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arXiv:2305.03190 · Physical Review Research

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

Ryusuke Hamazaki

Published Jan 8, 2024• 9 citations
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Abstract

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. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.013018

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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. Published by the American Physical Society 2024
Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control.While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive.Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones.We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the "gradient" of the observable, in contrast with conventional speed limits depending on the entire range of the observable.This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics.Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new trade-off relation between time and the quantum phase difference.Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound.We also find that, beyond expectation values of macroscopic observables, the speed of macroscopic quantum coherence can be bounded from above by our general approach.The newly obtained bounds are verified in transport phenomena in particle systems and nonequilibrium dynamics in many-body spin systems.We also demonstrate that our strategy can be applied for finding new speed limits for macroscopic transitions in stochastic systems, including quantum ones, where the bounds are expressed by the entropy production rate.Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.
In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.
Quantum speed limits establish fundamental bounds on the minimum time required for a quantum system to evolve from an initial state to a target state. Traditional formulations treat the evolution speed as a single global quantity governed by the system energy uncertainty or average energy. In this paper, we propose the theory of Quantum Speed-Limit Splitting, which posits that a quantum system undergoing complex dynamics does not necessarily evolve along a single speed limit, but may instead exhibit multiple competing, distinct state-evolution speed limits simultaneously. By defining a multi-valued velocity spectrum belonging to the set {v_1, v_2, ..., v_n}, we demonstrate that different geometric pathways in the underlying Hilbert space correspond to distinct temporal boundaries. We formulate the mathematical framework, replacing the traditional scalar evolution time with a set of multi-branch bounds {tau_1, tau_2, ...}. This theoretical extension provides a novel lens for viewing quantum dynamics, with profound implications for quantum control, thermodynamic cost, and the fundamental limits of information processing.
Quantum speed limits (QSLs) identify fundamental time scales of physical processes by providing lower bounds on the rate of change of a quantum state or the expectation value of an observable. We introduce a generalization of QSL for unitary operator flows, which are ubiquitous in physics and relevant for applications in both the quantum and classical domains. We derive two types of QSLs and assess the existence of a crossover between them, that we illustrate with a qubit and a random matrix Hamiltonian, as canonical examples. We further apply our results to the time evolution of autocorrelation functions, obtaining computable constraints on the linear dynamical response of quantum systems out of equilibrium and the quantum Fisher information governing the precision in quantum parameter estimation.
The presence of noise or the interaction with an environment can radically change the dynamics of observables of an otherwise isolated quantum system.We derive a bound on the speed with which observables of open quantum systems evolve.This speed limit is divided into Mandelstam and Tamm's original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, and both are generalized to open quantum systems.By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds on the speed of evolution.We prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits and that a preferred basis of speed operators serves to completely characterize the observables that saturate the speed limits.We use this construction to bound the effect of incoherent dynamics on the evolution of an observable and to find the Hamiltonian that gives the maximum coherent speedup to the evolution of an observable.
Leveraging quantum information geometry, we derive generalized quantum speed limits on the rate of change of the expectation values of observables. These bounds subsume and, for Hilbert space dimension ≥ 3 , tighten existing bounds—in some cases by an arbitrarily large multiplicative constant. Our theoretical results are supported by illustrative examples and an experimental demonstration using a superconducting qutrit. We also derive two upper bounds on the generalized quantum Fisher information in terms of the condition number of the density matrix. One of these bounds applies only to coherent dynamics and depends also on the variance of the Hamiltonian. The other bound depends also on the so-called Wigner-Yanase skew information. These bounds generalize well-known bounds on the symmetric logarithmic derivative quantum Fisher information and are tighter than the existing bounds for sufficiently mixed states (e.g., for sufficiently high temperature thermal states).