Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1

• Published in: Communications in mathematical physics Vol. 284; no. 1; pp. 263 - 280
• Main Authors: Hayden, Patrick, Winter, Andreas
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.11.2008; Springer
• Subjects: Classical and Quantum Gravitation; Complex Systems; Exact sciences and technology; Mathematical and Computational Physics; Mathematical methods in physics; Mathematical Physics; Other topics in mathematical methods in physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Entangle State; Entropy; Quantum Information Theory; Quantum Channel; Classical Capacity; Quantum information; Mathematical physics; Quantum theory; Information theory
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-008-0624-0

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 Rényi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Rényi 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.