The quantum speed limit (QSL) is the lower bound on the time required for a state to evolve to a desired final state under a given Hamiltonian evolution. Three well-known QSLs exist: the Mandelstam-Tamm (MT), Margolus-Levitin (ML), and dual ML (ML*) bounds. We consider one-dimensional systems that undergo a delocalization-localization transition in the presence of quasiperiodic and linear potential. By performing sudden quenches across the phase boundary, we find that the exact dynamics gets captured very well by QSLs. We show that the MT bound is always tighter in the short-time limit for any arbitrary state, while the optimal bound for the time of orthogonalization (time required to reach the orthogonal state) depends on the choice of the initial state. Further, for extreme quenches, we prove that the MT bound remains tighter for the time of orthogonalization and it can qualitatively describe the nonanalyticity in free energy for the dynamical quantum phase transition. We also demonstrate that the localization-delocalization transition point can be exactly identified from QSLs for two of the models of noninteracting fermions. Further, we also show that even for interacting systems, an ergodic to many-body localization transition can also be efficiently predicted from QSLs, whose computation cost is much less than other diagnostic tools.
The quantum speed limit (QSL) is the lower bound on the time required for a state to evolve to a desired final state under a given Hamiltonian evolution. Three well-known QSLs exist: the Mandelstam-Tamm (MT), Margolus-Levitin (ML), and dual ML (ML*) bounds. We consider one-dimensional systems that undergo a delocalization-localization transition in the presence of quasiperiodic and linear potential. By performing sudden quenches across the phase boundary, we find that the exact dynamics gets captured very well by QSLs. We show that the MT bound is always tighter in the short-time limit for any arbitrary state, while the optimal bound for the time of orthogonalization (time required to reach the orthogonal state) depends on the choice of the initial state. Further, for extreme quenches, we prove that the MT bound remains tighter for the time of orthogonalization and it can qualitatively describe the nonanalyticity in free energy for the dynamical quantum phase transition. We also demonstrate that the localization-delocalization transition point can be exactly identified from QSLs for two of the models of noninteracting fermions. Further, we also show that even for interacting systems, an ergodic to many-body localization transition can also be efficiently predicted from QSLs, whose computation cost is much less than other diagnostic tools.
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 speed limits (QSLs) impose fundamental constraints on the evolution speed of quantum systems. Traditionally, the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) bounds have been widely employed, relying on the standard deviation and mean of energy distribution to define the QSLs. However, these universal bounds only offer loose restrictions on the quantum evolution. Here we introduce the generalized ML bounds, which prove to be more stringent in constraining dynamic evolution, by utilizing moments of energy spectra of arbitrary orders, even noninteger orders. To validate our findings, we conduct experiments in a superconducting circuit, where we have the capability to prepare a wide range of quantum photonic states and rigorously test these bounds by measuring the evolution of the system and its photon statistics using quantum state tomography. While, in general, the MT bound is effective for short-time evolution, we identify specific parameter regimes where either the MT or the generalized ML bounds suffice to constrain the entire evolution. Our findings not only establish new criteria for estimating QSLs but also substantially enhance our comprehension of the dynamic evolution of quantum systems.
Abstract Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam–Tamm (MT) bound, the Margolus–Levitin (ML) bound, the Luo–Zhang bound, the Lee–Chau (LC) bound together with the dual ML bound introduced by Ness and coworkers. Here we show that the MT bound can be obtained by optimizing the LC bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter p. For any fixed p, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain pth moments related to the absolute value of energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We further compare the performance of this bound for the case of a fixed p as well as the case of optimizing over p with existing QSLs. We find that if the dimension of the underlying Hilbert space is ≲ 2000 , our unifying bound optimized over p can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.
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 $
The Mandelstam-Tamm and Margolus-Levitin quantum speed limits are two well-known evolution time estimates for isolated quantum systems. These bounds are usually formulated for fully distinguishable initial and final states, but both have tight extensions to systems that evolve between states with an arbitrary fidelity. However, the foundations of these extensions differ in some essential respects. The extended Mandelstam-Tamm quantum speed limit has been proven analytically and has a clear geometric interpretation. Furthermore, which systems saturate the limit is known. The derivation of the extended Margolus-Levitin quantum speed limit, on the other hand, is based on numerical estimates. Moreover, the limit lacks a geometric interpretation, and no complete characterization of the systems reaching it exists. In this paper, we derive the extended Margolus-Levitin quantum speed limit analytically and describe the systems that saturate the limit in detail. We also provide the limit with a symplectic-geometric interpretation, which indicates that it is of a different character than most existing quantum speed limits. At the end of the paper, we analyze the maximum of the extended Mandelstam-Tamm and Margolus-Levitin quantum speed limits and derive a dual version of the extended Margolus-Levitin quantum speed limit. The maximum limit is tight regardless of the fidelity of the initial and final states. However, the conditions under which the maximum limit is saturated differ depending on whether or not the initial state and the final state are fully distinguishable. The dual limit is also tight and follows from a time reversal argument. We describe the systems that saturate the dual quantum speed limit.
Quantum speed limits set the maximal pace of state evolution. Two well-known limits exist for a unitary time-independent Hamiltonian: the Mandelstam-Tamm and Margolus-Levitin bounds. The former restricts the rate according to the state energy uncertainty, while the latter depends on the mean energy relative to the ground state. Here we report on an additional bound that exists for states with a bounded energy spectrum. This bound is dual to the Margolus-Levitin one in the sense that it depends on the difference between the state's mean energy and the energy of the highest occupied eigenstate. Each of the three bounds can become the most restrictive one, depending on the spread and mean of the energy, forming three dynamical regimes which are accessible in a multilevel system. The new bound is relevant for quantum information applications, since in most of them, information is stored and manipulated in a Hilbert space with a bounded energy spectrum.