Semidefinite programming strong converse bounds for classical capacity

TL;DR

This paper develops semidefinite programming bounds for classical capacity of quantum channels, establishing strong converse results and improving capacity upper bounds, with applications to specific channels like amplitude damping.

Contribution

It introduces new SDP-based strong converse bounds for quantum channel capacities, enhancing understanding of capacity limits and zero-error capacities.

Findings

Derived SDP finite blocklength converse bounds for quantum channels.

Established strong converse bounds showing exponential decay of success probability.

Provided improved upper bounds on classical capacity of amplitude damping channels.

Abstract

We investigate the classical communication over quantum channels when assisted by no-signaling (NS) and positive-partial-transpose-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and the one-shot $\epsilon$-error capacity are formalized as semidefinite programs (SDPs). Based on this, we obtain improved SDP finite blocklength converse bounds of general quantum channels for entanglement-assisted codes and unassisted codes. Furthermore, we derive two SDP strong converse bounds for the classical capacity of general quantum channels: for any code with a rate exceeding either of the two bounds of the channel, the success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bounds, we derive an improved upper bound on the classical capacity of the amplitude damping channel. We also establish the strong converse property for the classical and private capacities of a new class of quantum channels. We finally study the zero-error setting and provide efficiently computable upper bounds on the one-shot zero-error capacity of a general quantum channel.