Optimising the relative entropy under semidefinite constraints

Abstract Finding the minimal relative entropy of two quantum states under semidefinite constraints is a pivotal problem located at the mathematical core of various applications in quantum information theory. An efficient method for providing provable upper and lower bounds is the central result of this work. Our primordial motivation stems from the essential task of estimating secret key rates for QKD from the measurement statistics of a real device. Further applications include the computation of channel capacities, the estimation of entanglement measures and many more. We build on a recently introduced integral representation of quantum relative entropy by [Frenkel, Quantum 7, 1102 (2023)] and provide reliable bounds as a sequence of semidefinite programs (SDPs). Our approach ensures provable sublinear convergence in the discretization, while also maintaining resource efficiency in terms of SDP matrix dimensions. Additionally, we can provide gap estimates to the optimum at each iteration stage.