Yet another additivity conjecture

Peter W. Shor

In this paper, I discuss the additivity conjecture in quantum information theory.The additivity conjecture was originally a set of at least four conjectures.These conjectures said that certain functions of quantum states and channels were additive 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 conjecture, in that counterexamples to a set of stronger conjectures have been found.These will be described briefly.

Keiji Matsumoto

Yoshinori Matsuo

Abstract Recently, it was proposed that there must be either large violation of the additivity conjecture or a set of disentangled states of the black hole in the AdS/CFT correspondence. In this paper, we study the additivity conjecture for quantum states of fields around the Schwarzschild black hole. In the eternal Schwarzschild spacetime, the entanglement entropy of the Hawking radiation is calculated assuming that the vacuum state is the Hartle-Hawking vacuum. In the additivity conjecture, we need to consider the state which gives minimal output entropy of a quantum channel. The Hartle-Hawking vacuum state does not give the minimal output entropy which is consistent with the additivity conjecture. We study the entanglement entropy in other static vacua and show that it is consistent with the additivity conjecture.

Keiji Matsumoto

Motohisa Fukuda, Christopher King, David K. Moser

A. A. Pomeransky

The additivity of both the entanglement of formation and the classical channel capacity is known to be a consequence of the strong superadditivity conjecture. We show that, conversely, the strong superadditivity conjecture follows from the additivity of the entanglement of formation; this means that the two conjectures are equivalent and that the additivity of the classical channel capacity is a consequence of them.