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

Research paper by Felix Leditzky, Cambyse Rouzé, Nilanjana Datta

Indexed on: 20 Apr '16Published on: 20 Apr '16Published in: Quantum Physics


The $\alpha$-sandwiched R\'enyi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for $\alpha\geq 1/2$. Here we derive a necessary and sufficient algebraic condition for equality in the data processing inequality. 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 (2013), p. 122201] 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 R\'enyi version of the Araki-Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb [Lett. Math. Phys. 101.1 (2012), pp. 1-11] about equality in the original Araki-Lieb inequality. Furthermore, we prove a general lower bound on a R\'enyi version of the entanglement of formation, and observe that it is saturated by states saturating the R\'enyi 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.