A quantum measurement is the process by which a 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 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 quantum state with a mathematical description of the measurement, using the Born rule.^[1]^[2]

For example, an electron can be described by a quantum state that assigns a 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 uncertainty principle implies that position and momentum cannot both be predicted with arbitrary precision at the same time.^[3]

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 collapse of the wavefunction, or more precisely by the Lüders rule in the case of projective measurements.^[4] More generally, quantum measurements can be represented using positive-operator-valued measures (POVMs) and Kraus operators.^[5]

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 measurement problem and the many interpretations of quantum mechanics.^[6]

Quantum measurement of spin via the Stern–Gerlach apparatus: the system is projected onto discrete eigenstates corresponding to measurement outcomes.

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.^[2] Physical quantities such as position, momentum, energy and angular momentum are represented by self-adjoint operators, traditionally called observables.^[1]

In finite-dimensional cases, such as the quantum theory of 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.^[1]

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 $ρ$ , then the probability of obtaining the outcome corresponding to projection operator $Π_{i}$ is given by the Born rule:

P (x_{i}) = tr (Π_{i} ρ) .

The expectation value of an observable $A$ in the state $ρ$ is

⟨ A ⟩ = tr (A ρ) .

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.^[2]^[7]

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.^[8]^[9]

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 ${F_{i}}$ satisfying

\sum_{i = 1}^{n} F_{i} = I .

If the system is in state $ρ$ , the probability of outcome $i$ is

P r o b (i) = tr (ρ F_{i}) .

For a pure state $| ψ ⟩$ , this becomes

P r o b (i) = ⟨ ψ | F_{i} | ψ ⟩ .

POVMs generalize the older concept of projection-valued measures and are indispensable in quantum information science, where they describe realistic and optimized measurement procedures.^[10]^[11]

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

E_{i} = A_{i}^{†} A_{i},

where the operators $A_{i}$ are Kraus operators. If outcome $i$ is obtained, then the post-measurement state is

ρ \to ρ^{'} = \frac{A_{i} ρ A_{i}^{†}}{tr (ρ E_{i})} .

An important special case is the Lüders rule. For a projective measurement with projection operators $Π_{i}$ , the state becomes

ρ \to ρ^{'} = \frac{Π_{i} ρ Π_{i}}{tr (ρ Π_{i})} .

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.^[12]^[13]^[14]

If the measurement result is not recorded, then summing over all possible outcomes gives a quantum channel:

ρ \to \sum_{i} A_{i} ρ A_{i}^{†} .

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

| ψ ⟩ = α | 0 ⟩ + β | 1 ⟩,

where $| α |^{2} + | β |^{2} = 1$ .

A measurement in the computational basis $(| 0 ⟩, | 1 ⟩)$ yields $| 0 ⟩$ with probability $| α |^{2}$ and $| 1 ⟩$ with probability $| β |^{2}$ .^[5]

More generally, an arbitrary qubit state can be represented by a point in the Bloch ball:

ρ = \frac{1}{2} (I + r_{x} σ_{x} + r_{y} σ_{y} + r_{z} σ_{z}),

where $σ_{x}$ , $σ_{y}$ and $σ_{z}$ are the Pauli matrices. Measurements of these Pauli observables correspond to measurements along different axes of the Bloch sphere.^[5]

Bell-basis measurement

For two qubits, an important projective measurement is the measurement in the Bell basis, consisting of four maximally entangled states:

\begin{aligned} | Φ^{+} ⟩ & = \frac{1}{\sqrt{2}} (| 0 ⟩_{A} \otimes | 0 ⟩_{B} + | 1 ⟩_{A} \otimes | 1 ⟩_{B}), \\ | Φ^{-} ⟩ & = \frac{1}{\sqrt{2}} (| 0 ⟩_{A} \otimes | 0 ⟩_{B} - | 1 ⟩_{A} \otimes | 1 ⟩_{B}), \\ | Ψ^{+} ⟩ & = \frac{1}{\sqrt{2}} (| 0 ⟩_{A} \otimes | 1 ⟩_{B} + | 1 ⟩_{A} \otimes | 0 ⟩_{B}), \\ | Ψ^{-} ⟩ & = \frac{1}{\sqrt{2}} (| 0 ⟩_{A} \otimes | 1 ⟩_{B} - | 1 ⟩_{A} \otimes | 0 ⟩_{B}) . \end{aligned}

Bell-basis measurements are central to quantum teleportation, entanglement swapping and many protocols in quantum information science.^[16]

Continuous-variable measurements

A standard example involving continuous observables is the quantum harmonic oscillator, defined by the Hamiltonian

H = \frac{p^{2}}{2 m} + \frac{1}{2} m ω^{2} x^{2} .

Its energy eigenstates satisfy

H | n ⟩ = E_{n} | n ⟩,

with eigenvalues

E_{n} = ℏ ω (n + \frac{1}{2}) .

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.^[17]

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.^[18]^[19]

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.^[20]^[21]

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,

σ_{x} σ_{p} \geq \frac{ℏ}{2},

meaning that no state can make both position and momentum simultaneously sharp.^[3]

This raised the question of whether quantum mechanics might be incomplete and whether more fundamental 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.^[22] Subsequent Bell test experiments have consistently supported the quantum predictions and ruled out local hidden-variable explanations.^[23]

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.^[24] Decoherence helps explain why measurements appear to yield definite outcomes, though it does not by itself settle all aspects of the measurement problem.^[25]

Quantum information and computation

Measurement plays a central role in quantum information science. The von Neumann entropy

S (ρ) = - tr (ρ \log ρ)

quantifies the uncertainty represented by a quantum state, and reduces to the Shannon entropy of the eigenvalue distribution of $ρ$ .^[5]

In the quantum circuit model, computation consists of a sequence of quantum gates followed by measurements, usually in the computational basis.^[26] In measurement-based quantum computation, measurements are not merely the final readout step but are the essential mechanism by which the computation proceeds.^[27]

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.^[28]^[29]

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.^[30] 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.^[31]^[32] Despite many proposals, no single interpretation has achieved universal acceptance.^[33]

Table of contents (137 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Measurement theory