We provide a complete characterization of all finite-dimensional quantum systems that saturate the Margolus-Levitin quantum speed limit at arbitrary Uhlmann-Jozsa fidelity. Employing a purification-based approach, we prove that mixed-state saturation occurs precisely when three structural criteria are fulfilled: the state's support is confined to the sum of two energy eigenspaces (the ground level and a single excited level); each eigenvector of the state with nonzero weight is a fixed superposition of one ground- and one excited-state energy eigenvector (determined by the minimizer of the objective function identified by Giovannetti et al.) and all such eigenvectors evolve in mutually orthogonal subspaces. These requirements impose a strict rank bound, ruling out saturation by any faithful state. For quantum bits, we derive a purity-resolved and tight Margolus-Levitin bound that reduces to the pure-state result in the limit of unit purity. Through a time-reversal argument, we further extend the dual Margolus-Levitin quantum speed limit to mixed states and establish the corresponding saturation conditions.
We provide a complete characterization of all finite-dimensional quantum systems that saturate the Margolus-Levitin quantum speed limit at arbitrary Uhlmann-Jozsa fidelity. Employing a purification-based approach, we prove that mixed-state saturation occurs precisely when three structural criteria are fulfilled: the state's support is confined to the sum of two energy eigenspaces (the ground level and a single excited level); each eigenvector of the state with nonzero weight is a fixed superposition of one ground- and one excited-state energy eigenvector (determined by the minimizer of the objective function identified by Giovannetti et al.) and all such eigenvectors evolve in mutually orthogonal subspaces. These requirements impose a strict rank bound, ruling out saturation by any faithful state. For quantum bits, we derive a purity-resolved and tight Margolus-Levitin bound that reduces to the pure-state result in the limit of unit purity. Through a time-reversal argument, we further extend the dual Margolus-Levitin quantum speed limit to mixed states and establish the corresponding saturation conditions.
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 describe how fast a system can evolve from one state to another. There are several well-known quantum speed limits such as the Mandelstam–Tamm and Margolus–Levitin bounds which were originally derived for pure states and are not always tight for mixed states. In this thesis a stronger bound is derived for mixed qubit states, which has been shown to be tight, by using additional information about the system. We chose to focus on a single qubit and use the Bloch representation to describe its evolution geometrically. Due to both the energy and the purity of the system being conserved under unitary time evolution, the Bloch vector moves in a circle. By expressing the fidelity between the initial and final state in terms of Bloch vectors, we can derive a stricter quantum speed limit for mixed qubit states. This new lower bound depends on both fidelity and purity and is tighter than the Margolus–Levitin bound. The result shows that taking conserved quantities like purity into account can lead to improved quantum speed limits and similar ideas might be useful for more complex systems as well.
Many quantum speed limits for isolated systems can be generalized to also apply to closed systems. This is, for example, the case with the well-known Mandelstam-Tamm quantum speed limit. Margolus and Levitin derived an equally well-known and ostensibly related quantum speed limit, and it seems to be widely believed that the Margolus-Levitin quantum speed limit can be similarly generalized to closed systems. However, a recent geometrical examination of this limit reveals that it differs significantly from most known quantum speed limits. In this paper, we show that, contrary to the common belief, the Margolus-Levitin quantum speed limit does not extend to closed systems in an obvious way. More precisely, we show that for every hypothetical bound of Margolus-Levitin type, there are closed systems that evolve with a conserved normalized expected energy between states with any given fidelity in a time shorter than the bound. We also show that for isolated systems, the Mandelstam-Tamm quantum speed limit and a slightly weakened version of this limit that we call the Bhatia-Davies quantum speed limit always saturate simultaneously. Both of these evolution time estimates extend straightforwardly to closed systems. We demonstrate that there are closed systems that saturate the Mandelstam-Tamm but not the Bhatia-Davies quantum speed limit.
Abstract For vanishing fidelity between initial and final states, two important quantum speed limits, the Mandelstam–Tamm limit (involving energy dispersion) and Margolus–Levitin one (involving excitation energy expectation value), have been derived. While the generalization of the former limit to the case of arbitrary fidelity is straightforward, the relevant generalization of the latter given in the seminal paper by Giovanetti et al. (Phys Rev A67:052109, 2003) was based on the conjectured equality of lower and upper bounds on the right-hand side of generalized Margolus–Levitin inequality, verified numerically up to seven digits. Only recently there appear two proofs of the conjecture. We provide below a very elementary new proof, based on the simplest tools from differential calculus. Thus, the generalized Margolus–Levitin speed limit can be derived much in the spirit of the original one valid for vanishing fidelity.
The Mandelstam-Tamm and Margolus-Levitin quantum speed limits bound how fast a state evolves, using the energy variance and the mean energy above the ground state. Speed limits on observables -- the change of an expectation value <A(t)> -- have so far used the variance (the Mandelstam-Tamm / quantum-Fisher-information lineage); the mean energy has not been brought to bear on the change of <A> in a fixed state. We close this branch. First, a no-go theorem: there is no state-independent linear mean-energy bound on the time to change an observable's expectation by Delta = |<A(T)> - <A(0)>|; the optimal state-independent exponent of Delta is exactly two. Second, the corresponding sharp quadratic bound, T (<H> - E_0) >= C_* Delta^2 / sigma_A^2, for every time-independent Hamiltonian H (ground energy E_0), every bounded observable A with spectral spread sigma_A = (lambda_max - lambda_min)/2, and every pure or mixed state, with the dimension-independent constant C_* = 1/(8 sin x_*) = 0.172506267461..., where x_* is the smallest positive root of tan(x/2) = x. The bound is tight, approached but not attained by a near-ground two-level family. We then give the exact energy-time/swing trade-off curve of which C_* is the small-swing slope -- tight at every swing, the observable analog of the Giovannetti-Lloyd-Maccone curve for states -- show the constant survives for mixed states via joint convexity of the trace distance, sharpen it for bandwidth-limited generators and several observables at once, and show the quadratic law degrades to a linear one when the initial state is an eigenvector of the observable. It completes the (mean-energy x observable) corner of the speed-limit landscape and, being quadratic, is most constraining where the linear variance bounds are weakest. Its cleanest physical home is the autonomous quantum clock, where it gives a coherent mean-energy resolution floor.
α>(ε)=α<(ε) For The Margolus-Levitin Quantum Speed Limit Bound
The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least πα(ε)/(2⟨E−E0⟩), where ⟨E−E0⟩ is the expected energy of the system relative to the ground state of the Hamiltonian and α(ε) is a function of the fidelity ε between the two state. For a long time, only a upper bound α>(ε) and lower bound α<(ε) are known although they agree up to at least seven significant figures. Lately, Hörnedal and Sönnerborn proved an analytical expression for α(ε), fully classified systems whose evolution times saturate the ML bound, and gave this bound a symplectic-geometric interpretation. Here I solve the same problem through an elementary proof of the ML bound. By explicitly finding all the states that saturate the ML bound, I show that α>(ε) is indeed equal to α<(ε). More importantly, I point out a numerical stability issue in computing α>(ε) and report a simple way to evaluate it efficiently and accurately.