Optimising the relative entropy under semi definite constraints – A new tool for estimating key rates in QKD

Finding the minimal relative entropy of two quantum states under semi definite constraints is a pivotal problem located at the mathematical core of various applications in quantum information theory. In this work, we provide a method that addresses this optimisation. Our primordial motivation stems form 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 from experimental data and many more. For all those tasks it is highly relevant to provide both, provable upper and lower bounds. An efficient method for this is the central result of this work. We build on a recently introduced integral representation of quantum relative entropy by P.E. Frenkel and provide reliable bounds as a sequence of semi definite programs (SDPs). Our approach ensures provable quadratic order convergence, while also maintaining resource efficiency in terms of SDP matrix dimensions. Additionally, we can provide gap estimates to the optimum at each iteration stage.