×

Conditional entropy in algebraic statistical physics. (English) Zbl 0587.46056

The notion of conditional entropy as entropy of a conditional state on a \(C^*\)-algebra \({\mathcal A}\) with respect to its \(C^*\)-subalgebra \({\mathcal A}_ 1\subset {\mathcal A}\) is introduced. It is proved that for a compatible state \(\sigma\) on \({\mathcal A}\) (which admits the conditional expectation of Umegaki-Takesaki) the mean conditional entropy in an a priori state \(\sigma_ 1\) on \({\mathcal A}\) is equal to the difference of the entropy of the tate \(\sigma\) on \({\mathcal A}\) and the entropy of the state \(\sigma_ 1\) on \({\mathcal A}_ 1\). The conditional entropy enables us to define the input-output information of a quantum communication channel in analogy to the classical Shannon formula.

MSC:

46L60 Applications of selfadjoint operator algebras to physics
46N99 Miscellaneous applications of functional analysis
94A40 Channel models (including quantum) in information and communication theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Billingsley, P., Ergodic Theory and Information (1965), Wiley: Wiley New York · Zbl 0141.16702
[2] Dixmier, J., Les algèbres d’opérateurs dans l’espace Hilbertien (1969), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0175.43801
[3] Evans, D. E.; Lewis, J. T., Communications of the Dublin Institute for Advanced Studies A (1977), No. 24, Dublin · Zbl 0375.46052
[4] Gudder, S.; Marchand, J.-P., Rep. Math. Phys., 12, 317 (1977)
[5] Naudts, J., Commun. Math. Phys., 37, 175 (1974)
[6] Sakai, S., \(C^∗\)-algebras and \(W^∗\)-algebras (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0219.46042
[7] Takesaki, M., J. Functional Analysis, 9, 306 (1972)
[8] Wehrl, A., Rev. Mod. Phys., 50, 221 (1980)
[9] Belavkin, V.P., Staszewski P.: On the Entropy of Dynamical Maps; Belavkin, V.P., Staszewski P.: On the Entropy of Dynamical Maps · Zbl 0526.46060
[10] Belavkin, V. P.; Staszewski, P., Ann. Inst. H. Poincaré, 37, 1, 51 (1982)
[11] Araki, H., Publ. RIMS Kyoto Univ., 11, 809 (1975⧸1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.