Tightening continuity bounds on entropies and bounds on quantum capacities
Seminar author:Michael Jabbour
Event date and time:05/02/2024 04:00:pm
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One of the most basic tasks in information theory is communication. The capacity of a quantum channel is the maximum rate at which information can be transmitted through it reliably (in the asymptotic limit) per use of the channel. Mathematically, the various definitions of capacities are expressed in terms of entropies. They are in general difficult to compute. However, they can be bounded with the help of continuity bounds on entropies. The latter are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove uniform continuity bounds for the Shannon and von Neumann entropies in terms of both the local- and total variation distances for the former, and both the operator norm- and trace distances for the latter. We then apply our results to compute upper bounds on channel capacities for channels that are $\varepsilon$–close in diamond norm and $\nu$–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. 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.