The MaxEnt extension of a quantum Gibbs family, convex geometry and geodesics

We discuss methods to analyze a quantum Gibbs family in the ultra-cold regime where the norm closure of the Gibbs family fails due to discontinuities of the maximum-entropy inference. The current discussion of maximum-entropy inference and irreducible correlation in the area of quantum phase transitions is a major motivation for this research. We extend a representation of the irreducible correlation from finite temperatures to absolute zero.Comment: 8 pages, 3 figures, 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 21-26 September 2014, Ch\^ateau du Clos Luc\'e, Amboise, Franc

A variation principle for ground spaces

The ground spaces of a vector space of hermitian matrices, partially ordered by inclusion, form a lattice constructible from top to bottom in terms of intersections of maximal ground spaces. In this paper we characterize the lattice elements and the maximal lattice elements within the set of all subspaces using constraints on operator cones. Our results contribute to the geometry of quantum marginals, as their lattices of exposed faces are isomorphic to the lattices of ground spaces of local Hamiltonians.Comment: 18 pages, 2 figures, version v3 has an improved exposition, v4 has a new non-commutative example and catches a glimpse of three qubit

Discontinuities in the Maximum-Entropy Inference

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.Comment: 8 pages, 3 figures, 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Garching, Germany, 15-20 July 201

Kippenhahn's construction revisited

Kippenhahn discovered that the numerical range of a complex square matrix is the convex hull of a plane real algebraic curve. Here, we present an example of a convex set, which has a similar algebraic description as the numerical range, whereas the analogue of Kippenhahn's construction fails regarding isolated, singular points of the curve. This example prompted us to carefully review Kippenhahn's assertion and to highlight aspects of a complete proof that was achieved with methods of convex geometry and real algebraic geometry.Comment: 10 pages, accepted for publication in the proceedings of IWOTA 202