Research

Tightening continuity bounds on entropies and bounds on quantum capacities

Michael G. Jabbour and Nilanjana Datta, IEEE Journal on Selected Areas in Information Theory 5, 645-658 (2024), arXiv:2310.17329 [quant-ph].

A continuity bound for the entropy of a state quantifies the amount by which the entropy changes when the underlying state changes by a small amount. The importance of continuity bounds lies in the fact that we usually lack precise knowledge of a state, in a particular information-processing task for instance. Uniform continuity bounds on entropies 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 a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances. We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.

Continuity bounds for quantum entropies arising from a fundamental entropic inequality

Koenraad Audenart, Bjarne Bergh, Nilanjana Datta, Michael G. Jabbour, Ángela Capel and Paul Gondolf, arXiv:2408.15306 [quant-ph] (2024).

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, ρ1 and ρ2. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator (ρ1 - ρ2). This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables.

Boson-fermion complementarity in a linear interferometer

Michael G. Jabbour and Nicolas J. Cerf, arXiv:2312.17709 [quant-ph] (2023).

Quantum interferences are a cornerstone of quantum physics but are notoriously difficult to characterize. We establish a fundamental relation combining bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer. It expresses a boson-fermion complementarity inherent to the particle statistics and oblivious to the details of the interaction. We believe it is the first time bosonic and fermionic transition probabilities appear in a unifying equation.