von Neumann entropy
E590895
Von Neumann entropy is a measure of quantum uncertainty or mixedness of a quantum state, generalizing classical Shannon entropy to density matrices in quantum mechanics and quantum information theory.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
information-theoretic quantity
ⓘ
quantum entropy measure ⓘ |
| additivityCondition | additive for product states: S(ρ⊗σ) = S(ρ)+S(σ) ⓘ |
| alternativeLogarithmBase | e ⓘ |
| alternativeUnit | nat ⓘ |
| appliesTo |
density matrix
ⓘ
density operator ⓘ |
| characterizes |
mixedness of a quantum state
ⓘ
quantum uncertainty ⓘ |
| classicalLimit | reduces to Shannon entropy for commuting density matrices ⓘ |
| context |
black hole entropy and holography
ⓘ
quantum thermodynamics ⓘ |
| definedOn | trace-class operators on Hilbert space ⓘ |
| definition | S(ρ) = -Tr(ρ log ρ) ⓘ |
| domain | quantum states ⓘ |
| equals | Shannon entropy of eigenvalue distribution of ρ ⓘ |
| field |
quantum information theory
ⓘ
quantum mechanics ⓘ statistical mechanics ⓘ |
| generalizes | Shannon entropy NERFINISHED ⓘ |
| isPositiveFor | mixed states ⓘ |
| isZeroFor | pure states ⓘ |
| maximumCondition | maximal for maximally mixed state on a given Hilbert space ⓘ |
| maximumValue | log d for maximally mixed state on d-dimensional Hilbert space ⓘ |
| monotoneUnder | completely positive trace-preserving maps for appropriate combinations ⓘ |
| namedAfter | John von Neumann NERFINISHED ⓘ |
| nonNegativity | S(ρ) ≥ 0 ⓘ |
| property |
concave in the density operator
ⓘ
unitarily invariant ⓘ |
| relatedConcept |
Rényi entropy
NERFINISHED
ⓘ
entanglement entropy ⓘ relative entropy ⓘ |
| requires |
spectral decomposition of density operator
ⓘ
trace operation ⓘ |
| satisfies |
Araki–Lieb inequality
NERFINISHED
ⓘ
strong subadditivity ⓘ subadditivity ⓘ |
| symbol |
S
ⓘ
S(ρ) ⓘ |
| typicalUnit | bit ⓘ |
| usedFor |
Holevo bound
NERFINISHED
ⓘ
defining coherent information ⓘ defining quantum conditional entropy ⓘ defining quantum mutual information ⓘ entanglement measures ⓘ quantum channel capacity formulas ⓘ quantum data compression ⓘ |
| usesLogarithmBase | 2 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.