Rabin oblivious transfer is the cryptographic task where Alice wishes to receive a bit from Bob but it may get lost with probability 1/2. In this work, we provide protocol designs which yield quantum protocols with improved security. Moreover, we provide a constant lower bound on any quantum protocol for Rabin oblivious transfer. To quantify the security of this task with asymmetric cheating definitions, we introduce the notion of cheating advantage which may be of independent interest in the study of other asymmetric cryptographic primitives.

The Pinsker inequality gives a simple method for lower bounding the relative entropy solely in terms of the total variation distance, which at times is easier to work with. We present a simple method for establishing Pinsker inequalities for$f$-divergences using a version of multivariate Taylor's theorem. By combining this with recent work on bounding$f$-divergences in terms of$\chi^{2}$-divergence, we establish that, under a large class of$f$-divergences, the rate at which finite-dimensional timehomogeneous Markov chains with a fixed, full rank stationary distribution converge to this distribution is upper bounded by the input-dependent contraction coefficient of the$\chi^{2}$-divergence. Given previous work, this bound can be tight, is the fastest it could converge using the theory of contraction coefficents, and is efficient to compute. We further extend these ideas to quantum timehomogeneous Markov chains measured under Petz$f$-divergences albeit without the same guarantees of computational efficiency or being the fastest contraction coefficient.

Divergences are a fundamental framework for measuring dissimilarity in information theory and statistics. Here, we use the multivariate Taylor's theorem to obtain new integral representations of twice-differentiable Bregman and$\boldsymbol{f}$-divergences in finite dimensions. This results in input-dependent bounds on many$f$-divergences in terms of the$\chi^{2}$-divergence as well as reverse Pinsker inequalities. As an application, we show how this provides upper bounds on the mixing time of irreducible, scrambling Markov chains under a large class of$f$divergences in terms of the input-dependent$\chi^{2}$contraction coefficient.