Weighted Bures Length Uncovers Quantum State Sensitivity

TL;DR

This paper demonstrates that the Weighted Bures Length metric reveals quantum state sensitivity over time, contrasting with traditional overlap measures, through numerical analysis of quantum cellular automaton dynamics.

Contribution

It introduces the Weighted Bures Length as a new metric to detect quantum state sensitivity, showing its effectiveness in numerical simulations of quantum automaton evolution.

Findings

WBL uncovers linear growth of perturbations in regular interactions.

WBL reveals exponential growth of perturbations in random interactions.

Quantum state sensitivity can be detected with WBL, unlike traditional overlap measures.

Abstract

The unitarity of quantum evolutions implies that the overlap between two initial states does not change in time. This property is commonly believed to explain the lack of state sensitivity in quantum theory, a feature that is prevailing in classical chaotic systems. However, a distance between two points in classical phase space is a completely different mathematical concept than an overlap distance between two points in Hilbert space. There is a possibility that state sensitivity in quantum theory can be uncovered with a help of some other metric. Here we show that the recently introduced Weighted Bures Length (WBL) achieves this task. In particular, we numerically study a cellular automaton-like unitary evolution of N qubits, known as Rule 54, and apply WBL to show that a single-qubit perturbation of a random initial state: (a) grows linearly in time under the nearest neighbour interaction on a cycle, (b) appears to grow exponentially in time under interaction given by a random bipartite graph.