The von Neumann entropy of a density matrix of dimension d, expressed in terms of the first d − 1 integer order Rényi entropies, is monotonically increasing in Rényi entropies of even order and decreasing in those of odd order.
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.
Abstract An alternative method is presented for extracting the von Neumann entropy − Tr(ρ ln ρ) from Tr(ρn) for integer n in a quantum system with density matrix ρ. Instead of relying on direct analytic continuation in n, the method uses a generating function − Tr{ρ ln[(1 − zρ)/(1 − z)]} of an auxiliary complex variable z. The generating function has a Taylor series that is absolutely convergent within |z| < 1, and may be analytically continued in z to z = −∞ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in n. Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.
We prove an integral representation of the α-z Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray z=cα, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders α above one, we require the divergence to be finite, whereas for orders below one support inclusion suffices. An explicit boundary correction removes the lower-order support assumption. The proof uses a compatible weighted Haagerup Lp scale and applies outside the data-processing region. For faithful density matrices, we express the derivative of the escort relative entropy with respect to the Rényi order in terms of a nonnegative fluctuation quantity. Differences of order-one derivatives of the divergence give quantum Fisher information and Wigner-Yanase skew information. For faithful bipartite density matrices, we also express optimized conditional α-z entropies at orders above one as averages of ordinary conditional entropies. Further applications concern coherent-state divergences in quantum field theory and the optimizing order for the strong converse exponent in hypothesis-testing.