Data processing for the sandwiched R\xe9nyi divergence: a condition for equality

Abstract

The alpha -sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for alpha ge 1/2. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the alpha -sandwiched Rényi divergence for all alpha ge 1/2. For the range alpha in [1/2,1), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.