Use this glossary to recall a term and see how it differs from nearby ideas. Each linked term leads to a fuller explanation with examples. The definitions use the Hilbert-space formulation; tensor-product entries assume a specified division into subsystems. The definition of a separable mixture used below is for finite-dimensional subsystems.
For symbol-level lookup, use the Symbol Map . For notation translation across representations, use the Representation Translation Table . For site-wide glossary entries, use the Reference Glossary .
Term and explanation
Meaning
Keep in mind
State The mathematical object that determines all probabilities for a specified system and preparation.
A state is not the same thing as a single measurement outcome.
Pure state A state represented by a ray, or equivalently by a rank-one density operator.
A vector representative is not unique because global phase is unobservable.
Mixed state A general state represented by a density operator that is not a rank-one projector.
A mixed state need not mean ignorance about a unique underlying pure state.
State vector A vector representative ∣ ψ ⟩ \lvert\psi\rangle ∣ ψ ⟩
The physical pure state is the ray, not one phase-fixed vector.
Ray The equivalence class of nonzero vectors differing by a nonzero scalar, usually by phase after normalization.
Global phase is not relative phase.
Wavefunction The position representation ψ ( x ) = ⟨ x ∣ ψ ⟩ \psi(x)=\langle x\vert\psi\rangle ψ ( x ) = ⟨ x ∣ ψ ⟩
It depends on the representation; a mixed state generally requires a density operator.
Basis A complete, linearly independent set used to expand vectors; Hilbert-space calculations usually use orthonormal bases.
Components change with the basis. Infinite expansions require a specified convergence notion.
Amplitude A complex number whose squared modulus enters a probability.
Amplitudes can interfere; probabilities are real and nonnegative.
Superposition A linear combination of state vectors relative to a chosen representation or basis.
The displayed superposition can change under a basis change.
Term and explanation
Meaning
Keep in mind
Operator A linear map A A A
Not every operator is an observable or is defined on the whole space.
Observable A physical quantity represented, for sharp measurements, by a self-adjoint operator.
Its spectral projections determine outcome probabilities; general measurements use POVMs.
Hermitian For a matrix, A † = A A^\dagger=A A † = A
State the domain and terminology: symmetric need not mean self-adjoint.
Self-adjoint An operator satisfying A = A † A=A^\dagger A = A † D ( A ) = D ( A † ) D(A)=D(A^\dagger) D ( A ) = D ( A † )
Equality of differential expressions alone does not establish equality of operators.
Spectrum The values at which A − λ I A-\lambda I A − λ I
For a self-adjoint operator this includes eigenvalues and possible continuous spectral values; not every spectral value has a normalizable eigenvector.
Eigenvalue A scalar a a a A ∣ ψ ⟩ = a ∣ ψ ⟩ A\lvert\psi\rangle=a\lvert\psi\rangle A ∣ ψ ⟩ = a ∣ ψ ⟩
A degenerate eigenvalue has more than one linearly independent eigenvector.
Eigenstate A pure state represented by an eigenvector; measuring that observable has a sharp outcome.
A density operator supported entirely in one eigenspace also gives that sharp outcome.
Projector Here an orthogonal projector: P 2 = P P^2=P P 2 = P P † = P P^\dagger=P P † = P
Degenerate outcomes use the full eigenspace, not a preferred basis vector.
Function of an operator An operator built from another operator through spectral or functional calculus.
It is not generally an entrywise function of a matrix in an arbitrary basis.
Term and explanation
Meaning
Keep in mind
Born rule The probability rule p ( i ) = Tr ( ρ E i ) p(i)=\operatorname{Tr}(\rho E_i) p ( i ) = Tr ( ρ E i )
A probability needs both a state and a specified measurement.
Probability density A function integrated over a region to produce a probability.
A density value is not itself a probability.
Expectation value The statistical average of an observable over repeated preparations.
It need not be one of the observable’s eigenvalues.
Variance The mean squared deviation Var ( A ) = ⟨ ( A − ⟨ A ⟩ ) 2 ⟩ \operatorname{Var}(A)=\langle(A-\langle A\rangle)^2\rangle Var ( A ) = ⟨( A − ⟨ A ⟩ ) 2 ⟩
It is nonnegative when defined; some infinite-dimensional states have divergent second moments.
Covariance The symmetrized two-observable statistic measuring how fluctuations co-vary in a state.
For noncommuting observables, ordered products and real covariance are different.
Commutator The operator difference [ A , B ] = A B − B A [A,B]=AB-BA [ A , B ] = A B − B A
Operator order matters.
Compatibility For sharp observables, existence of a joint measurement, characterized by commuting spectral projections.
For unbounded operators, a formal zero commutator on a small domain is insufficient.
Uncertainty relation A bound relating spreads of measurement statistics in a prepared state.
It is not merely a statement about detector disturbance.
Projective measurement A measurement whose outcome effects are mutually orthogonal projectors summing to identity.
The ideal Lüders update is an additional specification of the state change.
POVM effect A positive operator E i E_i E i ∑ i E i = I \sum_iE_i=I ∑ i E i = I
Effects need not be projectors and do not specify the post-measurement state.
State update The rule assigning the state after a measurement, either conditioned on an outcome or averaged over ignored outcomes.
The measurement procedure matters; outcome probabilities alone are insufficient.
Term and explanation
Meaning
Keep in mind
Hamiltonian The self-adjoint operator H ( t ) H(t) H ( t )
For a time-independent Hamiltonian, evolution forms the group e − i H t / ℏ e^{-iHt/\hbar} e − i H t /ℏ
Unitary operator A linear map satisfying U † U = U U † = I U^\dagger U=UU^\dagger=I U † U = U U † = I
In infinite dimensions, U † U = I U^\dagger U=I U † U = I
Stationary state For time-independent H H H [ ρ , H ] = 0 [\rho,H]=0 [ ρ , H ] = 0
A pure state is stationary when supported in one energy eigenspace, including a degenerate one.
Tensor product The construction used to build composite-system Hilbert spaces from subsystem Hilbert spaces.
Tensor-factor ordering must be fixed.
Product state A joint state ρ A B = ρ A ⊗ ρ B \rho_{AB}=\rho_A\otimes\rho_B ρ A B = ρ A ⊗ ρ B ∣ ψ A ⟩ ⊗ ∣ ψ B ⟩ \lvert\psi_A\rangle\otimes\lvert\psi_B\rangle ∣ ψ A ⟩ ⊗ ∣ ψ B ⟩
A mixture of product states need not itself be a product state.
Entangled state A state that is not a mixture of product states across the specified partition. For a pure state, this means its ket does not factor.
A correlated mixed state can be separable; nonproduct does not by itself imply entangled.
Local observable An observable acting nontrivially on one subsystem and as identity on the others.
Local predictions can be computed from a reduced state.
Term and explanation
Meaning
Keep in mind
Density operator A positive, trace-one operator ρ \rho ρ
It can describe a pure or mixed state, and does not identify a unique preparation ensemble.
Density matrix A matrix representation of a density operator in a chosen basis.
Matrix entries are representation-dependent.
Ensemble A preparation procedure that produces states with specified classical probabilities.
Different ensembles can yield the same density operator.
Trace rule The expectation formula ⟨ A ⟩ = Tr ( ρ A ) \langle A\rangle=\operatorname{Tr}(\rho A) ⟨ A ⟩ = Tr ( ρ A )
For unbounded observables, the state must have the required finite moments; trace one alone is insufficient.
Partial trace The operation ρ A = Tr B ρ A B \rho_A=\operatorname{Tr}_B\rho_{AB} ρ A = Tr B ρ A B A A A
It discards information about B B B
Reduced density operator The state assigned to a subsystem by taking a partial trace of the joint state.
For an entangled pure bipartite state, both reductions are mixed.
Purification A representation of a mixed state as a reduced state of a larger pure state.
Purifications are not unique.
Term and explanation
Meaning
Keep in mind
Postulate A structural assumption used to define the formalism and connect it to experiment.
Postulates do not by themselves settle interpretation debates.
Equivalent formulation A different mathematical presentation that gives the same operational predictions in its domain.
Equivalent does not mean notationally identical.
Assumption and scope The domain in which a statement of the formalism is meant to apply.
Nonrelativistic quantum mechanics is not a universal theory of all regimes.
Correspondence principle The requirement that quantum theory reproduce appropriate classical predictions in suitable regimes.
It is a guide, not a complete derivation of classical mechanics by itself.
Classical limit A regime in which quantum predictions approximate classical behavior for selected variables and scales.
There is no single universal knob called “the classical limit.”
Semiclassical limit A regime with a small dimensionless parameter such as ℏ / S \hbar/S ℏ/ S
Large action alone is insufficient near turning points or caustics; the approximation needs local validity conditions.
Decoherence Suppression of local interference as a system becomes correlated with uncontrolled degrees of freedom.
Decoherence does not by itself select a unique outcome. With a mixed environment, the correlations need not be entanglement.
Quantization A procedure for constructing quantum models from classical structures.
It is not a unique inverse of taking a classical limit.
A calculation gives ⟨ A ⟩ = 0.7 \langle A\rangle=0.7 ⟨ A ⟩ = 0.7 0 0 0 1 1 1
Solution No. An expectation value is an ensemble average, not necessarily a single-shot outcome. If the probability of outcome 1 1 1 0.7 0.7 0.7 0 0 0 0.3 0.3 0.3 0.7 0.7 0.7
A student says that a subsystem of an entangled pure state must also be pure because the full state is pure. Which glossary terms correct the statement?
Solution Use “entangled state,” “partial trace,” and “reduced density operator.” A subsystem state is obtained by tracing out the other subsystem. For a pure state entangled across the chosen bipartition, both reduced density operators are mixed. For example, the Bell state ( ∣ 00 ⟩ + ∣ 11 ⟩ ) / 2 (\lvert00\rangle+\lvert11\rangle)/\sqrt2 (∣ 00 ⟩ + ∣ 11 ⟩) / 2 I / 2 I/2 I /2 Tr [ ( I / 2 ) 2 ] = 1 / 2 \operatorname{Tr}[(I/2)^2]=1/2 Tr [( I /2 ) 2 ] = 1/2
Why is the phrase “take ℏ → 0 \hbar\to0 ℏ → 0
Solution ℏ \hbar ℏ ℏ \hbar ℏ
D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics
K. Roszak and Ł. Cywiński, “Characterization and measurement of qubit-environment entanglement generation during pure dephasing,” Physical Review A 92 , 032310 (2015), doi:10.1103/PhysRevA.92.032310 , arXiv:1506.05729 , Eqs. (5)–(6) — dephasing with a mixed environment need not create entanglement.
J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics
J. Watrous, The Theory of Quantum Information
