Quantifying quantum coherence and the deviation from the total probability formula

TL;DR

This work reframes quantum coherence as the extent to which quantum statistics deviate from the total probability formula, rather than as a resource-theoretic quantity. It introduces a distance between orthonormal bases and two axioms for basis-dependent coherence measures, showing that the $\ell^1$ and $\ell^2$ norms (and a new measure $\delta$) satisfy these axioms while a relative-entropy-based measure does not. The paper also proves that, in large-dimensional systems, the squared $\ell^2$-coherence in a Haar-random basis concentrates to the state purity $\mathrm{tr}(\rho^2)$, providing a basis-agnostic sense of typical coherence tied to purity. Overall, the ontology-driven framework links physical interpretation with concrete, basis-dependent coherence quantifications and opens avenues to compare with resource-theoretic measures across bases.

Abstract

We propose a novel approach to quantify quantum coherence which, contrary to the previous ones, does not rely on resource theory but rather on ontological considerations. In this framework, coherence is understood as the ability for a quantum system's statistics to deviate from the total probability formula. After motivating the importance of the total probability formula in quantum foundations, we propose a new set of axioms that a measure of coherence should satisfy, and show that it defines a class of measures different from the main previous proposal. Finally, we prove a general result about the dependence of the l2-coherence norm on the basis of interest, and show that it is well approximated by the square root of the purity in most bases.

Theorems & Definitions (10)

• Proposition 3.1
• Lemma 3.2: Almost equality in quadratic Jensen's inequality
• proof : Proof of Lemma
• proof : Proof of Proposition \ref{['inequalities']}
• Definition 4.1: Measure of coherence
• Theorem 4.2
• proof
• Proposition 5.1
• proof
• Remark 5.2