bibkey: wehrl1978entropy authors: Alfred Wehrl year: 1978 title: “General properties of entropy” doi: 10.1103/RevModPhys.50.221 url: https://doi.org/10.1103/RevModPhys.50.221 claim: “The von Neumann entropy of a density operator is the infimum of the Shannon entropies of its diagonals over complete orthonormal bases.” strata_touched:

• D5/S3/Quantum/QuantumChannels/FockAttenuator/Entropy
• D5/S3/Quantum/QuantumChannels/PhaseCovariantAttenuatorCoherentRefutation license: citation-only triage: anchor

General properties of entropy

A. Wehrl, Rev. Mod. Phys. 50, 221–260 (1978).

For a positive trace-one operator $\rho$ on a separable Hilbert space, its von Neumann entropy is the extended nonnegative spectral sum ${\sum }_j-{\lambda }_j\log {\lambda }_j$, with $0\log 0=0$. For every complete orthonormal basis $b$, its diagonal probabilities have the form $q_i={\sum }_j|⟨b_i,e_j⟩{|}^2{\lambda }_j$ in an eigenbasis $e$. Concavity and nonnegative summation give ${\sum }_i-q_i\log q_i\ge {\sum }_j-{\lambda }_j\log {\lambda }_j$. An eigenbasis, completed by a basis of the kernel, attains equality. This also applies when the entropy is infinite. The characterization therefore identifies spectral entropy with the infimum of diagonal Shannon entropies over complete countable bases.

The formal attenuator model adopts the latter definition. Its finite diagonal spectral formula is kernel-proved using concavity, the subprobability row bound and Parseval column sums. Its arbitrary-unitary invariance is checked by transporting complete Hilbert bases. The general spectral characterization for arbitrary density operators is cited, not kernel-proved. Bibliographic metadata and the standard eigenbasis argument support this reference; the full text of Wehrl’s review has not been inspected.

Locator

DOI and bibliographic metadata: https://doi.org/10.1103/RevModPhys.50.221. The full text has not been inspected; the general spectral characterization is literature-attested and is not claimed as a kernel theorem.