Semidefinite programming converse bounds for classical communication over quantum channels

Abstract

We study the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes. We first show that both the optimal success probability of a given transmission rate and one-shot-error capacity can be formalized as semidefinite programs (SDPs) when assisted by NS or NS∩PPT codes. Based on this, we derive SDP finite blocklength converse bounds for general quantum channels, which also reduce to the converse bound of Polyanskiy, Poor, and Verdú for classical channels. Furthermore, we derive an SDP strong converse bound for the classical capacity of a general quantum channel: for any code with a rate exceeding this bound, the optimal success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bound, we derive improved upper bounds to the classical capacity of the amplitude damping channels and also establish the strong converse property for a new class of quantum channels.

Amortization does not enhance the max-Rains information of a quantum channel

“…We note that these choices are somewhat similar to those made in the proof of [21], Proposition6, and they can be understood roughly via (11) as a postselected teleportation of the optimal operators of…”

“…For the benefit of the reader, we give technical details of this application in section 4 . The quantity  ( ) R max has already been shown in [21]to be efficiently computable via a semi-definite program, and in section 4, we explain how  ( ) R max is both 'singleletter' and 'strong converse'.…”

“…We note here that the equality in (8) solves an open question posed in the conclusion of [12], and we set our result in the context of the prior result of [12] and other literature in section 6. The max-Rains information of a quantum channel is a special case of a quantity known as the sandwiched Rényi-Rains information [19]and was recently shown to be equal to an information quantity discussed in [20,21] and based on semi-definite programming (SDP). To prove our main technical result (the equality in (8)), we critically make use of the tools and framework developed in the recent works [20][21][22].…”

“…The max-Rains information of a quantum channel is a special case of a quantity known as the sandwiched Rényi-Rains information [19]and was recently shown to be equal to an information quantity discussed in [20,21] and based on semi-definite programming (SDP). To prove our main technical result (the equality in (8)), we critically make use of the tools and framework developed in the recent works [20][21][22]. In particular, we employ SDP duality [23] and the well known Choi isomorphism to establish our main result, with the proof consisting of just a few lines once the framework from [20][21][22]is set in place.…”

“…To prove our main technical result (the equality in (8)), we critically make use of the tools and framework developed in the recent works [20][21][22]. In particular, we employ SDP duality [23] and the well known Choi isomorphism to establish our main result, with the proof consisting of just a few lines once the framework from [20][21][22]is set in place.…”