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

Abstract

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 $3\times 3$ matrices.

27 pages

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Cited by in corpus (9)

• Classification of joint numerical ranges of three hermitian matrices of size three
• Measures of quantum correlations in infinite-dimensional systems
• Kippenhahn's Theorem for joint numerical ranges and quantum states
• Maximum-entropy inference and inverse continuity of the numerical range
• A new signature of quantum phase transitions from the numerical range
• On discontinuity of information characteristics of quantum systems and channels
• Pre-images of extreme points of the numerical range, and applications
• A variation principle for ground spaces
• Matrix systems, algebras, and open maps