Semidefinite programming converse bounds for quantum communication

TL;DR

This paper develops efficiently computable semidefinite programming bounds for quantum communication, providing new limits on quantum channel capacity and improving upon previous bounds with practical applications.

Contribution

It introduces new SDP-based converse bounds for quantum communication, including a strong converse bound equal to max-Rains information, enhancing capacity estimation methods.

Findings

Derived one-shot SDP bounds improving previous results

Established an SDP strong converse bound for quantum capacity

Proved the SDP bound equals the max-Rains information

Abstract

We derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one-shot and asymptotic regime. First, we derive one-shot semidefinite programming (SDP) converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes, Nat. Commun. 7, 2016]. As applications, we study quantum communication over depolarizing channels and amplitude damping channels with finite resources. Second, we find an SDP strong converse bound for the quantum capacity of an arbitrary quantum channel, which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases. Furthermore, we prove that the SDP strong converse bound improves the partial transposition bound introduced by Holevo and Werner. Third, we prove that this SDP strong converse bound is equal to the so-called max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP strong converse bound is weaker than the Rains information, but it is efficiently computable for general quantum channels.