Section
Quantum Foundations
Quantum mechanical foundations for statistical mechanics
Core idea
Operators
- Complex Hilbert space (finite-dim)
- Bounded operator
- Self-adjoint operator (observable)
- Spectrum (finite-dimensional)
- Trace of an operator
A finite-dimensional complex Hilbert space is a complex vector space of finite dimension equipped with an inner product
that is linear in one argument and conjugate-linear in the other (conventions vary), positive definite, and induces a norm . In finite dimension, every normed vector space is complete, so “Hilbert space” is automatic once the inner product is given.
This is the basic setting for finite-dimensional quantum mechanics: pure states are unit vectors up to a global phase, and more generally states are represented by Density Operator.
Let be a (complex) Hilbert space. A linear operator is bounded if there exists a constant such that
The smallest such constant is the operator norm
Bounded operators are the standard class of operators used to model physical transformations and observables in many finite-dimensional quantum settings.
Let be a complex Hilbert space and let be a bounded operator (see Bounded Operator Hilbert). The operator is self-adjoint (or Hermitian) if
where is the adjoint defined by
In (finite-dimensional) quantum mechanics, self-adjoint operators are used to represent observables (measurable quantities).
Let be a finite-dimensional complex Hilbert space (Complex Hilbert Space Finite) and let be self-adjoint (Self Adjoint Operator Observable).
Let be a finite-dimensional complex Hilbert space and let be a linear operator. The trace of , denoted , is defined by choosing any orthonormal basis and setting
This value is independent of the chosen orthonormal basis.
This notion agrees with the usual matrix trace (see Trace) after identifying with its matrix in an orthonormal basis.
Let be a complex Hilbert space (typically finite-dimensional in basic quantum theory). A density operator (also called a density matrix) is an operator such that:
- Positivity: , meaning for all .
- Unit trace: , where is the operator trace (Trace Operator).
In finite dimension, these conditions are equivalent to being a positive semidefinite matrix with trace .
Let be a (finite-dimensional) complex Hilbert space (see complex-hilbert-space-finite). A pure quantum state can be specified in either of two equivalent ways:
- State vector (ray): a unit vector with , where and represent the same physical state (global phase is unobservable).
- Density operator (projector): the rank-one operator
which is a density-operator (positive semidefinite with trace ).
Equivalent characterizations (finite dimension)
For a density operator on , the following are equivalent:
- is pure.
- has rank .
- (idempotent projector).
- (maximal “purity” among states).
- The eigenvalues of are .
Expectation values
If is an observable (a self-adjoint operator; see self-adjoint-operator-observable), then in the pure state ,
A mixed quantum state on a finite-dimensional complex Hilbert space is a density-operator that is not pure (see pure-state-quantum). Concretely, is mixed iff it cannot be written as for any unit vector .
Equivalent characterizations (finite dimension)
For a density operator , the following are equivalent:
- is mixed.
- has rank .
- .
- .
- The von-neumann-entropy satisfies .
Convex mixture form
Every density operator admits a decomposition as a convex combination of pure states:
This decomposition is generally not unique. A distinguished choice is the spectral decomposition, where the are orthonormal eigenvectors and the are eigenvalues.
How mixed states arise
Mixed states appear in (at least) two mathematically distinct ways:
- Classical uncertainty (statistical mixture): the system is prepared in pure state with probability .
- Reduced state of a larger system: if is a state on , then the subsystem state obtained by the partial-trace can be mixed even when is pure (entanglement).
Special cases
- Maximally mixed state: on , which has the largest von Neumann entropy .
Let and be finite-dimensional complex Hilbert spaces, and let be an operator on the tensor product . The partial trace over is the unique linear map
characterized by the identity
where is the usual trace (see trace-operator).
Let be a density-operator on a finite-dimensional Hilbert space . The von Neumann entropy of is
Here is defined by functional calculus via the spectral decomposition of (see spectrum-self-adjoint-finite). By convention, eigenvalues equal to contribute .
Let and be density-operators on a finite-dimensional Hilbert space . The quantum relative entropy (also called Umegaki relative entropy) is defined by
The support condition is needed because is not finite on the kernel of . The operators and are defined via spectral calculus (see spectrum-self-adjoint-finite).
Classical (commuting) case
If and commute, they are simultaneously diagonalizable and the formula reduces to the classical relative entropy (Kullback-Leibler divergence) of their eigenvalue distributions; compare relative-entropy-kl-divergence.
Key properties
- Nonnegativity (Klein inequality): , with equality iff .
- Not symmetric: typically ; it is not a metric.
- Data processing inequality: for any completely positive trace-preserving map , In particular, taking (see partial-trace) gives
Relation to von Neumann entropy
If and is the maximally mixed state, then
where is the von-neumann-entropy.
Let and be Hermitian (self-adjoint) matrices (equivalently, finite-dimensional self-adjoint operators; see self-adjoint-operator-observable). The Golden-Thompson inequality states that
where denotes the matrix exponential and is the trace (see trace-operator).
A normalized state vector in a complex Hilbert space is a vector with . It lies on the unit sphere.
In quantum mechanics, and represent the same pure state, because all transition probabilities depend only on squared inner-product magnitudes. The phase-independent state is equivalently represented by the rank-one projector .
For a normalized vector in a complex Hilbert space, the rank-one projector onto its span is the operator
It is an orthogonal projection, is positive semidefinite, is idempotent, and has trace one. Multiplying by a complex phase leaves unchanged, so the projector represents the associated one-dimensional complex line rather than a chosen vector on it.
On a finite-dimensional complex Hilbert space , a positive operator-valued measure or POVM with finite outcome set is a family of positive semidefinite operators satisfying
For a density operator , outcome has probability . Positivity makes these probabilities nonnegative, and the normalization makes them sum to one.
Orthogonal projective measurements are a special case; general POVM effects need not be projections or mutually orthogonal.
An operator on a complex Hilbert space is positive semidefinite, written , if
for every vector . Positivity implies that is self-adjoint. In finite dimension it is equivalent to all eigenvalues of being real and nonnegative, and also equivalent to a factorization .
Positive operators are the effects in a POVM and, after trace normalization, the density operators of quantum theory.
A bounded operator on a complex Hilbert space is normal if it commutes with its adjoint:
Self-adjoint, unitary, and orthogonal projection operators are normal. In finite dimension, normality is equivalent to unitary diagonalizability.
The spectral theorem extends this structure to infinite-dimensional Hilbert spaces through a projection-valued spectral measure. This extra structure implies that a non-scalar normal operator on a Hilbert space has nontrivial closed invariant subspaces.