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arXiv:2601.10662

Geometric Aspects of Entanglement Generating Hamiltonian Evolutions

Carlo Cafaro, James Schneeloch

quant-phSubmitted Jan 15, 2026• Announced Jan 16, 2026• 0 citations• Revised Jun 6, 2026 (v2)
v1 · Jan 15, 2026v2 · Jun 6, 2026
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Abstract

We examine the pertinent geometric characteristics of entanglement that arise from stationary Hamiltonian evolutions transitioning from separable to maximally entangled two-qubit quantum states. From a geometric perspective, each evolution is characterized by means of geodesic efficiency, speed efficiency, and curvature coefficient. Conversely, from the standpoint of entanglement, these evolutions are quantified using various metrics, such as concurrence, entanglement power, and entangling capability. Overall, our findings indicate that time-optimal evolution trajectories are marked by high geodesic efficiency, with no energy resource wastage, no curvature (i.e., zero bending), and an average path entanglement that is less than that observed in time-suboptimal evolutions. Additionally, when analyzing separable-to-maximally entangled evolutions between nonorthogonal states, time-optimal evolutions demonstrate a greater short-time degree of nonlocality compared to time-suboptimal evolutions between the same initial and final states. Interestingly, the reverse is generally true for separable-to-maximally entangled evolutions involving orthogonal states. Our investigation suggests that this phenomenon arises because suboptimal trajectories between orthogonal states are characterized by longer path lengths with smaller curvature, which are traversed with a higher energy resource wastage compared to suboptimal trajectories between nonorthogonal states. Consequently, a higher initial degree of nonlocality in the unitary time propagators appears to be essential for achieving the maximally entangled state from a separable state. Furthermore, when assessing optimal and suboptimal evolutions...

Journal reference
Int. J. Theor. Phys. 65, 177 (2026)
Comments
27 pages, 2 figures, 3 tables

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We examine the pertinent geometric characteristics of entanglement that arise from stationary Hamiltonian evolutions transitioning from separable to maximally entangled two-qubit quantum states. From a geometric perspective, each evolution is characterized by means of geodesic efficiency, speed efficiency, and curvature coefficient. Conversely, from the standpoint of entanglement, these evolutions are quantified using various metrics, such as concurrence, entanglement power, and entangling capability. Overall, our findings indicate that time-optimal evolution trajectories are marked by high geodesic efficiency, with no energy resource wastage, no curvature (i.e., zero bending), and an average path entanglement that is less than that observed in time-suboptimal evolutions. Additionally, when analyzing separable-to-maximally entangled evolutions between nonorthogonal states, time-optimal evolutions demonstrate a greater short-time degree of nonlocality compared to time-suboptimal evolutions between the same initial and final states. Interestingly, the reverse is generally true for separable-to-maximally entangled evolutions involving orthogonal states. Our investigation suggests that this phenomenon arises because suboptimal trajectories between orthogonal states are characterized by longer path lengths with smaller curvature, which are traversed with a higher energy resource wastage compared to suboptimal trajectories between nonorthogonal states. Consequently, a higher initial degree of nonlocality in the unitary time propagators appears to be essential for achieving the maximally entangled state from a separable state. Furthermore, when assessing optimal and suboptimal evolutions...
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.
Abstract In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.
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 compare Krylov's state complexity with an information-geometric (IG) measure of complexity for the quantum evolution of two-level systems. Focusing on qubit dynamics on the Bloch sphere, we analyze evolutions generated by stationary and nonstationary Hamiltonians, corresponding to geodesic and nongeodesic trajectories. We formulate Krylov complexity in geometric terms, both instantaneously and in a time-averaged sense, and contrast it with an IG complexity of quantum evolutions characterized in terms of efficiency and curvature. We show that the two measures reflect fundamentally different aspects of quantum dynamics: Krylov's state complexity quantifies the directional spread of the evolving state relative to the initial state, whereas the IG complexity captures the effective volume explored along the trajectory on the Bloch sphere. This geometric distinction explains their inequivalent behavior and highlights the complementary nature of state-based and information-geometric notions of complexity in quantum systems.
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 occur along nonpredetermined paths of shortest length in an optimal time. Alternatively, optimal evolutions can happen along predefined paths with no waste of energy resources and 100100%100 speed efficiency. Unfortunately, realistic physical scenarios typically result in less-than-ideal evolutions. In this paper we study different families of suboptimal qubit Hamiltonians, both stationary and time varying, for which the so-called geodesic efficiency and the speed efficiency of the corresponding quantum evolutions are less than one. Furthermore, after proposing an alternative hybrid efficiency measure constructed out of the two previously mentioned efficiency quantifiers, we provide illustrative examples where the average departures from time optimality and 100100%100 speed efficiency are globally captured over a limited time period. In particular, due to this hybrid measure, quantum evolutions are partitioned in four categories: geodesic unwasteful, nongeodesic unwasteful, geodesic wasteful, and nongeodesic wasteful. Finally, we discuss Hamiltonians specified by magnetic-field configurations, both stationary and nonstationary, yielding optimal hybrid efficiency, that it, both time optimality and 100100%100 speed efficiency, over a finite time interval.