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

Abstract

We offer the first operational interpretation of the $\alpha$-z relative entropies, a measure of distinguishability between two quantum states introduced by Jakšić 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.

The third version contains some improvements to the exposition of Section 4.3, and a correction to the proof of Theorem 4. Accepted for publication in Communications in Mathematical Physics

References in corpus (6)

• Quantum Relative Lorenz Curves
• Multiple-copy entanglement transformation and entanglement catalysis
• Abstract Vergleichsstellensätze for preordered semifields and semirings I
• Asymptotic relative submajorization of multiple-state boxes
• Matrix majorization in large samples
• Matrix Majorization in Large Samples with Varying Support Restrictions