Von Neumann defined the informational entropy of a density matrix ρ by the expression S(ρ)=−tr(ρ ln ρ). [J. Von Neumann, Die Mathematischen Grundlagen der Quantummechanik (Springer‐Verlag, Berlin, 1932)]. Here, starting from the definitions of Shannon entropy and of Renyi entropy of random variables, and using some (reasonable) rules of inference, two expressions are obtained for the quantum entropy of a given square (deterministic) matrix, irrespective of any probabilistic framework and/or any randomization technique. These definitions apply directly to linear operators on Hilbert spaces even when they are not of positive trace class, and they contain Von Neumann entropy as a special case. These new measures of uncertainty could provide new approaches to some problems related to structure complexity.
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November 1991
Research Article|
November 01 1991
Quantum entropies of nonprobabilistic matrices
Guy Jumarie
Guy Jumarie
Department of Mathematics and Computer Science, Université du Québec à Montréal, P.O. Box 8888, St A, Montreal, Quebec, H3C 3P8, Canada
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J. Math. Phys. 32, 2967–2971 (1991)
Article history
Received:
January 18 1991
Accepted:
April 30 1991
Citation
Guy Jumarie; Quantum entropies of nonprobabilistic matrices. J. Math. Phys. 1 November 1991; 32 (11): 2967–2971. https://doi.org/10.1063/1.529039
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