Quantifying the difference between many-body quantum states

TL;DR

This paper introduces weighted distances as new measures to better compare complex many-body quantum states, accounting for measurement difficulty and providing bounds on quantum process costs.

Contribution

It proposes weighted distances and the weighted Bures length, offering improved tools for analyzing many-body quantum states and quantum processes.

Findings

Weighted distances quantify measurement difficulty between quantum states.

Weighted Bures length bounds the experimental cost of quantum transformations.

New measures improve understanding of resource conversion in quantum systems.

Abstract

The quantum state overlap is the textbook measure of the difference between two quantum states. Yet, it is inadequate to compare the complex configurations of many-body systems. The problem is inherited by the widely employed quantum state fidelity and related distances. We introduce the weighted distances, a new class of information-theoretic measures that overcome these limitations. They quantify how hard it is to discriminate between two quantum states of many particles, factoring in the structure of the required measurement apparatus. Therefore, they can be used to evaluate both the theoretical and the experimental performances of complex quantum devices. We also show that the newly defined "weighted Bures length" between the input and output states of a quantum process is a lower bound to the experimental cost of the transformation. The result uncovers an exact quantum limit to our ability to convert physical resources into computational ones.