# Canonical divergence for measuring classical and quantum complexity

## TL;DR

This paper introduces a new canonical divergence that generalizes classical and quantum complexity measures, unifying them with known divergences like Kullback-Leibler and quantum relative entropy.

## Contribution

It proposes a unified divergence measure applicable to both classical probability measures and quantum states, extending information geometry tools.

## Key findings

- Divergence coincides with Kullback-Leibler on probability simplices.
- Reduces to quantum relative entropy on density operators.
- Provides a unified framework for classical and quantum complexity measures.

## Abstract

A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.