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arXiv:2307.09118 · Physical Review Letters

Quantum Speed Limit for States and Observables of Perturbed Open Systems

Benjamin Yadin, Satoya Imai, Otfried Gühne

Published Jun 7, 2024• 13 citations
arXivHTMLPDF

Abstract

Quantum speed limits provide upper bounds on the rate with which a quantum system can move away from its initial state. Here, we provide a different kind of speed limit, describing the divergence of a perturbed open system from its unperturbed trajectory. In the case of weak coupling, we show that the divergence speed is bounded by the quantum Fisher information under a perturbing Hamiltonian, up to an error which can be estimated from system and bath timescales. We give three applications of our speed limit. First, it enables experimental estimation of quantum Fisher information in the presence of decoherence that is not fully characterized. Second, it implies that large quantum work fluctuations are necessary for a thermal system to be driven quickly out of equilibrium under a quench. Moreover, it can be used to bound the response to perturbations of expectation values of observables in open systems.

DOI
10.1103/physrevlett.132.230404

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Quantum speed limits provide upper bounds on the rate with which a quantum system can move away from its initial state. Here, we provide a different kind of speed limit, describing the divergence of a perturbed open system from its unperturbed trajectory. In the case of weak coupling, we show that the divergence speed is bounded by the quantum Fisher information under a perturbing Hamiltonian, up to an error which can be estimated from system and bath timescales. We give three applications of our speed limit. First, it enables experimental estimation of quantum Fisher information in the presence of decoherence that is not fully characterized. Second, it implies that large quantum work fluctuations are necessary for a thermal system to be driven quickly out of equilibrium under a quench. Moreover, it can be used to bound the response to perturbations of expectation values of observables in open systems.
Abstract We present a new quantum speed limit (QSL) for open quantum systems governed by Markovian dynamics. Our approach is based on analyzing the time derivative of the Bures angle between the initial pure state and its time-evolved state, leading to an analytically computable upper bound on the evolution speed. This bound naturally decomposes into three distinct physical contributions: a unitary term, a dissipative deformation term, and a fluctuation term. Building on this structure, we establish a general inequality that connects the QSL to the quantum Fisher information (QFI) in the short-time regime. This reveals a fundamental trade-off between evolution speed and estimation precision, and clarifies how decoherence can both accelerate and constrain information acquisition in open quantum systems.
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).
Quantum asymmetry and coherence are genuinely quantum resources that are essential to realize quantum advantage in information technologies. However, all quantum processes are fundamentally constrained by quantum speed limits, which raises the question on the corresponding bounds on the rate of consumption of asymmetry and coherence. In the present work, we derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable. This quantum speed limit can be directly observed in experiment through weak value measurement and provides a lower bound to the quantum Fisher information about the parameter conjugate to the observable. It can be further related to quantum coherence relative to the eigenbasis of the observable. We obtain a complementary relation for the speed of three mutually unbiased observables for a single qubit. As an application, we derive a notion of a quantum thermodynamic speed limit.
In the context of quantum speed limits, it has been shown that the minimum time required to cause a desired state conversion via the open quantum dynamics can be estimated using the entropy production. However, the established entropy-based bounds tend to be loose, making it difficult to accurately estimate the minimum time for evolution. In this research, we have combined the knowledge of the entropy-based speed limits with that of the resource theory of asymmetry (RTA) and provided much stricter inequalities. Our results show that the limitation on the change rate of states and expectation values can be divided into two parts: quantum coherence for energy (i.e., asymmetry) contributed by the system and the heat bath and the classical entropy-increasing effect from the bath. As a result, our inequalities demonstrate that the difference in the speed of evolution between classical and quantum open systems, i.e., the quantum enhancement in speed, is determined by the quantum Fisher information, which measures quantum fluctuations of energy and serves as a standard resource measure in the resource theory of asymmetry. We further show that a similar relation holds for the rate of change of expectation values of physical quantities.
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.