Exact characterizations for quantum conditional mutual information and some other entropies

TL;DR

This paper provides exact mathematical characterizations of quantum conditional mutual information and related entropies, offering sharp equalities that clarify their fundamental properties in quantum information theory.

Contribution

It introduces precise, equality-based characterizations of quantum entropies, enhancing understanding of their structure regardless of the entropy's magnitude.

Findings

Provides exact equalities for quantum mutual information and other entropies.

Transforms entropy definitions into summations of explicit, well-behaved terms.

Summations converge rapidly and absolutely, ensuring mathematical robustness.

Abstract

Lieb and Ruskai's strong subadditivity theorem, which shows that the conditional mutual information must be nonnegative, is fundamental in quantum theory. It has numerous applications, such as in quantum error correction. When the mutual information is zero, the Petz recovery map can be used to reconstruct the quantum channel. When the mutual information is small, one seeks to define an optimal recovery channel. To this end, a mathematical characterization of the mutual information is desirable. We address this problem by providing an exact characterization of the mutual information, along with characterizations for other entropies. Our controls are sharp, leaving no room for improvement, in the sense that we provide equalities, regardless of whether the mutual information (or remainder) is small or large. We transform the definitions of these entropies into a summation of explicitly constructed terms, and the definition of each term obviously demonstrates the desired positivity/convexity/concavity. The summation converges rapidly and absolutely in a chosen elementary norm.