Semidefinite Programs for Completely Bounded Norms

John Watrous

doi:10.4086/toc.2009.v005a011

Abstract

The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.

Highlights

Linear mappings from one space of operators to another play an important role in quantum information theory
A natural question that arises is: what norms give rise to the most physically meaningful notions of distance? As is argued in [16], the answer to this question may depend on the problem at hand—but perhaps the most natural and widely applicable choice within quantum information theory is the completely bounded trace norm, known as the diamond norm
This norm was first used in the setting of quantum information by Kitaev [21], who used it mainly as a tool in studying quantum error correction and fault-tolerance

Summary

Introduction

Linear mappings from one space of operators (or matrices) to another play an important role in quantum information theory. To the author’s knowledge there were only two papers written prior to this one, namely [38] and [20], that presented methods to compute the completely bounded spectral or trace norm of a given mapping Both papers describe iterative methods, and analyze the complexity of each iteration of these methods, but do not analyze their rates of convergence. A second semidefinite programming formulation of the completely bounded trace norm is presented, based on the competitive quantum game framework of [18] This formulation is slightly simpler, but is valid only for mappings that are the difference between two quantum channels—which happens to be an important special case in quantum information. The first example concerns an alternate characterization of the completely bounded trace norm and the second illustrates a precise sense in which two known characterizations of the fidelity function (given by Uhlmann’s theorem and Alberti’s theorem) are dual statements to one another

Background

Strict dual feasibility

A semidefinite program for the completely bounded trace norm

A simpler semidefinite program for quantum channel distance

Connections with known results