157. Physics:Quantum Electroweak theory

158. Physics:Quantum Standard Model

159. Physics:Quantum triviality

160. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

161. Physics:Quantum Statistical mechanics

162. Physics:Quantum Partition function

163. Physics:Quantum Distribution functions

164. Physics:Quantum Liouville equation

165. Physics:Quantum Kinetic theory

166. Physics:Quantum Boltzmann equation

167. Physics:Quantum BBGKY hierarchy

168. Physics:Quantum Relaxation and thermalization

169. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (17) Back to index

170. Physics:Quantum Band structure

171. Physics:Quantum Fermi surfaces

172. Physics:Quantum Landau levels

173. Physics:Quantum fractional Hall effect

174. Physics:Quantum Semiconductor physics

175. Physics:Quantum Phonons

176. Physics:Quantum Electron-phonon interaction

177. Physics:Quantum Exchange interaction

178. Physics:Quantum Superconductivity

179. Physics:Quantum Topological phases of matter

180. Physics:Quantum anyon

181. Physics:Quantum well

182. Physics:Quantum spin liquid

183. Physics:Quantum spin Hall effect

184. Physics:Quantum phase transition

185. Physics:Quantum critical point

186. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

187. Physics:Quantum Fusion reactions and Lawson criterion

188. Physics:Quantum Plasma (fusion context)

189. Physics:Quantum Magnetic confinement fusion

190. Physics:Quantum Inertial confinement fusion

191. Physics:Quantum Plasma instabilities and turbulence

192. Physics:Quantum Tokamak core plasma

193. Physics:Quantum Tokamak edge physics and recycling asymmetries

194. Physics:Quantum Stellarator

15. Timeline (8) Back to index

195. Physics:Quantum mechanics/Timeline

196. Physics:Quantum mechanics/Timeline/Pre-quantum era

197. Physics:Quantum mechanics/Timeline/Old quantum theory

198. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

199. Physics:Quantum mechanics/Timeline/Quantum field theory era

200. Physics:Quantum mechanics/Timeline/Quantum information era

201. Physics:Quantum mechanics/Timeline/Quantum technology era

202. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

203. Physics:Quantum topology

204. Physics:Quantum battery

205. Physics:Quantum Supersymmetry

206. Physics:Quantum Black hole thermodynamics

207. Physics:Quantum Holographic principle

208. Physics:Quantum gravity

209. Physics:Quantum De Sitter invariant special relativity

210. Physics:Quantum Doubly special relativity

211. Physics:Quantum arithmetic geometry

212. Physics:Quantum unsolved problems

213. Physics:Quantum Yang-Mills mass gap

214. Physics:Quantum gravity problem

215. Physics:Quantum black hole information paradox

216. Physics:Quantum dark matter problem

217. Physics:Quantum neutrino mass problem

218. Physics:Quantum matter-antimatter asymmetry problem

References

Author: Harold Foppele

Source attribution: Physics:Quantum mechanics measurements