René Schwonnek

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.

We present a numerical framework for the reliable estimation of conditional von Neumann entropy in device-independent quantum cryptography and randomness extraction sceratios. By leveraging a hierarchy of semidefinite programs derived from the Navascués--Pironio--Acín (NPA) hierarchy, our method efficiently computes entropy bounds based solely on observed statistics, under the assumption that quantum mechanics holds true. Our approach is built on a recent integral representation presented by [Frenkel, Quantum 7, 1102 (2023)] and extends the landscape of methods for computing provable lower bounds on the conditional von Neumann entropy. Notably, it requires approximately half as many support variables compared to the Brown--Fawzi--Fawzi method, with the additional advantage that these variables can be chosen projectively. The method facilitates the derivation of provable bounds on extractable randomness even in noisy scenarios and aligns with modern entropy accumulation theorems. This makes our framework a versatile tool for practical quantum cryptographic protocols, broadening the possibilities for secure communication in untrusted environments.