Tightening continuity bounds for entropies and bounds on quantum capacities

Abstract

Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between a pair of probability distributions or quantum states, typically, the total variation distance or trace distance. However, if an additional distance measure between the probability distributions or states is known, then the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality proven in [I. Sason, IEEE Trans. Inf. Th., 59, 7118 (2013)]. We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. The bound is tight when the quotient of the trace distance by the operator norm distance is an integer. We then apply our results to compute upper bounds on the quantum- and private classical capacities of channels. We begin by refining the concept of approximate degradable channels, namely, $\epsilon$-degradable channels, which are, by definition, $\epsilon$-close in diamond norm to their complementary channel when composed with a degrading channel. To this end, we introduce the notion of $(\epsilon ,\nu )$-degradable channels; these are $\epsilon$-degradable channels that are, in addition, $\nu$-close in completely bounded spectral norm to their complementary channel, when composed with the same degrading channel. This allows us to derive improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. We illustrate our results by obtaining a new upper bound on the quantum capacity of the qubit depolarizing channel.

arXiv comment34 pages, 5 figures. v2: Removed Lemma 3 (which was already known), added some references, improved presentation and removed some calculation details. v3: Some slight corrections to the conditions in which the results hold

References in corpus (18)

• Mixed State Entanglement and Quantum Error Correction
• The capacity of the noisy quantum channel
• Quantum Communication With Zero-Capacity Channels
• Quantum Channel Capacity of Very Noisy Channels
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• Continuity of quantum conditional information
• Contractivity of positive and trace preserving maps under $L_p$ norms
• Approaches for approximate additivity of the Holevo information of quantum channels
• Continuity of quantum channel capacities
• Structured Codes Improve the Bennett-Brassard-84 Quantum Key Rate
• Approximate Degradable Quantum Channels
• Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of quantum channels
• Useful states and entanglement distillation
• Quantum and private capacities of low-noise channels
• Quantum flags, and new bounds on the quantum capacity of the depolarizing channel
• Additive Extensions of a Quantum Channel
• Entropy Bounds for Discrete Random Variables via Maximal Coupling

Cited by in corpus (2)

• The dynamical $\alpha$-Rényi entropies of local Hamiltonians grow at most linearly in time
• Continuity of entropies via integral representations