On convex optimization problems in quantum information theory

Abstract

Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is usually impossible. As inspired by earlier investigations into the relative entropy of entanglement [Phys. Rev. A 78 032310 (2008)], we introduce a general method to solve the converse problem rather than find explicit solutions. That is, given a matrix in a convex set, we determine a family of convex functions that are minimized at this point. This method allows us find explicit formulae for the relative entropy of entanglement and the Rains bound, two well-known upper bounds on the distillable entanglement, and yields interesting information about these quantities, such as the fact that they coincide in the case where at least one subsystem of a multipartite state is a qubit.

15 pages. (version 2: minor edits -- updated to published version)

References in corpus (6)

• The resource theory of quantum reference frames: manipulations and monotones
• Generalized relative entropies and the capacity of classical-quantum channels
• Closed formula for the relative entropy of entanglement
• A limit of the quantum Renyi divergence
• Analytical formula connecting entangled state and the closest disentangled state
• Phase-asymmetry resource interconversion via estimation

Cited by in corpus (15)

• Quantum Resource Theories
• Convex geometry of quantum resource quantification
• Increasing relative nonclassicality quantified by standard entanglement potentials by dissipation and unbalanced beam splitting
• Computing Quantum Channel Capacities
• Nonadditivity of Rains' bound for distillable entanglement
• Exponential clustering of bipartite quantum entanglement at arbitrary temperatures
• Entanglement manipulation and distillability beyond LOCC
• Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
• Hierarchy of correlation quantifiers comparable to negativity
• Upper bounds for relative entropy of entanglement based on active learning
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• On directional derivatives of trace functionals of the form $A\mapsto\Tr(Pf(A))$
• Testing the Monogamy Relations via Rank-2 Mixtures
• Detecting Entanglement by Pure Bosonic Extension
• Online learning of a panoply of quantum objects