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arXiv:2204.06476 · Physical Review A

Unified entropies and quantum speed limits for nonunitary dynamics

Diego Paiva Pires

Published Jul 5, 2022• 12 citations
arXivHTMLPDF

Abstract

We discuss a class of quantum speed limits (QSLs) based on unified quantum ($\ensuremath{\alpha},\ensuremath{\mu}$)-entropy for nonunitary physical processes. The bounds depend on both the Schatten speed and the smallest eigenvalue of the evolved state and the two-parametric unified entropy. We specialize these results to quantum channels and non-Hermitian evolutions. In the first case, the QSL depends on the Kraus operators related to the quantum channel, while in the second case the speed limit is recast in terms of the non-Hermitian Hamiltonian. To illustrate these findings, we consider a single-qubit state evolving under the (i) amplitude damping channel and (ii) the nonunitary dynamics generated by a parity-time-reversal symmetric non-Hermitian Hamiltonian. The QSL is nonzero at earlier times, while it becomes loose as the smallest eigenvalue of the evolved state approaches zero. Furthermore, we investigate the interplay between unified entropies and speed limits for the reduced dynamics of quantum many-body systems. The unified entropy is upper bounded by the quantum fluctuations of the Hamiltonian of the system, while the QSL is nonzero when entanglement is created by the nonunitary evolution. Finally, we apply these results to the XXZ model and verify that the QSL asymptotically decreases as the system size increases. Our results find applications to nonequilibrium thermodynamics, quantum metrology, and equilibration of many-body systems.

DOI
10.1103/physreva.106.012403

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We discuss a class of quantum speed limits (QSLs) based on unified quantum (\ensuremathα,\ensuremathμ\ensuremath{\alpha},\ensuremath{\mu}\ensuremathα,\ensuremathμ)-entropy for nonunitary physical processes. The bounds depend on both the Schatten speed and the smallest eigenvalue of the evolved state and the two-parametric unified entropy. We specialize these results to quantum channels and non-Hermitian evolutions. In the first case, the QSL depends on the Kraus operators related to the quantum channel, while in the second case the speed limit is recast in terms of the non-Hermitian Hamiltonian. To illustrate these findings, we consider a single-qubit state evolving under the (i) amplitude damping channel and (ii) the nonunitary dynamics generated by a parity-time-reversal symmetric non-Hermitian Hamiltonian. The QSL is nonzero at earlier times, while it becomes loose as the smallest eigenvalue of the evolved state approaches zero. Furthermore, we investigate the interplay between unified entropies and speed limits for the reduced dynamics of quantum many-body systems. The unified entropy is upper bounded by the quantum fluctuations of the Hamiltonian of the system, while the QSL is nonzero when entanglement is created by the nonunitary evolution. Finally, we apply these results to the XXZ model and verify that the QSL asymptotically decreases as the system size increases. Our results find applications to nonequilibrium thermodynamics, quantum metrology, and equilibration of many-body systems.
We present a class of generalized entropic quantum speed limits based on ααα-zzz-Rényi relative entropy, a real-valued, contractive, two-parameter family of distinguishability measures. The quantum speed limit (QSL) falls into the class of Mandelstam-Tamm bounds, and applies to finite-dimensional quantum systems that undergo a general physical process, i.e., their effective dynamics can be modeled by unitary or nonunitary evolutions. The results cover pure or mixed, separable, and entangled probe quantum states. The QSL time depends on the smallest and largest eigenvalues of the probe and instantaneous states of the system, and its evaluation requires low computational cost. In addition, it is inversely proportional to the time-average of the Schatten speed of the instantaneous state, which in turn is fully characterized by the considered dynamics. We specialize our results to the case of unitary and nonunitary evolutions. In the former case, the QSL scales with the inverse of the energy fluctuations, while the latter depends on the Schatten 111-norm of the rate of change of the quantum channel Kraus operators. We illustrate our findings for single-qubit and two-qubit states, and unitary and nonunitary evolutions. Our results may find applications in the study of entropic uncertainty relations, quantum metrology, and also entanglement entropies signaled by generalized entropies.
Quantum speed limits (QSLs) establish intrinsic bounds on the minimum time required for the evolution of quantum systems. We present a class of QSLs formulated in terms of the two-parameter Sharma-Mittal entropy (SME), applicable to finite-dimensional systems evolving under general nonunitary dynamics. In the single-qubit case, the QSLs for both quantum channels and non-Hermitian dynamics are analyzed in detail. For many-body systems, we explore the role of SME-based bounds in characterizing the reduced dynamics and apply the results to the XXZ spin chain model. These entropy-based QSLs characterize fundamental limits on quantum evolution speeds and may be employed in contexts including entropic uncertainty relations, quantum metrology, coherent control and quantum sensing.
We discuss quantum speed limits (QSLs) for finite-dimensional quantum systems undergoing general physical processes. These QSLs were obtained using two families of entropic measures, namely the square root of the Jensen-Shannon divergence, which in turn defines a faithful distance of quantum states, and the square root of the quantum Jeffreys divergence. The results apply to both closed and open quantum systems, and are evaluated in terms of the Schatten speed of the evolved state, as well as cost functions that depend on the smallest and largest eigenvalues of both initial and instantaneous states of the quantum system. To illustrate our findings, we focus on the unitary and nonunitary dynamics of mixed single-qubit states. In the first case, we obtain speed limits aˋ la\textit{à la}aˋ la Mandelstam-Tamm that are inversely proportional to the variance of the Hamiltonian driving the evolution. In the second case, we set the nonunitary dynamics to be described by the noisy operations: depolarizing channel, phase damping channel, and generalized amplitude damping channel. We provide analytical results for the two entropic measures, present numerical simulations to support our results on the speed limits, comment on the tightness of the bounds, and provide a comparison with previous QSLs. Our results may find applications in the study of quantum thermodynamics, entropic uncertainty relations, and also complexity of many-body systems.
Quantum speed limit (QSL) for open quantum systems in the non-Markovian regime is analyzed. We provide the lower bound for the time required to transform an initial state to a final state in terms of thermodynamic quantities such as the energy fluctuation, entropy production rate and dynamical activity. Such bound was already analyzed for Markovian evolution satisfying detailed balance condition. Here we generalize this approach to deal with arbitrary evolution governed by time-local generator. Our analysis is illustrated by three paradigmatic examples of qubit evolution: amplitude damping, pure dephasing, and the eternally non-Markovian evolution.
The quantum speed limit (QSL) refers to the maximum speed of a quantum system to evolve from an initial state to its orthogonal states. The bound on the QSL for Hermitian systems, for example the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) as well as Sun-Zheng(SZ) bound, was studied respectively from the perspectives of average value and variance of the system Hamiltonian as well as the geometry of the system. While the compactness of the MT-type, ML-type and SZ-type bounds has been examined well for Hermitian systems, a compact QSL for non-Hermitian systems has not been well studied. In this work, based on the biorthogonal basis theory we derive two distinct and tighter bounds on the QSL for non-Hermitian systems, which correspond to the MT and ML bounds for Hermitian systems. We show that the shortest evolution time corresponding to the two bounds of the non-Hermitian system can be attained by certain initial states, showing the compactness and tightness of our bounds. These initial states dubbed fastest initial states(FIS) are different from that in Hermitian systems. A bound close to QSL for non-FIS is presented and comparison of our bound with others in literature is performed. To illustrate our results, we present a minimal non-Hermitian system to show QSL, and the condition for the shortest evolution time is derived analytically using the present theory.
Quantum mechanics dictates bounds for the minimal evolution time between predetermined initial and final states. Several of these quantum-speed-limit (QSL) bounds were derived for nonunitary dynamics using different approaches. Here, we perform a systematic analysis of the most common QSL bounds in the damped Jaynes-Cummings model, covering the Markovian and non-Markovian regimes. We show that only one of the analyzed bounds cleaves to the essence of the QSL theory outlined in the pioneer works of Mandelstam and Tamm and of Margolus and Levitin in the context of unitary evolutions. We also show that all QSL bounds analyzed reflect the fact that in our model non-Markovian effects speed up quantum evolution. However, it is not possible to infer Markovian or non-Markovian behavior of the dynamics by analyzing only the QSL bounds.