Abstract. Inequalities of Mandelstam-Tamm and Margolus-Levitin type provide lower bounds on the time it takes for a quantum system to evolve from one state into another. Knowledge of such bounds, called quantum speed limits, is of utmost importance in virtually all areas of physics, where determination of the minimum time required for a quantum process is of interest. Most Mandelstam-Tamm and Margolus-Levitin inequalities found in the literature have been derived from growth estimates for the Bures length, which is a statistical distance measure. In this paper we derive such inequalities by differential geometric methods, and we compare the obtained quantum speed limits with those involving the Bures length. We also characterize the Hamiltonians which optimize the evolution time for generic finite-level quantum systems. 1.
Abstract. Inequalities of Mandelstam-Tamm and Margolus-Levitin type provide lower bounds on the time it takes for a quantum system to evolve from one state into another. Knowledge of such bounds, called quantum speed limits, is of utmost importance in virtually all areas of physics, where determination of the minimum time required for a quantum process is of interest. Most Mandelstam-Tamm and Margolus-Levitin inequalities found in the literature have been derived from growth estimates for the Bures length, which is a statistical distance measure. In this paper we derive such inequalities by differential geometric methods, and we compare the obtained quantum speed limits with those involving the Bures length. We also characterize the Hamiltonians which optimize the evolution time for generic finite-level quantum systems. 1.
Inequalities of Mandelstam–Tamm (MT) and Margolus–Levitin (ML) type provide lower bounds on the time that it takes for a quantum system to evolve from one state into another. Knowledge of such bounds, called quantum speed limits, is of utmost importance in virtually all areas of physics, where determination of the minimum time required for a quantum process is of interest. Most MT and ML inequalities found in the literature have been derived from growth estimates for the Bures length, which is a statistical distance measure. In this paper we derive such inequalities by differential geometric methods, and we compare the quantum speed limits obtained with those involving the Bures length. We also characterize the Hamiltonians which optimize the evolution time for generic finite-level quantum systems.
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.
Quantum speed limits are rigorous estimates on how fast a state of a quantum system can depart from the initial state in the course of quantum evolution. Most known quantum speed limits, including the celebrated Mandelstam-Tamm and Margolus-Levitin ones, are general bounds applicable to arbitrary initial states. However, when applied to mixed states of many-body systems, they, as a rule, dramatically overestimate the speed of quantum evolution and fail to provide meaningful bounds in the thermodynamic limit. Here we derive a quantum speed limit for a closed system initially prepared in a thermal state and evolving under a time-dependent Hamiltonian. This quantum speed limit exploits the structure of the thermal state and, in particular, explicitly depends on the temperature. In a broad class of many-body setups it proves to be drastically stronger than general quantum speed limits.
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.