De 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.
De Anthony Leverrier
De Zbigniew Puchała
De Benoit Collins
De Satya Majumdar
De Motohisa Fukuda
De Nilanjana Datta
De Daniel Stilck Franca
De Radosław Adamczak
De Madalin Guta
De Antonio Acín
De Ashley Montanaro
De Ping Zhong
De James Mingo
De Camille Male
De Albert Schwarz
De Clément Delcamp
De Clément Delcamp
De Giuseppe Savaré
