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arXiv:2108.05936 · Physical Review A

Relative purity, speed of fluctuations, and bounds on equilibration times

Diego Paiva Pires, Thiago Rodrigues de Oliveira

Published Nov 24, 2021• 9 citations
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Abstract

We discuss the local equilibration of closed systems using the relative purity, a paradigmatic information-theoretic distinguishability measure that finds applications ranging from quantum metrology to quantum speed limits. First we obtain an upper bound on the average size of the fluctuations on the relative purity: it depends on the effective dimension resembling the bound obtained with the trace distance. Second, we investigate the dynamics of relative purity and its rate of change as a probe of the speed of fluctuations around the equilibrium. In turn, such speed captures the notion of how fast some nonequilibrium state approaches the steady state under the local nonunitary dynamics, somehow giving the information of the quantum speed limit towards the equilibration. We show that the size of fluctuations depends on the quantum coherences of the initial state with respect to the eigenbasis of the Hamiltonian, also addressing the role played by the correlations between system and reservoir into the averaged speed. Finally, we have derived a family of lower bounds on the time of evolution between these states, thus obtaining an estimate for the equilibration time at the local level. These results could be of interest to the subjects of equilibration, quantum speed limits, and also quantum metrology.

DOI
10.1103/physreva.104.052223

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We discuss the local equilibration of closed systems using the relative purity, a paradigmatic information-theoretic distinguishability measure that finds applications ranging from quantum metrology to quantum speed limits. First we obtain an upper bound on the average size of the fluctuations on the relative purity: it depends on the effective dimension resembling the bound obtained with the trace distance. Second, we investigate the dynamics of relative purity and its rate of change as a probe of the speed of fluctuations around the equilibrium. In turn, such speed captures the notion of how fast some nonequilibrium state approaches the steady state under the local nonunitary dynamics, somehow giving the information of the quantum speed limit towards the equilibration. We show that the size of fluctuations depends on the quantum coherences of the initial state with respect to the eigenbasis of the Hamiltonian, also addressing the role played by the correlations between system and reservoir into the averaged speed. Finally, we have derived a family of lower bounds on the time of evolution between these states, thus obtaining an estimate for the equilibration time at the local level. These results could be of interest to the subjects of equilibration, quantum speed limits, and also quantum metrology.
We review various bounds concerning out-of-equilibrium dynamics in few-level and many-body quantum systems. We primarily focus on closed quantum systems but will also mention some related results for open quantum systems and classical stochastic systems. We start from the speed limits, the universal bounds on the speeds of (either quantum or classical) dynamical evolutions. We then turn to review the bounds that address how good and how long would a quantum system equilibrate or thermalize. Afterward, we focus on the stringent constraint set by locality in many-body systems, rigorously formalized as the Lieb–Robinson bound. We also review the bounds related to the dynamics of entanglement, a genuine quantum property. Apart from some other miscellaneous topics, several notable error bounds for approximated quantum dynamics are discussed. While far from comprehensive, this topical review covers a considerable amount of recent progress and thus could hopefully serve as a convenient starting point and up-to-date guidance for interested readers.
Finding the equilibration time scale is an important open question in studying the equilibration of quantum systems. There are many kinds of systems that are unable to achieve equilibration, such as Anderson insulator, many-body localization systems and some integrable systems. For those systems that can reach equilibration, it was proved that there exists a general equilibration time scale. But these upper bounds are unrealistically long, and the lower bounds are also unrealistically short. How to get an accurate and general equilibration time scale is still unclear. In this paper, we study local equilibration time scales in quantum lattice systems. With the relation between equilibration and entanglement entropy, we define a new criterion for equilibration. This criterion is based on Rényi entropy, which is simpler for calculation. Moreover, the production of Rényi entropy is highly dependent on the spreading of information, and hence the tools developed in quantum information theory can help a lot. For the upper bound of equilibration time, we use the normal criterion of the time average ofthe fluctuation of the observation. This equilibration criterion is assessed with concrete observable operator, hence it is more accurate than the criterion of Rényi entropy. But this criterion is more complicated for calculation. With an appropriate assumption about the initial state, we present a new upper bound of equilibration time. Since the results are not constrained by the Hamiltonian of the system, this bound can be applied to various situations. If we are concerned with the local equilibration, we can limit the observation to a small region. If the whole system is big enough, the local observation would always find that the rest part is staying at the canonical ensemble, so that the local equilibration time will not increase with the size of whole system. When the local region is small enough, the upper bound of equilibration time can be much shorter. The local Rényi entropy has close relation to the propagation of information. The limitation of the speed of information propagation in a quantum lattice system can be given by the Lieb-Robinson bound, with which we evaluate the production rate of local 2-Rényi entropy. With these, we give a new lower bound of equilibration time for systems whose interactions are respectively short-range, exponentially-decaying and long-range. In earlier works, the lower bound of equilibration time is equal to the Lieb-Robinson time of the local system. In our rigorous proof, however, the lower bound is related to the strength of hopping, which also decides the Lieb-Robinson velocity. This means the equilibration time is related to the Lieb-Robinson time indeed. But the strictly equal relation may not hold. These new bounds are important to understanding the process of quantum equilibration.
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.
Bounds to the speed of evolution of a quantum system are of fundamental interest in quantum metrology, quantum chemical dynamics, and quantum computation. We derive a time-energy uncertainty relation for open quantum systems undergoing a general, completely positive, and trace preserving evolution which provides a bound to the quantum speed limit. When the evolution is of the Lindblad form, the bound is analogous to the Mandelstam-Tamm relation which applies in the unitary case, with the role of the Hamiltonian being played by the adjoint of the generator of the dynamical semigroup. The utility of the new bound is exemplified in different scenarios, ranging from the estimation of the passage time to the determination of precision limits for quantum metrology in the presence of dephasing noise.
We develop a unified dynamical-spectral framework for equilibration in isolated quantum systems based on a subspace coarse-graining approach. Central to our formulation is the Leakage Fidelity Function (LFF), defined as the probability that a unitarily evolving state escapes the support of its initial subspace. This quantity provides a direct, operational measure of information flow and memory loss without invoking ensemble assumptions or perturbative arguments. We derive universal bounds on temporal fluctuations of the LFF, in terms of the spectral gap structure and the square of the effective dimension, evincing that large spectral delocalization suppresses fluctuations and guarantees equilibration on average. By introducing spectral power distributions and associated entropic measures, we establish a quantitative link between phase mixing, gap participation, and dynamical stability. We further investigate the equilibration timescale by connecting the LFF to quantum speed limits, thereby revealing the average time required for equilibration. Our results provide a state-dependent, geometrically transparent perspective on how spectral complexity and subspace information leakage jointly govern irreversibility in closed quantum many-body systems.