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Radiant intensity describes how a source distributes its radiant flux over directions. A small cone of solid angle $\Delta \Omega$ carries approximately $I\Delta \Omega$ when $I$ varies little across it; integrating $I$ over all emitting directions recovers the total radiant flux. For a source with isotropy emitting $4\pi \mathrm{W}$ over a full sphere, $I=1\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and a $0.01\mathrm{sr}$ cone carries $0.01\mathrm{W}$. A planar source satisfying Lambert's cosine law instead has direction-dependent radiant intensity, even though its radiance is direction-independent.

Radiant flux is the power carried by electromagnetic radiation: radiation energy transferred per unit time, measured in watts, $\mathrm{W}=\mathrm{J}{\mathrm{s}}^{-1}$. Specify what is counted: all radiation emitted by a source, radiation within one beam, or radiation intercepted by a photodetector. Writing $Q(t)$ for the cumulative energy crossing the specified boundary givesThe first expression is an instantaneous derivative; the second is an average over the stated time interval. If $0.020\mathrm{J}$ arrives during $0.010\mathrm{s}$, the mean radiant flux is $2\mathrm{W}$. The photodetector's absorbed power equals this incident radiant flux only if it absorbs all of the incident radiation.

Keeping radiant flux fixed while widening a beam reduces its irradiance, the incident power per receiving area. Radiance additionally resolves the propagation direction and has units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$; an optical camera can collect different radiant fluxes from sources with the same radiance but different apparent solid angles. Luminous flux, measured in lumens, weights the optical spectrum by the human eye's standardized response instead of counting energy equally at every wavelength. Equal radiant fluxes need not give equal luminous fluxes.^[ref]

Radiant power also differs from the optical power of an optical lens, which is reciprocal focal length in dioptres. Likewise, astronomical radiative flux is power per area. Stating the units prevents these shared words from hiding different physical quantities.

The effective recombination coefficient for a spectral line counts recombinations and subsequent radiative cascades producing that line. At sufficiently low density, its photon production rate per volume is $n_e{n}_{\mathrm{parent}}{\alpha }_{\ell }^{\mathrm{eff}}$. It depends on gas temperature, optical-depth assumptions such as Case B recombination, and sometimes density. It differs from the total rate coefficient removing the parent ion.

A recombination line is a spectral line produced after a free electron is captured, often through a subsequent radiative cascade among bound levels. Its emissivity is computed with an effective recombination coefficient, distinct from the free-bound recombination continuum.

A radiation-dominated universe has expansion controlled mainly by radiation in cosmology. When spatial curvature and the cosmological constant are negligible, the Friedmann equation and density scaling $\rho \propto a^{-4}$ give the displayed scale factor, with Hubble parameter $H=1/(2t)$.

A correlated pair of quarks. Its color, flavor, spin and orbital states are separate degrees of freedom and must collectively satisfy Fermi statistics. In flavor SU(3), the antisymmetric pair space is ${\Lambda }^{2}3\cong \overline{3}$; contraction with the invariant Levi-Civita symbol realizes the isomorphism. The antisymmetric $ud$ flavor pair is an isospin singlet with flavor hypercharge $2/3$. Combining two such flavor-antitriplet pairs with a strange antiquark permits the highest weight of the pentaquark antidecuplet; this flavor construction does not itself establish a bound state.

Partial-wave scattering decomposes a scattering state into partial waves with definite orbital angular momentum. For a central potential, each channel has its own scattering phase shift and Partial-wave S-matrix.

A scattering state describes an incoming preparation and its outgoing components after an interaction. In a stationary calculation, it is typically a generalized energy eigenstate with specified incoming boundary data, rather than a square-integrable bound state.

For a short-range potential, its distant form separates freely propagating incoming and scattered waves. The outgoing boundary condition selects how a localized disturbance radiates away, and the resulting relative coefficient determines a scattering amplitude. A physical normalizable packet is formed by superposing stationary scattering states over an energy or momentum range. This distinction allows finite particle probabilities and localized preparations without treating an infinite plane wave as a normalized state inside the physical Hilbert space.

For one-dimensional scattering from a real potential, write the left exterior wavefunction as $e^{ikx}+r{e}^{-ikx}$ with $k>0$. The reflected-to-incident probability-current ratio is $R=|r{|}^2$, since both components have the same speed and their currents have opposite signs. This probability differs from the complex reflection amplitude $r$. With transmission into a right exterior wavenumber $q$, the transmission probability is $T=(q/k)|t{|}^2$ and conservation of current gives $R+T=1$ when there is no absorption.

The quantum rotor with angle period $2\pi$ has both the spectral kernel $(2\pi {)}^{-1}{\sum }_n{e}^{in\Delta -i\hslash n^{2}T/(2I)}$ and the displayed winding representation. The Poisson summation formula converts one into the other by a Gaussian integral. Each winding sector has a constant-velocity classical path and action $I(\Delta +2\pi m{)}^2/(2T)$. Quadratic fluctuations give the same prefactor in every sector, so this is exact. Real time is understood by $T\to T-i0$, which fixes the square-root branch and convergence prescription.

A projective measurement on one half of a Bell state prepares the complex-conjugate basis on the other half, conditional on the outcome. Measuring Alice in the columns of $U^{\ast }$ prepares Bob in the columns of $U$. Ignoring the equally likely outcomes leaves Bob's reduced density matrix equal to $I/2$ in either basis.

If a positive operator-valued measure distinguishes normalized pure states $|u⟩,|v⟩$ with certainty, the effect $E$ for the first answer satisfies $⟨u|E|u⟩=1$ and $⟨v|E|v⟩=0$. Positivity of $E$ and $I-E$ implies $E|v⟩=0$ and $E|u⟩=|u⟩$, because a positive operator with zero expectation annihilates the vector. Consequently $⟨u|v⟩=⟨u|E|v⟩=0$. Conversely, orthogonal states can be measured with their rank-one linear projections. A common unitary operator preserves this inner product and cannot remove the obstruction.

Measurement backaction is the effect of a measurement in quantum mechanics on later predictions for the measured system. A quantum instrument with one Kraus operator $K_m$ per recorded outcome hasThe conditional density operator ${\rho }_m$ exists only when $p_m>0$. Its nonlinear normalization expresses conditioning; the unnormalized map $\rho \mapsto K_m\rho K_m^{†}$ is linear. More generally, an unresolved outcome can contain several Kraus operators: replace the numerator by ${\sum }_a{K}_{ma}\rho K_{ma}^{†}$ and take its trace for $p_m$. ^[ref].

Keeping the outcome gives a selective quantum measurement. Discarding it gives a nonselective quantum measurement with quantum channel $\rho \mapsto {\sum }_m{p}_m{\rho }_m$. The latter describes the whole measured population without its labels, whereas each ${\rho }_m$ describes a selected subpopulation. Neither description requires the measured quantity to have possessed its eventual value before the interaction.

For a concrete test, prepare the Pauli X eigenstate $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. Its $X=+1$ probability is one. An ideal $Z$ projective measurement gives zero or one with equal probability, leaving $|0⟩$ or $|1⟩$ under the Lüders rule. Discarding the result leaves the maximally mixed state $I/2$. A subsequent $X$ measurement now gives either sign with probability one half, even if the $Z$ result was kept. This observable change in statistics is backaction. Measuring $X$ on the original $|+⟩$ instead leaves it unchanged: disturbance depends on both the preparation and the instrument. Measurement-based quantum feedback uses the recorded outcome to choose a subsequent control operation, so both parts of the instrument matter.

A photon in a quantum superposition of a safe path and an absorbing path can sometimes certify a live absorber without being absorbed. Recombination is chosen so that one output is impossible if the absorber is absent. A detection at that output is then an unambiguous successful test. The measurement back-action of a surviving live-absorber trial changes the interference pattern.

In 1993, Avshalom Elitzur and Lev Vaidman asked how to identify a working bomb whose sensor would be triggered by even one photon. Their theoretical proposal places the sensor in one arm of a Mach-Zehnder interferometer, adjusted so that destructive interference keeps one detector dark when both arms are unobstructed. A click there can identify a working bomb while leaving it unexploded. The motivation is to obtain information about a fragile object without destroying it; the bomb makes that operational claim explicit without committing to an interpretation of quantum mechanics.^[ref]

An ideal position measurement of a normalized wavefunction $\psi$ assigns a probability to each spatial region $A$ through the Born rule: $P(A)={\int }_A|\psi (x){|}^{2}dx$. This specifies outcome statistics; it does not by itself specify the post-measurement state. A physical detector has finite spatial resolution, which must be included in its measurement model.

An ancilla bit is an auxiliary bit providing workspace in reversible computation. In a quantum circuit, an ancilla qubit provides quantum workspace; a general quantum ancilla need not be a single bit.

A Werner state is a bipartite density operator invariant under $U\otimes U$ for every local unitary operator $U$. An isotropic quantum state instead has invariance under $U\otimes \overline{U}$; these are different families.

Quantum state discrimination uses a measurement in quantum mechanics to identify which candidate density operator was prepared. Minimum-error and unambiguous discrimination impose different objectives; quantum binary hypothesis testing is the two-state case.

For density operators $\rho ,\sigma$ on the same finite-dimensional Hilbert space, quantum fidelity measures how close their quantum states are. This article uses the unsquared convention:Each square root is the positive square root of an operator, and $\Vert \cdot {\Vert }_1$ is the trace norm. Quantum fidelity is symmetric in its inputs; it is one exactly when $\rho =\sigma$, and zero exactly when $\rho \sigma =0$, meaning that the support of a positive operator for one state is orthogonal to that for the other. Larger quantum fidelity means greater similarity. The separate squared quantum fidelity is $F_{\mathrm{sq}}=F^2$. ^[ref]

For two normalized pure states, quantum fidelity is the magnitude of their inner product, $F=|⟨\psi |\phi ⟩|$. Thus $|0⟩$ compared with itself has $F=1$, while $|0⟩$ and $|1⟩$ have $F=0$. For $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, comparison with $|0⟩$ gives $F=1/\sqrt{2}$, $F_{\mathrm{sq}}=1/2$, and trace distance $D=1/\sqrt{2}$ by the pure-state trace distance and fidelity identity.

Why compare density operators rather than their entries? A change of coordinates should not make the same physical preparations look more or less alike. Unitary invariance of trace distance and fidelity makes quantum fidelity independent of a common basis choice. Multiplying a pure state vector by a global phase leaves its density operator unchanged, so its quantum fidelity with the original state is one. These features make quantum fidelity useful for checking whether quantum channels, state preparations, and Schumacher compression preserve a target quantum state.

The connection to Uhlmann's theorem supplies another beautiful interpretation: quantum fidelity is the largest absolute inner product between pure vectors obtained from a purification of a density operator for each state, using a sufficiently large common reference space. Uhlmann's 1976 paper extended transition probability to algebraic states, including those of a C-star algebra; for density operators its transition probability corresponds to our squared quantum fidelity. The measurement characterization of quantum fidelity gives a complementary description directly in terms of observable outcome statistics. ^[ref]

An encoding stores $n$ classical bits in $m$ qubits so that a receiver who later chooses a bit index can recover that bit with a specified success probability. The sender cannot tailor the encoding to the unknown requested index. This task permits approximate recovery of any one bit rather than simultaneous recovery of all bits, and must specify whether additional communication, shared randomness, or entanglement is allowed.