We establish a comprehensive theoretical framework for coherent quantum speed limits (QSLs), deriving fundamental bounds on the rate of quantum evolution that explicitly isolate the contribution of quantum coherence. By applying Hölder's inequality for matrix norms to the Liouville-von Neumann equation, we construct two infinite families of QSLs for general unitary dynamics. These bounds are characterized by coherence measures based on Schatten $p$-norms and Hellinger distance, respectively, defined with respect to the instantaneous energy eigenbasis. Unlike traditional Mandelstam-Tamm bounds, our approach disentangles the quantum state's coherence structure from the Hamiltonian's energy scale. Using the Landau-Zener model accelerated by shortcuts to adiabaticity, we demonstrate that coherence functions as a critical kinematic resource: achieving faster evolution entails maintaining a state with high coherence relative to the instantaneous basis. Our results provide a resource-theoretic perspective on time-energy uncertainty, offering insights into the fundamental limits of quantum control and information processing.
We establish a comprehensive theoretical framework for coherent quantum speed limits (QSLs), deriving fundamental bounds on the rate of quantum evolution that explicitly isolate the contribution of quantum coherence. By applying Hölder's inequality for matrix norms to the Liouville-von Neumann equation, we construct two infinite families of QSLs for general unitary dynamics. These bounds are characterized by coherence measures based on Schatten p-norms and Hellinger distance, respectively, defined with respect to the instantaneous energy eigenbasis. Unlike traditional Mandelstam-Tamm bounds, our approach disentangles the quantum state's coherence structure from the Hamiltonian's energy scale. Using the Landau-Zener model accelerated by shortcuts to adiabaticity, we demonstrate that coherence functions as a critical kinematic resource: achieving faster evolution entails maintaining a state with high coherence relative to the instantaneous basis. Our results provide a resource-theoretic perspective on time-energy uncertainty, offering insights into the fundamental limits of quantum control and information processing.
Quantum speed limits (QSLs) impose fundamental constraints on the evolution speed of quantum systems. For systems with time-independent Hamiltonians, the QSLs are intricately linked to the energy spectra of the initial state, which signify the generalization of the time-energy uncertainty relation. Based on arbitrary-order norms of the Hamiltonian, dual energies from inverted spectra, and the standard deviation of energy, different types of QSLs have been proposed, revealing three dynamical parameter regimes of quantum state evolution. Despite the theoretical advancements, a comprehensive experimental examination remains elusive. In this research, by introducing unified bounds suitable for all evolution times, we present the experimental verification to the QSLs in various scenarios with a superconducting circuit. We explore the dynamical evolution with regard to each parameter regime as well as the critical point by carefully engineering the initial-state population of the superconducting resonator. We test our unified bounds over finite-level states and infinite-level classical and nonclassical Gaussian states, which validates the criteria for the QSLs. Our work substantially deepens our comprehension of unitary quantum evolution.
Quantum speed limits (QSLs) constrain the minimum time for quantum state evolution, typically derived from energy bounds like the Margolus-Levitin (ML) and Mandelstam-Tamm (MT) relations. In open systems, decoherence—driven by environmental interactions—may impose additional constraints. This work investigates whether decoherence times fundamentally limit QSLs in spin-chain systems, extending ML/MT bounds to open dynamics. By modeling spin chains with Lindblad equations and simulating Heisenberg-type interactions, we analyze the interplay between energy fluctuations, decoherence rates, and evolution times. Experimental validation via trapped-ion or superconducting-qubit arrays could test energy-time uncertainty in many-body systems. Results suggest decoherence times establish a lower QSL bound proportional to [to verify], offering a platform to probe open-system quantum dynamics. ## 1. Introduction Quantum speed limits (QSLs) define the fastest possible evolution between distinguishable quantum states, critical for quantum computing and information processing. Closed-system QSLs, such as the ML [arXiv:quant-ph/9804014] and MT [DOI:10.1103/PhysRev.41.1019] bounds, rely on energy constraints. However, open systems—subject to decoherence—require extended frameworks. Decoherence times, governed by environmental couplings, may impose additional QSLs, yet their interplay with ML/MT bounds remains unexplored. Spin chains, realizable in trapped ions [DOI:10.1038/s41586-019-1394-6] and superconducting circuits, offer a many-body testbed for studying these effects. This work bridges open-system dynamics with QSL theory, assessing whether decoherence times set a fundamental lower bound and enabling experimental verification of energy-time uncertainty. ## 2. Background The ML bound states that a quantum system requires at least $ t_{\text{QSL}} \geq \hbar / (2\Delta E) $, where $ \Delta E $ is the energy spread. The MT bound $ t_{\text{QSL}} \geq \hbar / (2E) $, with $
We present a class of generalized entropic quantum speed limits based on α-z-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 1-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) 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 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.
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 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.