Mark M. Wilde

Many fundamental quantities in quantum information processing are instances of quantum divergences - functionals on quantum states that satisfy natural axioms grounded in information-theoretic principles. Recently, a new class of divergences - the f-divergences - has gained prominence in quantum information theory and received operational interpretations, while being long established in the classical setting. Furthermore, Frenkel showed that the Umegaki relative entropy is a special case of a quantum f-divergence for the function f(x) = x log x; building on this, Hirche et al. introduced a parameterized family of f-divergences that, in appropriate regimes, recovers the sandwiched and Petz relative entropies as regularizations. Taken together, these results reveal a tight link between the best-understood quantum divergences - the Umegaki, Petz, and sandwiched relative entropies - on a technical level and the general class of f-divergences, thereby strongly motivating a program that connects f-divergences to concrete quantum information tasks as already started by Cheng et al. In this contribution, we develop a variational formulation that approximates general quantum f-divergences to arbitrary precision. These approximations yield (i) efficient evaluation of the quantum relative entropy of channels and already used as the core numerical method in quantum many body physics and (ii) computation of asymptotic key rates in DIQKD in particular in the scenario of two switches in routed Bell scenarios.

In this paper, we develop the resource theory of quantum secret key. Operating under the assumption that entangled states with zero distillable key do not exist, we define the key cost of a quantum state, and device. We study its properties through the lens of a quantity that we call the key of formation. The main result of our paper is that the regularized key of formation is an upper bound on the key cost of a quantum state. The core protocol underlying this result is privacy dilution, which converts states containing ideal privacy into ones with diluted privacy. Next, we show that the key cost is bounded from below by the regularized relative entropy of entanglement, which implies the irreversibility of the privacy creation-distillation process for a specific class of states. We further focus on mixed-state analogues of pure quantum states in the domain of privacy, and we prove that a number of entanglement measures are equal to each other for these states, similar to the case of pure entangled states. The privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation, and basic consequences for quantum devices are also provided.