  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.
