There, and Back Again: Quantum Theory and Global Optimisation

Abstract

We consider a problem in quantum theory that can be formulated as an optimisation problem and present a global optimisation algorithm for solving it, the foundation of which relies in turn on a theorem from quantum theory. To wit, we consider the maximal output purity ${\nu }_q$ of a quantum channel as measured by Schatten $q$-norms, for integer $q$. This quantity is of fundamental importance in the study of quantum channel capacities in quantum information theory. To calculate ${\nu }_q$ one has to solve a non-convex optimisation problem that typically exhibits local optima. We show that this particular problem can be approximated to arbitrary precision by an eigenvalue problem over a larger matrix space, thereby circumventing the problem of local optima. The mathematical proof behind this algorithm relies on the Quantum de Finetti theorem, which is a theorem used in the study of the foundations of quantum theory. We expect that the approach presented here can be generalised and will turn out to be applicable to a larger class of global optimisation problems. We also present some preliminary numerical results, showing that, at least for small problem sizes, the present approach is practically realisable.

6 pages; RevTex 4; minor change to major result; major change to minor result

References in corpus (8)

• Unknown Quantum States: The Quantum de Finetti Representation
• A complete family of separability criteria
• Coding Theorems for Quantum Channels
• Additivity for a class of unital qubit channels
• On some additivity problems in quantum information theory
• Maximization of capacity and p-norms for some product channels
• On the multiplicativity conjecture for quantum channels
• Comments on multiplicativity of maximal p-norms when p = 2