By Stephan Weis
A convex set C is stable if the midpoint map (x,y) - (x+y)/2 is open. For compact C the Vesterstrøm–O’Brien theorem asserts that C is stable if and only if the barycentric map from the set of all Borel probability measures to C is open. Equivalently, the convex hull of any continuous function on C is continuous. We briefly discuss aspects of an extension of this theorem to certain non-compact sets [1]. An example is the set of quantum states (density operators on a separable Hilbert space) where continuity properties of entropic characteristics have been obtained from the stability. In the main part of the talk we present consequences of the stability for a finite-dimensional quantum system. 1) One result is a sufficient condition for the discontinuity of a maximum-entropy inference map under linear constraints (MaxEnt map) in term of the geometry of the linear image of the set of quantum states [2]. A corollary is that the irreducible three-party correlation of three qubits is discontinuous at the GHZ-state. 2) A second result is a continuity characterization of a MaxEnt map defined by a two-dimensional family of linear constraints [3]. A corollary is that the non-analyticity of the ground state energy of a one-parameter Hamiltonian is witnessed by a discontinuity of the MaxEnt map constrained on expectation values of the two energy terms.
By Anthony Leverrier
By Zbigniew Puchała
By Benoit Collins
By Satya Majumdar
By Motohisa Fukuda
By Anand Natarajan
By Nilanjana Datta
By Daniel Stilck Franca
By Ke Li
By Radosław Adamczak
By Madalin Guta
By Antonio Acín
By Ashley Montanaro
By Ping Zhong
By James Mingo
By Camille Male
By Clément Delcamp
By Claudio Dappiaggi
By Alexander Strohmaier
By Misha Gromov
By Céline Duval
By Olivier Benoist
