Current Developments in Mathematics

The Additivity Conjecture in Quantum Information Theory

Peter W. Shor

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In this paper, I discuss the additivity conjecture in quantum information theory. The additivity conjecture was orig- inally a set of at least four conjectures. These conjectures said that certain functions of quantum states and channels were addi- tive under tensor products. While some of these conjectures were previously known to be stronger than others, they have recently all been proved equivalent. This conjecture is a very intriguing mathematical question which the best efforts of a large number of quantum information theorists have not been able to resolve for nearly a decade. It is a mathematically elegant question that is one of the most important open questions in the field of quantum information and computation. This paper then is intended to be both an exposition of the conjecture, its background, and some of the methods that have been used to yield partial results for it, as well as a plea for help in resolving this conjecture. Very recently (summer 2007), substantial progress has been made on this conjec- ture, in that counterexamples to a set of stronger conjectures have been found. These will be described briefly.

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Current Developments in Mathematics, Volume 2005 (2007), 173-189.

First available in Project Euclid: 10 October 2008

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Shor, Peter W. The Additivity Conjecture in Quantum Information Theory. Current Developments in Mathematics 2005 (2007), 173--189.

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