Conditions for Large-Sample Majorization of Pairs of Flat States in Terms of $\alpha$-z Relative Entropies

TL;DR

This paper provides an operational interpretation of $oldsymbol{ extalpha}$-z relative entropies, linking them to conditions for large-sample quantum state transformations and deriving optimal conversion rates.

Contribution

It establishes that $oldsymbol{ extalpha}$-z relative entropies characterize when pairs of flat quantum states can be transformed into each other in the large-sample limit, revealing their independence and operational significance.

Findings

Transformations exist if and only if all $ extalpha$-z relative entropies for $ extalpha<1$ are ordered.

Provides an expression for the optimal rate of flat state pair conversion.

Introduces real-algebraic techniques involving preordered semirings for analysis.

Abstract

We offer the first operational interpretation of the $\alpha$-z relative entropies, a measure of distinguishability between two quantum states introduced by Jak\v{s}i\'c et al. and Audenaert and Datta. We show that these relative entropies appear when formulating conditions for large-sample or catalytic relative majorization of pairs of flat states and certain generalizations of them. Indeed, we show that such transformations exist if and only if all the $\alpha$-z relative entropies for $\alpha$<1 of the two pairs are ordered. In this setting, the $\alpha$ and z parameters are truly independent from each other. These results also yield an expression for the optimal rate of converting one flat state pair into another. Our methods use real-algebraic techniques involving preordered semirings and certain monotone homomorphisms and derivations on them.