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Section
Quantum mechanical foundations for statistical mechanics
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
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).
The spectrum of is the set of scalars for which is not invertible. In finite dimension this is exactly the set of eigenvalues of , and self-adjointness makes every spectral value real.
Let be a trace-class operator on a complex Hilbert space, using the inner product convention linear in its first argument. For any orthonormal basis , the trace is
The sum is absolutely convergent and independent of the basis. For an arbitrary index set, it is the limit over finite subsets; at most countably many summands are nonzero. In finite dimension every linear operator is trace-class, and this formula equals the matrix trace.
Let be a complex Hilbert space. A density operator (also called a density matrix) is a positive trace-class operator such that:
In finite dimension, trace-class is automatic, so these conditions say that is a positive semidefinite matrix of trace one.
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:
which is a density-operator (positive semidefinite with trace ).
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 .
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).
A qubit is a quantum system whose state Hilbert space is two-dimensional over , hence isomorphic after choosing an orthonormal basis to
A pure state is a ray in , equivalently a rank-one density operator, and a general state is a positive trace-one operator .
A qutrit is a quantum system whose state Hilbert space is three-dimensional over , hence isomorphic after choosing an orthonormal basis to
Its pure states are rays in , and its general states are positive trace-one operators .
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 by spectral calculus.
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.
Let be a measurable space and a complex Hilbert space. A positive operator-valued measure or POVM is a map
such that:
The third condition is countable additivity in the weak operator topology.
Let be a complex Hilbert space. A bounded linear operator is positive semidefinite, written , if
for every . This condition implies that is self-adjoint and is equivalent to the existence of a bounded operator with . In finite dimension it is also equivalent to every eigenvalue of being nonnegative.
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.
An antiunitary operator between complex Hilbert spaces and is a conjugate-linear bijection such that
when the inner product is linear in its first argument. Equivalently, is a surjective conjugate-linear isometry. This differs from a unitary operator, which is linear.
This permanent provenance and reading index records 125 new concept knowls, the new nonassociative-algebra subject page, and one expanded existing entry. The expansion follows John Baez's Three Generations in E7 and John C. Baez and Paul Schwahn's The Standard Model Gauge Group from the Exceptional Jordan Algebra. The order below starts with reusable prerequisites and ends with the papers' principal constructions and theorems.