concept

quantum measurement

Quantum measurement is the process of observing a quantum system, collapsing its wavefunction and extracting information.
23 related articles · 6 formulas · 6 who worked here

Description

Quantum measurement is the process of obtaining classical information about a quantum system, radically changing its state. According to quantum mechanics, before measurement the system is in a superposition of possible values of a physical quantity; measurement irreversibly projects it onto one of these values. The probability of the outcome is determined by the squared modulus of the projection of the initial state onto the corresponding eigenstate. A key feature is that the result is probabilistic even with full knowledge of the wave function, and the act of measurement itself influences the system (observer effect).

History

The problem was recognized by the founders of quantum mechanics: Bohr and Heisenberg in 1927–28 introduced the concept of wave function reduction. The Einstein–Bohr debate (1927–1935) revealed the paradoxical nature. Von Neumann in 1932 formalized two types of processes: unitary evolution and collapse. In 1964, Bell proved that any hidden-variable theory reproducing quantum predictions must be nonlocal. Aspect's experiments in the 1980s confirmed quantum nonlocality and emphasized the role of measurement. Decoherence theory (Zeh, 1970s) provided an understanding of the loss of interference, but the measurement problem remains open.

How it works

Consider measuring the spin of an electron. Spin is described as a two-level system: |ψ⟩ = a|↑⟩ + b|↓⟩. When passing through a Stern–Gerlach apparatus (a magnet with an inhomogeneous field), the electron's trajectory is deflected, entangling the spin state with spatial position. As a result, entanglement with the environment occurs, effectively implementing the measurement. The spin operator Ŝ_z with eigenstates |↑⟩ and |↓⟩ yields values +ħ/2 and –ħ/2 with probabilities |a|² and |b|². After measurement, the state collapses into the corresponding one. The Heisenberg uncertainty principle limits the simultaneous knowledge of non-commuting observables, such as spin components.

In practice

Quantum sensors that detect gravitational waves and devices that see through walls by changes in quantum states are built on this.

Fun fact

In the delayed-choice quantum eraser experiment, it appears that the decision of what to measure now can change the particle's behavior in the past.

Concept graph →

Formulas

$$D(\psi_1, \psi_2) = \sqrt{1 - |\langle\psi_1|\psi_2\rangle|^2}$$
Distance between quantum states based on their overlap.
What is in the formula
Dvariabledistance between quantum states — quantum distance
ψ1variablefirst quantum state — wave function
ψ2variablesecond quantum state — wave function
⟨ψ1|ψ2⟩operatorcomputes the complex amplitude of one state in another — inner product
|·|operatorgives the magnitude of the complex amplitude — absolute value
√operatortakes the non-negative root — square root
t(|ψ_1\rangle,|ψ_2\rangle)=\sqrt{1-|\langleψ_1|ψ_2\rangle|^2} · 2505.00777v1
$$b_N \sim N^k$$
Quantity scales as power of system size.
What is in the formula
b_Nvariablequantity of interest for system size N
Nvariablesystem size
kvariablescaling exponent
b_n\simn^2 · 2506.01957v3
α(t)=α_0a(t)^4 · 2606.31571
σ\simv_{{dw}}^3 · 2607.20340v1
$$\rho = \sum_i p_i |i\rangle\langle i|$$
A quantum state's density matrix becomes diagonal after measurement.
What is in the formula
ρvariableDensity matrix — density matrix
p_ivariableProbability of state |i⟩ — probability
|i⟩variableEigenstate of the density matrix — quantum state
⟨i|operatorDual vector (bra) of |i⟩ — bra operator
d_{a'}=\begin{pmatrix}\cos^2\frac{θ}{2}&0\0&\sin^2\frac{θ}{2}\end{pmatrix} · 2505.00777v1
$$F(U,V) = \frac{1}{N} |\operatorname{Tr}(U^{\dagger}V)|$$
Measure of similarity between two unitary matrices.
What is in the formula
UvariableFirst unitary matrix — unitary matrix
VvariableSecond unitary matrix — unitary matrix
NvariableDimension of the Hilbert space — dimension
TroperatorSum of diagonal elements of a matrix — trace operator
†operatorConjugate transpose of a matrix — hermitian conjugate
| |operatorMagnitude of a complex number — absolute value
f=\frac{1}{n}|{tr}(u^{\dagger}v)| · 2505.01609v2
$$S_t = \\max(0, S_{t-1} + z_t - k)$$
Cumulative sum statistic for detecting changes in a process.
What is in the formula
S_tvariableCUSUM statistic at time t
S_{t-1}variableCUSUM statistic at previous time
z_tvariableObservation or standardized value at time t
kvariableAllowance or reference value (positive constant)
maxoperatorTakes the maximum of two numbers — maximum
s_t=\max(0,s_{t-1}+z_t-k) · 2607.00331
$$F(\rho, \sigma) = \operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$$
Quantum fidelity measures similarity between two quantum states.
What is in the formula
FvariableQuantum fidelity between states — fidelity
ρvariableDensity matrix of first quantum state — density matrix
σvariableDensity matrix of second quantum state — density matrix
TroperatorSum of diagonal elements of a matrix — trace operator
√operatorPrincipal square root of a positive semidefinite matrix — matrix square root
k({x}_n,{x}_m)=\operatorname{tr}[ρ_a({x}_n)ρ_a({x}_m)] · 2607.10656

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