Strong converse for the quantum capacity of the erasure channel for almost all codes

Abstract

A strong converse theorem for channel capacity establishes that the error probability in any communication scheme for a given channel necessarily tends to one if the rate of communication exceeds the channel's capacity. Establishing such a theorem for the quantum capacity of degradable channels has been an elusive task, with the strongest progress so far being a so-called "pretty strong converse". In this work, Morgan and Winter proved that the quantum error of any quantum communication scheme for a given degradable channel converges to a value larger than $1/\sqrt{2}$ in the limit of many channel uses if the quantum rate of communication exceeds the channel's quantum capacity. The present paper establishes a theorem that is a counterpart to this "pretty strong converse". We prove that the large fraction of codes having a rate exceeding the erasure channel's quantum capacity have a quantum error tending to one in the limit of many channel uses. Thus, our work adds to the body of evidence that a fully strong converse theorem should hold for the quantum capacity of the erasure channel. As a side result, we prove that the classical capacity of the quantum erasure channel obeys the strong converse property.

15 pages, submission to the 9th Conference on the Theory of Quantum Computation, Communication, and Cryptography (TQC 2014)

References in corpus (11)

• On quantum Renyi entropies: a new generalization and some properties
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Quantum Communication With Zero-Capacity Channels
• Quantum marginal problem and representations of the symmetric group
• On the strong converses for the quantum channel capacity theorems
• Fundamental bound on the reliability of quantum information transmission
• The decoupling approach to quantum information theory
• Multiplicativity of completely bounded $p$-norms implies a strong converse for entanglement-assisted capacity
• Strong converse for the classical capacity of the pure-loss bosonic channel
• The capacity of a quantum channel for simultaneous transmission of classical and quantum information
• Uncertainty relations for multiple measurements with applications

Cited by in corpus (2)

• The platypus of the quantum channel zoo
• Geometric Rényi Divergence and its Applications in Quantum Channel Capacities