2008/07/30 by Patrick Hayden, Andreas Winter
Computer Science · Mathematics · Physics and Astronomy · #cs.IT #math-ph #math.IT #math.MP #quant-ph
paper · pdf · doi:10.1007/s00220-008-0624-0
published as Comm. Math. Phys. 284(1):263-280, 2008. · Merger of arXiv:0707.0402 and arXiv:0707.3291 containing new and improved analysis of counterexamples. 17 pages
arxiv created 2008/07/30 · arxiv updated 2012/07/06
For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p >1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p=1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.