Stephan Weis

Entropy distance: New quantum phenomena

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.

Geometry of the Set of Mixed Quantum States: An Apophatic Approach

The set of quantum states consists of density matrices of order N, which are hermitian, positive and normalized by the trace condition. We analyze the structure of this set in the framework of the Euclidean geometry naturally arisingin the space of hermitian matrices. For \( N\,\,=\,\,2 \) this set is the Bloch ball, embedded in \( \mathbb{R}^3 \). For \( N\,\,\geq \,\,3\) this set of dimensionality \( N^2 \,\,-\,1 \) has a much richer structure. We study its properties and at first advocate an apophatic approach, which concentrates on characteristics not possessed by this set. We also apply more constructive techniques and analyze twodimensional cross-sections and projections of the set of quantum states. They are dual to each other. At the end we make some remarks on certain dimension dependent properties.

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

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

Maximizing the Divergence from a Hierarchical Model of Quantum States

3\times 3

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

matrices.

Discontinuities in the maximum-entropy inference

We study many-party correlations quantified in terms of the Umegaki relative entropy (divergence) from a Gibbs family known as a hierarchical model. We derive these quantities from the maximum-entropy principle which was used earlier to define the closely related irreducible correlation. We point out the differences between quantum states and probability vectors which exist in hierarchical models, in the divergence from a hierarchical model and in local maximizers of this divergence. The differences are, respectively, missing factorization, discontinuity and reduction of uncertainty. We discuss global maximizers of the mutual information of separable qubit states.

Classification of joint numerical ranges of three hermitian matrices of size three

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.

Pre-images of extreme points of the numerical range, and applications

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.

A new signature of quantum phase transitions from the numerical range

Abstract The joint numerical range W ( F ) of three hermitian 3-by-3 matrices F = ( F 1 , F 2 , F 3 ) is a convex and compact subset in R 3 . Generically we find that W ( F ) is a three-dimensional oval. Assuming dim ⁡ ( W ( F ) ) = 3 , every one- or two-dimensional face of W ( F ) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W ( F ) using random matrices and dual varieties.

Erratum to: Continuity of the Maximum-Entropy Inference

We extend the pre-image representation of exposed points of the numerical range of a matrix to all extreme points. With that we characterize extreme points which are multiply generated, having at least two linearly independent pre-images, as the extreme points which are Hausdorff limits of flat boundary portions on numerical ranges of a sequence converging to the given matrix. These studies address the inverse numerical range map and the maximum-entropy inference map which are continuous functions on the numerical range except possibly at certain multiply generated extreme points. This work also allows us to describe closures of subsets of 3-by-3 matrices having the same shape of the numerical range.