Quantum velocity limits for multiple observables: Conservation laws, correlations, and macroscopic systems

How multiple observables mutually influence their dynamics has been a crucial issue in statistical mechanics. We here introduce a new concept, "quantum velocity limits," to establish a quantitative and rigorous theory for nonequilibrium quantum dynamics for multiple observables. Quantum velocity limits are universal inequalities for a vector that 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 conventional speed limits for a single observable. Moreover, quantum velocity limits are conceptually distinct from the conventional speed limits because we need to introduce the velocity vector and solve an optimization problem for multiple variables to obtain them. 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, while most of the previous speed limits provide only upper bounds; (iii) there exists a notable nonequilibrium tradeoff relation, stating that speeds of uncorrelated observables, e.g., anticommuting observables, cannot be simultaneously large; (iv) velocity limits for local observables in locally interacting many-body systems are described by the fluctuation of a local Hamiltonian, which is convergent even in the thermodynamic limit, with a nontrivial finite-size correction. 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. Our newly found velocity limits ubiquitously apply not only to unitary quantum dynamics but to classical and quantum stochastic dynamics, offering a key step towards universal theory of far-from-equilibrium dynamics for multiple observables.