It is remarkable that Heisenberg’s position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg’s uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time.
It is remarkable that Heisenberg’s position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg’s uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time.
The concept of quantum acceleration limit has been recently introduced for any unitary time evolution of quantum systems under arbitrary nonstationary Hamiltonians. While Alsing and Cafaro used the Robertson uncertainty relation in their derivation, employed the Robertson–Schrödinger uncertainty relation to find the upper bound on the temporal rate of change of the speed of quantum evolutions. In this paper, we provide a comparative analysis of these two alternative derivations for quantum systems specified by an arbitrary finite-dimensional projective Hilbert space. Furthermore, focusing on a geometric description of the quantum evolution of two-level quantum systems on a Bloch sphere under general time-dependent Hamiltonians, we find the most general conditions needed to attain the maximal upper bounds on the acceleration of the quantum evolution. In particular, these conditions are expressed explicitly in terms of two three-dimensional real vectors, the Bloch vector that corresponds to the evolving quantum state and the magnetic field vector that specifies the Hermitian Hamiltonian of the system. For pedagogical reasons, we illustrate our general findings for two-level quantum systems in explicit physical examples characterized by specific time-varying magnetic field configurations. Finally, we briefly comment on the extension of our considerations to higher-dimensional physical systems in both pure and mixed quantum states.
In the geometry of quantum-mechanical processes, the time-varying curvature coefficient of a quantum evolution is specified by the magnitude squared of the covariant derivative of the tangent vector to the state vector. In particular, the curvature coefficient measures the bending of the quantum curve traced out by a parallel-transported pure quantum state that evolves in a unitary fashion under a nonstationary Hamiltonian that specifies the Schr\"odinger evolution equation. In this paper, we present an exact analytical expression of the curvature of a quantum evolution for a two-level quantum system immersed in a time-dependent magnetic field. Specifically, we study the dynamics generated by a two-parameter nonstationary Hermitian Hamiltonian with unit speed efficiency. The two parameters specify the constant temporal rates of change of the polar and azimuthal angles used in the Bloch sphere representation of the evolving pure state. To better grasp the physical significance of the curvature coefficient, showing that the quantum curve is nongeodesic since the geodesic efficiency of the quantum evolution is strictly less than 1 and tuning the two Hamiltonian parameters, we compare the temporal behavior of the curvature coefficient with that of the speed and the acceleration of the evolution of the system in projective Hilbert space. Furthermore, we compare the temporal profile of the curvature coefficient with that of the square of the ratio between the parallel and transverse magnetic field intensities. Finally, we discuss the challenges in finding exact analytical solutions when extending our geometric approach to higher-dimensional quantum systems that evolve unitarily under an arbitrary time-dependent Hermitian Hamiltonian.
Recently, the notion of a quantum acceleration limit has been proposed for any unitary time evolution of quantum systems governed by arbitrary nonstationary Hamiltonians. This limit articulates that the rate of change over time of the standard deviation of the Hamiltonian operator — representing the acceleration of quantum evolution within projective Hilbert space — is constrained by the standard deviation of the time-derivative of the Hamiltonian, expressed as σ ̇ H ≤ σ H ̇ . In this paper, we extend our earlier findings to encompass any observable A within the framework of unitary quantum dynamics, leading to the inequality σ ̇ A ≤ σ v A . This relationship signifies that the speed of the standard deviation of any observable is limited by the standard deviation of its associated velocity-like observable v A . Finally, for pedagogical purposes, we illustrate the relevance of our inequality by providing clear examples. We choose suitable observables related to the unitary dynamics of two-level quantum systems, as well as a harmonic oscillator within a finite-dimensional Fock space.
We use geometric concepts originally proposed by Anandan and Aharonov [Phys. Rev. Lett. 65, 1697 (1990)] to show that the Farhi-Gutmann time-optimal analog quantum search evolution between two orthogonal quantum states is characterized by unit efficiency dynamical trajectories traced on a projective Hilbert space. In particular, we prove that these optimal dynamical trajectories are the shortest geodesic paths joining the initial and the final states of the quantum evolution. In addition, we verify they describe minimum uncertainty evolutions specified by an uncertainty inequality that is tighter than the ordinary time-energy uncertainty relation. We also study the effects of deviations from the time-optimality condition from our proposed Riemannian geometric perspective. Furthermore, after pointing out some physically intuitive aspects offered by our geometric approach to quantum searching, we mention some practically relevant physical insights that could emerge from the application of our geometric analysis to more realistic time-dependent quantum search evolutions. Finally, we briefly discuss possible extensions of our paper to the geometric analysis of the efficiency of thermal trajectories of relevance in quantum computing tasks.
In optimal quantum-mechanical evolutions, motion can take place along paths of minimal length within an optimal time frame. Alternatively, optimal evolutions may occur along established paths without any waste of energy resources and achieving 100% speed efficiency. Unfortunately, realistic physical scenarios often lead to less-than-ideal evolutions that demonstrate suboptimal efficiency, nonzero curvature, and a high level of complexity. In this paper, we provide an exact analytical expression for the curvature of a quantum evolution pertaining to a two-level quantum system subjected to various time-dependent magnetic fields. Specifically, we examine the dynamics produced by a two-parameter nonstationary Hermitian Hamiltonian with unit speed efficiency. To enhance our understanding of the physical implications of the curvature coefficient, we analyze the curvature behavior in relation to geodesic efficiency, speed efficiency, and the complexity of the quantum evolution (as described by the ratio of the difference between accessible and accessed Bloch-sphere volumes for the evolution from initial to final state to the accessible volume for the given quantum evolution). Our findings indicate that, generally, efficient quantum evolutions exhibit lower complexity compared to inefficient ones. However, we also note that complexity transcends mere length. In fact, longer paths that are sufficiently curved can demonstrate a complexity that is less than that of shorter paths with a lower curvature coefficient.
The evolution speed in projective Hilbert space is considered for Hermitian Hamiltonians and for non-Hermitian (NH) ones. Based on the Hilbert–Schmidt norm and the spectral norm of a Hamiltonian, resource-related upper bounds on the evolution speed are constructed. These bounds are valid also for NH Hamiltonians and they are illustrated for an optical NH Hamiltonian and for a NH -symmetric matrix Hamiltonian. Furthermore, the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective Hilbert space. A recipe for the construction of time-dependent Hamiltonians which ensure 100% speed efficiency is given. Generally these efficient Hamiltonians are NH but there is a Hermitian efficient Hamiltonian as well. Finally, the extremal case of a NH non-diagonalizable Hamiltonian with vanishing energy difference is shown to produce a 100% efficient evolution with minimal resources consumption.