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

TL;DR

This paper establishes a precise algebraic condition for equality in the data processing inequality of the sandwiched Re9nyi divergence for all b1 b5 b7 1/2, with applications to entropic inequalities and quantum information measures.

Contribution

It provides the first necessary and sufficient condition for equality in the data processing inequality for the sandwiched Re9nyi divergence across all b1 b5 b7 1/2, generalizing previous results.

Findings

Derived a condition for equality in the data processing inequality for b1 b5 b7 1/2.

Formulated a Re9nyi Araki-Lieb inequality and analyzed equality cases.

Proved bounds on Re9nyi entanglement of formation and fidelity saturation.

Abstract

The $\alpha$-sandwiched R\'enyi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for $\alpha\geq 1/2$. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the $\alpha$-sandwiched R\'enyi divergence for all $\alpha\geq 1/2$. 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 (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 attained 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.