Quantum equilibration and measurements -- bounds on speeds, Lyapunov exponents, and transport coefficients obtained from the uncertainty relations and their comparison with experimental data

TL;DR

This paper derives fundamental bounds on quantum many-body system dynamics, such as speeds and transport coefficients, using uncertainty relations, and compares these bounds with experimental data to understand equilibration and measurement times.

Contribution

It introduces a direct method to obtain exact bounds on quantum dynamics and transport properties, extending previous conjectures and providing a framework for experimental comparison.

Findings

Bounds on speeds, Lyapunov exponents, and transport coefficients are derived.

Some bounds are comparable to experimental measurements, indicating tightness.

The results suggest minimal times for thermalization and measurement stabilization.

Abstract

We discuss our recent study of local quantum mechanical uncertainty relations in quantum many body systems. These lead to fundamental bounds for quantities such as the speed, acceleration, relaxation times, spatial gradients and the Lyapunov exponents. We additionally obtain bounds on various transport coefficients like the viscosity, the diffusion constant, and the thermal conductivity. Some of these bounds are related to earlier conjectures, such as the bound on chaos by Maldacena, Shenker and Stanford while others are new. Our approach is a direct way of obtaining exact bounds in fairly general settings. We employ uncertainty relations for local quantities from which we strip off irrelevant terms as much as possible, thereby removing non-local terms. To gauge the utility of our bounds, we briefly compare their numerical values with typical values available from experimental data. In various cases, approximate simplified variants of the bounds that we obtain can become fairly tight, i.e., comparable to experimental values. These considerations lead to a minimal time for thermal equilibrium to be achieved. Building on a conjectured relation between quantum measurements and equilibration, our bounds, far more speculatively, suggest a minimal time scale for measurements to stabilize to equilibrium values.