Tightening continuity bounds for entropies and bounds on quantum capacities

TL;DR

This work tightens entropy continuity bounds by exploiting dual-distance information: a refined classical bound for Shannon entropy leveraging both $\mathrm{TV}$ and $\mathrm{LO}$, and a quantum analogue for von Neumann entropy leveraging both trace and operator distances. The refined bounds enable stronger, efficiently computable upper bounds on quantum and private classical channel capacities through the introduction of $(\varepsilon,\nu)$-degradable channels, with SDPs facilitating practical computation. A key application demonstrates a tighter quantum-capacity bound for the qubit depolarizing channel, illustrating the potential for sharper capacity estimates in noisy quantum communications. The results connect tight entropy inequalities to operational capacities and pave the way for further refinements via unstabilized norms and related optimization frameworks.

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, $\varepsilon$-degradable channels, which are, by definition, $\varepsilon$-close in diamond norm to their complementary channel when composed with a degrading channel. To this end, we introduce the notion of $(\varepsilon,ν)$-degradable channels; these are $\varepsilon$-degradable channels that are, in addition, $ν$-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.

Figures (3)

• Figure 1: (a) depicts upper and lower bounds for the quantum capacity $Q(\mathcal{E}_p)$ of a qubit depolarizing channel $\mathcal{E}_p$ in the low-noise regime $p \in [0, 0.025]$ for which \ref{['eq:boundQSutter']} (dashed blue curve) was the previously known tightest upper bound. The channel coherent information given in \ref{['eq:cohInfoDepol']} denotes a lower bound on $Q(\mathcal{E}_p)$ (dotted black curve). The solid red line depicts our new upper bound given by \ref{['eq:boundQnu']}. (b): a magnification of the box represented on Fig. \ref{['fig:depol1']}.
• Figure 2: (a): Plots of the quantities defined in \ref{['eq:epdia']} and \ref{['eq:ep1nu']}. (b): Plot of $\varepsilon^{(p)}_1$ and comparison with the quantity $2 \nu^{(p)} d_E/(\nu^{(p)} d_E + 3)$. We see that the condition $\varepsilon^{(p)}_1 \leq 2 \nu^{(p)} d_E/(\nu^{(p)} d_E + 3)$ is indeed satisfied.
• Figure 3: (a) and (b): Graphs of \ref{['eq:RHSBound']} for $d=10$. The two figures show the same plot from two different angles. It can be seen that if one fixes a value of $\nu$, the corresponding value of $f_{10}(\varepsilon,\nu)$ will not necessarily be increasing with $\varepsilon$. (c): Plots of \ref{['eq:RHSBound']} for $d=10$, for different fixed values of $\nu$. As it can be seen from the plots, the function $f_{d}(\varepsilon,\nu)$ is not monotonically increasing in $\varepsilon$ for a fixed value of $\nu$.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Lemma 1: Sason Sason2013
• Theorem 3
• Remark 1
• Theorem 4
• Lemma 2
• proof : Proof of Lemma \ref{['lem:constDimQ']}
• Definition 1: Definition 4 of Sutter2017, $\varepsilon$--degradable
• Definition 2: $(\varepsilon,\nu)$--degradable