“Pretty Strong” Converse for the Quantum Capacity of Degradable Channels

Abstract

We exhibit a possible road towards a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a "pretty strong" converse: When the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 1 √ 2 , asymptotically. A similar result can be shown for the private (classical) capacity.Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse: The abovementioned asymptotic jump of the fidelity at the quantum capacity is then from 1 down to 0.Index Terms-quantum information, private classical information, channel coding, strong converse, smooth entropies, errorrate trade-off I. INTRODUCTIONCommunication via noisy channels is one of the information processing tasks by which, following the fundamental work of Shannon [42], we have learned to quantify information and noise. One of the most important models considered from these early days of information theory is that of a discrete memoryless channel, for which Shannon gave his famous single-letter formula for the capacity (i.e., the maximum communication rate achievable by asymptotically error-free block coding).The analogous model in quantum Shannon theory is the memoryless quantum channel N ⊗n (for asymptotically large integer n), given by a completely positive and trace preserving (cptp) map N : L(A ′ ) → L(B), with Hilbert spaces A ′ and B that we assume to be finite dimensional throughout this paper.The quantum capacity Q(N ) of N is informally defined as the maximum rate at which quantum information can be

Entanglement Cost of Quantum Channels

“…For many channels, there are also upper bounds known [5], [22], [23], but a general formula for the strong converse classical capacity is still lacking. Understanding the strong converse capacity for sending quantum information over a quantum channel turns out to be an even more elusive problem, and only partial results are known [5], [24], [25]. Here, we make progress by showing that any coding scheme sending quantum information (using free forward, backward, or two-way classical communication) at an asymptotic rate higher than the entanglement cost must have an exponentially small fidelity.…”

Preface to the Second Edition

“…Berta et al (2013) proved that a quantity called the entanglement cost of a quantum channel is a strong converse rate for quantum communication. Morgan and Winter (2014) established what they called a "pretty strong converse" for the quantum capacity of degradable channels, meaning that there is a sharp transition in the fidelity from one to 1/2, when the rate of communication goes from below to above the quantum capacity (this is in the limit of many channel uses). demonstrated that randomly selected codes with a communication rate exceeding the quantum capacity of the quantum erasure channel lead to a fidelity that decreases exponentially fast as the number of channel uses increases (a strong converse would however demonstrate that this behavior occurs for all codes).…”

“…Beigi et al (2015) and established second-order achievability characterizations for quantum capacity. Beigi et al (2015) did so by making use of a "Petz recovery map" decoder and with a version of the decoupling theorem from Morgan and Winter (2014). also gave a second-order converse for quantum communication by making use of the Rains bound, and they obtained an exact second-order characterization of quantum communication for dephasing channels.…”

The Structure and Quantum Capacity of a Partially Degradable Quantum Channel

“…It immediately shows that the output system σ E P of channel N AE is a boundentangled state; in other words, the complementary channel N AE is an entanglement-binding channel. We note that for the complementary channel N AE , for the product of the input and output space dimensions d in (N AE ) and d out (N AE ), the condition d in (N AE ) · d out (N AE ) > 6 has to be satisfied [1], [14], [15], otherwise the degrading map D E→E on the output E of the complementary channel N AE cannot produce bound-entangled system σ E P . Since this condition trivially could be satisfied, it does not have any further impact on the proof.…”

“…The PD channels also have a clear and well-defined structure and offer several benefits for the evaluation of quantum capacity. The channel structure that makes the quantum capacity of a PD channel additive in arbitrary dimensions has to be equipped with an entanglement-binding [14], [15] complementary channel. PD channels could be degradable (degradable PD channels) and anti-degradable (anti-degradable PD channels) ones, and also could be conjugate degradable (conjugate-PD channels) channels.…”