Information measures reveal when classical and quantum states convert
Classical and Quantum Information Measures Characterizing Convertibility: An Axiomatic Approach
Information Theory
Summary
This thesis studies how to measure differences between groups of classical or quantum states using mathematical formulas called divergences. The researchers found exact formulas that tell when one group of states can be converted into another, covering both classical data and certain quantum cases. In particular, for quantum states made of pure components, they showed how a family of known quantum measures relates to a type of ordering called majorization. This work gives new ways to understand how information changes when transforming data or quantum systems.
What this means in practice
- •For quantum computing engineers: Determine when a specific quantum state can be transformed into another using known entropy measures, aiding quantum algorithm design.
- •For data compression developers: Use new classical divergence measures to analyze and optimize data compression schemes where data distributions differ widely.
A theory result. No direct application yet.
Authors
Frits Verhagen
Abstract
In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all $α$-$z$ relative entropies for $α<1$. This demonstrates the first operational interpretation of these relative entropies.