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arXiv:2204.02884 · New Journal of Physics

Speed limits of the trace distance for open quantum system

Shin-ichi Nakajima, Yasuhiro Utsumi

Published Sep 1, 2022• 15 citations
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Abstract

Abstract We investigate the speed limit of the state transformation in open quantum systems described by the Lindblad type quantum master equation. We obtain universal bounds of the total entropy production described by the trace distance between the initial and final states in the interaction picture. Our bounds can be tighter than the bound of Vu and Hasegawa (2021 Phys. Rev. Lett. 126 010601) which measures the distance by the eigenvalues of the initial and final states: this distance is less than or equal to the trace distance. For this reason, our results can significantly improve Vu–Hasegawa’s bound. The trace distance in the Schrödinger picture is bounded by a sum of the trace distance in the interaction picture and the trace distance for unitary dynamics described by only the Hamiltonian in the quantum master equation.

DOI
10.1088/1367-2630/ac8eca

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Abstract We investigate the speed limit of the state transformation in open quantum systems described by the Lindblad type quantum master equation. We obtain universal bounds of the total entropy production described by the trace distance between the initial and final states in the interaction picture. Our bounds can be tighter than the bound of Vu and Hasegawa (2021 Phys. Rev. Lett. 126 010601) which measures the distance by the eigenvalues of the initial and final states: this distance is less than or equal to the trace distance. For this reason, our results can significantly improve Vu–Hasegawa’s bound. The trace distance in the Schrödinger picture is bounded by a sum of the trace distance in the interaction picture and the trace distance for unitary dynamics described by only the Hamiltonian in the quantum master equation.
We study the quantum speed limit for open quantum systems described by the Lindblad master equation. The obtained inequality shows a trade-off relation between the operation time and the physical quantities such as the energy fluctuation and the entropy production. We further identify a quantity characterizing the speed of the state transformation, which appears only when we consider the open system dynamics in the quantum regime. When the thermal relaxation is dominant compared to the unitary dynamics of the system, we show that this quantity is approximated by the energy fluctuation of the counter-diabatic Hamiltonian which is used as a control field in the shortcuts to adiabaticity protocol. We discuss the physical meaning of the obtained quantum speed limit and try to give better intuition about the speed in open quantum systems.
We extend the speed limit of a distance between two states evolving by different generators for quantum systems [K. Suzuki and K. Takahashi, Phys. Rev. Res. 2, 032016(R) (2020)] to the classical stochastic processes described by the master equation. We demonstrate that the trace distance between arbitrary evolving states is bounded from above by using a geometrical metric. The geometrical bound reduces to the Fisher information metric for the distance between the time-evolved state and the initial state. We compare the bound in relaxation and annealing processes with a different type of bound known for nonequilibrium thermodynamical systems. For dynamical processes such as annealing and pumping processes, the distance between the time-evolved state and the instantaneous stationary state becomes a proper choice and the bound is represented by the Fisher information metric of the stationary state. The metric is related to the counterdiabatic term defined from the time dependence of the stationary state.
Starting from a geometric perspective, we derive a quantum speed limit for arbitrary open quantum evolution, which could be Markovian or non-Markovian, providing a fundamental bound on the time taken for the most general quantum dynamics. Our methods rely on measuring angles and distances between (mixed) states represented as generalized Bloch vectors. We study the properties of our bound and present its form for closed and open evolution, with the latter in both Lindblad form and in terms of a memory kernel. Our speed limit is provably robust under composition and mixing, features that largely improve the effectiveness of quantum speed limits for open evolution of mixed states. We also demonstrate that our bound is easier to compute and measure than other quantum speed limits for open evolution, and that it is tighter than the previous bounds for almost all open processes. Finally, we discuss the usefulness of quantum speed limits and their impact in current research.
The speed limit on entanglement is defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary and arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For the unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.
Existing quantum speed limits for controlled open quantum systems depend on the specified trajectory. For example, lower bounds on quantum annealing times in the presence of dissipation depend explicitly on the chosen annealing schedule. Recently, schedule-independent speed limits have been derived for annealing in the closed quantum system setting (SciPost Phys. 18, 159 (2025)). In this work, we generalize these results to open quantum systems, deriving schedule-independent lower bounds for quantum annealing times in systems described by a Lindblad master equation. We analyze the interplay between coherent control and dissipation in single- and two-qubit examples, demonstrating that the derived lower bounds capture key scaling behavior with respect to the strength of the dissipator. Finally, we apply the bound to thermal state preparation and show that the bound matches the expected asymptotic behavior for an Ising model in the high temperature limit.
In the Schr\"odinger picture, the state of a quantum system evolves in time and the quantum speed limit describes how fast the state of a quantum system evolves from an initial state to a final state. However, in the Heisenberg picture the observable evolves in time instead of the state vector. Therefore, it is natural to ask how fast an observable evolves in time. This can impose a fundamental bound on the evolution time of the expectation value of quantum-mechanical observables. We obtain the quantum speed limit time bound for observable for closed systems, open quantum systems, and arbitrary dynamics. Furthermore, we discuss various applications of these bounds. Our results can have several applications ranging from setting the speed limit for operator growth, correlation growth, quantum thermal machines, quantum control, and many-body physics.