We present an information geometric characterization of quantum driving schemes specified by su(2;C) time-dependent Hamiltonians in terms of both complexity and efficiency concepts. Specifically, starting from pure output quantum states describing the evolution of a spin-1/2 particle in an external time-dependent magnetic field, we consider the probability paths emerging from the parametrized squared probability amplitudes of quantum origin. The information manifold of such paths is equipped with a Riemannian metrization specified by the Fisher information evaluated along the parametrized squared probability amplitudes. By employing a minimum action principle, the optimum path connecting initial and final states on the manifold in finite time is the geodesic path between the two states. In particular, the total entropy production that occurs during the transfer is minimized along these optimum paths. For each optimum path that emerges from the given quantum driving scheme, we evaluate the so-called information geometric complexity (IGC) and our newly proposed measure of entropic efficiency constructed in terms of the constant entropy production rates that specify the entropy minimizing paths being compared. From our analytical estimates of complexity and efficiency, we provide a relative ranking among the driving schemes being investigated. Moreover, we determine that the efficiency and the temporal rate of change of the IGC are monotonic decreasing and increasing functions, respectively, of the constant entropic speed along these optimum paths. Then, after discussing the connection between thermodynamic length and IGC in the physical scenarios being analyzed, we briefly examine the link between IGC and entropy production rate. Finally, we conclude by commenting on the fact that an higher entropic speed in quantum transfer processes seems to necessarily go along with a lower entropic efficiency together with a higher information geometric complexity.
We present an information geometric characterization of quantum driving schemes specified by su(2;C) time-dependent Hamiltonians in terms of both complexity and efficiency concepts. Specifically, starting from pure output quantum states describing the evolution of a spin-1/2 particle in an external time-dependent magnetic field, we consider the probability paths emerging from the parametrized squared probability amplitudes of quantum origin. The information manifold of such paths is equipped with a Riemannian metrization specified by the Fisher information evaluated along the parametrized squared probability amplitudes. By employing a minimum action principle, the optimum path connecting initial and final states on the manifold in finite time is the geodesic path between the two states. In particular, the total entropy production that occurs during the transfer is minimized along these optimum paths. For each optimum path that emerges from the given quantum driving scheme, we evaluate the so-called information geometric complexity (IGC) and our newly proposed measure of entropic efficiency constructed in terms of the constant entropy production rates that specify the entropy minimizing paths being compared. From our analytical estimates of complexity and efficiency, we provide a relative ranking among the driving schemes being investigated. Moreover, we determine that the efficiency and the temporal rate of change of the IGC are monotonic decreasing and increasing functions, respectively, of the constant entropic speed along these optimum paths. Then, after discussing the connection between thermodynamic length and IGC in the physical scenarios being analyzed, we briefly examine the link between IGC and entropy production rate. Finally, we conclude by commenting on the fact that an higher entropic speed in quantum transfer processes seems to necessarily go along with a lower entropic efficiency together with a higher information geometric complexity.
We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su(2;C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external time-dependent magnetic field that specifies the su(2;C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.
We present an information geometric analysis of both entropic speeds and entropy production rates arising from geodesic evolution on manifolds parametrized by pure quantum states. In particular, we employ pure states that emerge as outputs of suitably chosen [Formula: see text] time-dependent Hamiltonian operators that characterize analog quantum search algorithms of specific types. The [Formula: see text] Hamiltonian models under consideration are specified by external time-dependent magnetic fields within which spin-[Formula: see text] test particles are immersed. The positive definite Riemannian metrization of the parameter manifold is furnished by the Fisher information function. The Fisher information function is evaluated along parametrized squared probability amplitudes obtained from the temporal evolution of these spin-[Formula: see text] test particles. A minimum action approach is then utilized to induce the transfer of the quantum system from its initial state to its final state on the parameter manifold over a finite temporal interval. We demonstrate in an explicit manner that the minimal (that is, optimum) path corresponds to the shortest (that is, geodesic) path between the initial and final states. Furthermore, we show that the minimal path serves also to minimize the total entropy production occurring during the transfer of states. Finally, upon evaluating the entropic speed as well as the total entropy production along optimal transfer paths within several scenarios of physical interest in analog quantum searching algorithms, we demonstrate in a transparent quantitative manner a correspondence between a faster transfer and a higher rate of entropy production. We therefore conclude that higher entropic speed is associated with lower entropic efficiency within the context of quantum state transfer.
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.
We study the complexity of both time-optimal and time sub-optimal quantum Hamiltonian evolutions connecting arbitrary source and a target states on the Bloch sphere equipped with the Fubini-Study metric. This investigation is performed in a number of steps. First, we describe each unitary Schrödinger quantum evolution by means of the path length, the geodesic efficiency, the speed efficiency, and the curvature coefficient of its corresponding dynamical trajectory linking the source state to the target state. Second, starting from a classical probabilistic setting where the so-called information geometric complexity can be employed to describe the complexity of entropic motion on curved statistical manifolds underlying the physics of systems when only partial knowledge about them is available, we transition into a deterministic quantum setting. In this context, after proposing a definition of the complexity of a quantum evolution, we present a notion of quantum complexity length scale. In particular, we discuss the physical significance of both quantities in terms of the accessed (i.e., partial) and accessible (i.e., total) parametric volumes of the regions on the Bloch sphere that specify the quantum mechanical evolution from the source to the target states. Third, after calculating the complexity measure and the complexity length scale for each one of the two quantum evolutions, we compare the behavior of our measures with that of the path length, the geodesic efficiency, the speed efficiency, and the curvature coefficient. We find that, in general, efficient quantum evolutions are less complex than inefficient evolutions. However, we also observe that complexity is more than length. Indeed, longer paths that are sufficiently bent can exhibit a behavior that is less complex than that of shorter paths with a smaller curvature coefficient.
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.
We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.