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arXiv:1007.5464 · Journal of Mathematical Physics

Entropy distance: New quantum phenomena

Stephan Weis, Andreas Knauf

Published Oct 1, 2012• 30 citations
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Abstract

We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance, and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology, and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.

DOI
10.1063/1.4757652

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We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance, and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology, and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.
We study the continuity of an abstract generalization of the maximum-entropy inference—a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for 3 × 3 matrices.
We revisit the maximum-entropy inference of the state of a finite-level quantum system under linear constraints. The constraints are specified by the expected values of a set of fixed observables. We point out the existence of discontinuities in this inference method. This is a pure quantum phenomenon since the maximum-entropy inference is continuous for mutually commuting observables. The question arises why some sets of observables are distinguished by a discontinuity in an inference method which is still discussed as a universal inference method. In this paper we make an example of a discontinuity and we explain a characterization of the discontinuities in terms of the openness of the (restricted) linear map that assigns expected values to states.
This work generalizes the interaction measure of multi-information known in probability theory to finite-level quantum systems.This is done in the more general context of the entropy distance from an exponential family.One of the most well-known measures of stochastic dependence is multi-information, applied in various fields including Neuroscience and Statistical Mechanics.Multi-information is the entropy distance from an exponential family.In statistics, an exponential family is a very familiar parametric model, it admits a simple geometric description of entropy distance and maximum-likelihood estimation by mean values.But a complete description of these concepts requires to study extensions of a family, an investigation that was started by N. N. Čencov and O. Barndorff-Nielsen and continued principally by I. Csiszár and F. Matúš.Very little is known about extensions of an exponential family for a finite-level quantum system.In this thesis we consider mean value parameters of the statistic of such a family and a suitable extension thereof.Generalizing probability theory, we prove that the parameters describe the entropy distance from the family and they parametrize its rIclosure consisting of the points approximated in relative entropy.The dimension function of a local maximizer of entropy distance is bounded by the dimension of the family.We show that the rI -closure of a Gibbs family consists of the maximum entropy ensembles.A new and generic phenomenon of a non-abelian exponential family is the appearance of non-exposed faces of the convex mean value set.We prove that mean values of the closure of an e-geodesic included in the family meet only the relative interior of exposed faces.Unlike in finite probability spaces there are examples with a discontinuous entropy distance, the continuity being equivalent to equality of rI -closure and topological closure of the family.Examples suggest that the topology of an exponential family is related to the topology of associated projector lattices and to open projections and symmetrizations of state spaces.We conclude that multi-information for a quantum system is a continuous function equal to the entropy distance from a factorizable family.For analysis of a factorizable family we supply a partial classification of convex exponential families.As a perspective to a dynamical situation we examine a measure of temporal interaction for abelian systems, which is related to multi-information.i Being able to write this thesis in the Mathematical Physics group at Friedrich-Alexander-Universität Erlangen-Nürnberg was a great opportunity.In the first place I want to thank my supervisor Prof. Andreas Knauf for the constant and friendly support in conversations about ideas and literature both in mathematics and physics and for his encouragement to a systematic problem solving.
In scope of CPCPCP -convexity theory for C∗C^{*}C∗ -algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational stmcture of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.
An exponential family is a manifold of generalized Gibbs state of the form exp(H)/Tr(exp(H)), where H belongs to a vector space of (possibly non-commutative) hermitian matrices. Generalized Gibbs states are important in small-scale thermodynamics, they represent equilibrium states regarding several conserved quantities that admit novel operations without heat dissipation [1]. Quantum information theory and condensed matter physics consider a space of local Hamiltonians acting on spins. The entropy distance from this exponential family is a measure of many-body complexity [2,3,4]. This talk is concerned with the geometry and topology of an exponential family and its entropy distance [5]. The maximum-entropy inference map parametrizes the exponential family. This map is continuous in the interior of its domain, the joint numerical range [6]. We describe the points of discontinuity in terms of open mapping theorems and eigenvalue crossings. Because of the discontinuity, the inference map and the entropy distance cannot be approximated through interior points. Instead, it is necessary to study faces (flat portions on the boundary) of the joint numerical range. With local Hamiltonians, this requires studying the faces of the set of quantum marginals. References. [1] M. Lostaglio, D. Jennings, and T. Rudolph,New Journal of Physics 19, 043008 (2017). [2] N. Ay, E. Olbrich, N. Bertschinger, and J. Jost,Chaos 21, 037103 (2011). [3] S. Niekamp, T. Galla, M. Kleinmann, and O. Gühne, J. Phys. A-Math. Gen. 46, 125301 (2013). [4] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter,New York: Springer, 2019. [5] S. Weis, Journal of Convex Analysis 21, 339-399 (2014). [6] L. Rodman, I. M. Spitkovsky, A. Szkoła, and S. Weis,Journal of Mathematical Physics 57, 015204 (2016).