Multiplicativity of Completely Bounded p-Norms Implies a Strong Converse for Entanglement-Assisted Capacity

2013/10/31 by Manish K. Gupta, Mark M. Wilde · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Bounded function #Computability, Logic, AI Algorithms #Converse #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematical economics #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum entanglement #Quantum mechanics #cs.IT #math.IT #quant-ph

paper · pdf · doi:10.1007/s00220-014-2212-9

published as Communications in Mathematical Physics, vol. 334, no. 2, pages 867-887 (March 2015) · 21 pages, final version accepted for publication in Communications in Mathematical Physics

arxiv created 2014/08/11 · openalex publication_date 2014/10/20 · arxiv updated 2015/02/10 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05

Abstract

The fully quantum reverse Shannon theorem establishes the optimal rate of noiseless classical communication required for simulating the action of many instances of a noisy quantum channel on an arbitrary input state, while also allowing for an arbitrary amount of shared entanglement of an arbitrary form. Turning this theorem around establishes a strong converse for the entanglement-assisted classical capacity of any quantum channel. This paper proves the strong converse for entanglement-assisted capacity by a completely different approach and identifies a bound on the strong converse exponent for this task. Namely, we exploit the recent entanglement-assisted "meta-converse" theorem of Matthews and Wehner, several properties of the recently established sandwiched Renyi relative entropy (also referred to as the quantum Renyi divergence), and the multiplicativity of completely bounded p-norms due to Devetak et al. The proof here demonstrates the extent to which the Arimoto approach can be helpful in proving strong converse theorems, it provides an operational relevance for the multiplicativity result of Devetak et al., and it adds to the growing body of evidence that the sandwiched Renyi relative entropy is the correct quantum generalization of the classical concept for all α>1.

Citations

• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Correcting Quantum Errors with Entanglement
• A Mathematical Theory of Communication
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Distance measures to compare real and ideal quantum processes
• Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding
• A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
• Quantum computations: algorithms and error correction
• Simpler semidefinite programs for completely bounded norms
• Trace inequalities and quantum entropy: an introductory course
• A relation between completely bounded norms and conjugate channels
• Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem

Cited by

• Completely Bounded Qusi-Norms, Their Mutiplicativity, and New Additivity Results of Quantum Channels
• Refined Strong Converse for the Constant Composition Codes
• Relative entropy for von Neumann subalgebras