Semidefinite Programming Strong Converse Bounds for Classical Capacity

Abstract

We investigate the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and the one-shot -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.

Quantum Channel Simulation and the Channel’s Smooth Max-Information

“…Note that the channel simulation cost operationally provides converse for the channel capacity. However this approach does not provide a tighter bound than the NS-assisted capacity in the one-shot and asymptotic setting (see, e.g., [18,26,28]).…”

Lower and Upper Bounds on Nonunital Qubit Channel Capacities

“…The upper bound from Ref. [19] is plotted in dash-dotted line, the upper bound from Ref. [20] is plotted in dashed line.…”

“…In this paper, we partially fill the gap and find lower and upper bounds on the capacity C(Φ) of qubit channels Φ. We also compare these bounds with the known ones that can be computed with the help of semidefinite programming [19,20]. We demonstrate that for some channels our upper bound outperforms the previously known upper bounds.…”

On Converse Bounds for Classical Communication Over Quantum Channels

“…For classical channels, the Matthews-Wehner bound is exactly equal to the one-shot classical capacity assisted by NS codes [11]. For quantum channels the one-shot ε-error capacity assisted by NS codes is given by [10] C (1)…”

“…Here J M is the Choi-Jamiołkowski matrix of M and T B means the partial transpose on system B. We note that β(·) for a quantum channel N is faithful in the sense that β(J N ) = 1 if and only if C(N ) = 0 [10]. Thus the set V β contains all the constant channels, which makes it reasonable, to some extent, to introduce the set V β here.…”