Quantum Velocity Limits for Multiple Observables: Conservation Laws, Correlations, and Macroscopic Systems

TL;DR

This paper introduces quantum velocity limits that provide universal bounds on the dynamics of multiple observables in quantum systems, revealing how conservation laws, correlations, and locality influence their evolution.

Contribution

The paper develops a new theoretical framework for quantum velocity limits applicable to multiple observables, incorporating conservation laws and correlations, with implications for non-equilibrium dynamics.

Findings

Velocity limits are tighter with knowledge of other observables or conserved quantities.

Conservation laws and correlations can improve bounds on observable speeds.

Velocity limits remain finite in the thermodynamic limit for local systems.

Abstract

How multiple observables mutually influence their dynamics has been a crucial issue in statistical mechanics. We introduce a new concept, "quantum velocity limits," to establish a quantitative and rigorous theory for non-equilibrium quantum dynamics for multiple observables. Quantum velocity limits are universal inequalities for a vector the describes velocities of multiple observables. They elucidate that the speed of an observable of our interest can be tighter bounded when we have knowledge of other observables, such as experimentally accessible ones or conserved quantities, compared with the conventional speed limits for a single observable. We first derive an information-theoretical velocity limit in terms of the generalized correlation matrix of the observables and the quantum Fisher information. The velocity limit has various novel consequences: (i) conservation law in the system, a fundamental ingredient of quantum dynamics, can improve the velocity limits through the correlation between the observables and conserved quantities; (ii) speed of an observable can be bounded by a nontrivial lower bound from the information on another observable; (iii) there exists a notable non-equilibrium tradeoff relation, stating that speeds of uncorrelated observables, e.g., anti-commuting observables, cannot be simultaneously large; (iv) velocity limits for any observables on a local subsystem in locally interacting many-body systems remain convergent even in the thermodynamic limit. Moreover, we discover another distinct velocity limit for multiple observables on the basis of the local conservation law of probability current, which becomes advantageous for macroscopic transitions of multiple quantities.