Journal of Mathematical Physics, since 1960, publishes some of the best papers from outstanding mathematicians and physicists. It was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods suitable for such applications and for the formulation of physical theories.
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We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of MüllerLennert et al. [“On quantum Rényi entropies: A new definition, some properties,” J. Math. Phys.54, 122203 (2013); eprint arXiv:1306.3142v1; see also eprint arXiv:1306.3142].

Sandwiched (quantum) αRényi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglementbreaking channels,” preprint arXiv:1306.1586 (2013)] and MüllerLennert et al. [“On quantum Rényi entropies: a new definition, some properties and several conjectures,” preprint arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular, we show that sandwiched αRényi divergence satisfies the data processing inequality for all values of α > 1. Moreover we prove that αHolevo information, a variant of Holevo information defined in terms of sandwiched αRényi divergence, is superadditive. Our results are based on Hölder's inequality, the RieszThorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.

The Rényi entropies constitute a family of information measures that generalizes the wellknown Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasientropies and Renner's conditional min, max, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, minentropy, collision entropy, and the maxentropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including dataprocessing inequalities, a duality relation, and an entropic uncertainty relation.

We present an analytical study for the scattering amplitudes (Reflection R and Transmission T), of the periodic symmetric optical potential confined within the region 0 ⩽ x ⩽ L, embedded in a homogeneous medium having uniform potential W 0. The confining length L is considered to be some integral multiple of the period π. We give some new and interesting results. Scattering is observed to be normal (T^{2} ⩽ 1, R^{2} ⩽ 1) for V 0 ⩽ 0.5, when the above potential can be mapped to a Hermitian potential by a similarity transformation. Beyond this point (V 0 > 0.5) scattering is found to be anomalous (T^{2}, R^{2} not necessarily ⩽1). Additionally, in this parameter regime of V 0, one observes infinite number of spectral singularities E SS at different values of V 0. Furthermore, for L = 2nπ, the transition point V 0 = 0.5 shows unidirectional invisibility with zero reflection when the beam is incident from the absorptive side (Im[V(x)] < 0) but with finite reflection when the beam is incident from the emissive side (Im[V(x)] > 0), transmission being identically unity in both cases. Finally, the scattering coefficients R^{2} and T^{2} always obey the generalized unitarity relation : , where subscripts R and L stand for right and left incidence, respectively.

We investigate an algebraic model for the quantum oscillator based upon the Lie superalgebra , known as the Heisenberg–Weyl superalgebra or “the algebra of supersymmetric quantum mechanics,” and its Fock representation. The model offers some freedom in the choice of a position and a momentum operator, leading to a free model parameter γ. Using the technique of Jacobi matrices, we determine the spectrum of the position operator, and show that its wavefunctions are related to Charlier polynomials C n with parameter γ^{2}. Some properties of these wavefunctions are discussed, as well as some other properties of the current oscillator model.