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arXiv:2408.07794

Characterizing Optimal-speed unitary time evolution of pure and quasi-pure quantum states

Rodríguez, John A. Mora, Brian Grajales, Marcelo Terra Cunha, Lino Grama

Published Aug 14, 2024• 0 citations
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Abstract

We present a characterization of the Hamiltonians that generate optimal-speed unitary time evolution and the associated dynamical trajectory, where the initial states are either pure states or quasi-pure quantum states. We construct the manifold of pure states as an orbit under the conjugation action of the Lie group $\SU(n)$ on the manifold of one-dimensional orthogonal projectors, obtaining an isometry with the flag manifold $\SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1 ))$. From this construction, we show that Hamiltonians generating optimal-speed time evolution are fully characterized by equigeodesic vectors of $\SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1))$. We later extend that result to quasi-pure quantum states.

DOI
10.48550/arxiv.2408.07794

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We present a characterization of the Hamiltonians that generate optimal-speed unitary time evolution and the associated dynamical trajectory, where the initial states are either pure states or quasi-pure quantum states. We construct the manifold of pure states as an orbit under the conjugation action of the Lie group \SU(n)\SU(n)\SU(n) on the manifold of one-dimensional orthogonal projectors, obtaining an isometry with the flag manifold \SU(n)/S(U(1)×U(n−1))\SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1 ))\SU(n)/S(U(1)×U(n−1)). From this construction, we show that Hamiltonians generating optimal-speed time evolution are fully characterized by equigeodesic vectors of \SU(n)/S(U(1)×U(n−1))\SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1))\SU(n)/S(U(1)×U(n−1)). We later extend that result to quasi-pure quantum states.
We investigate the dynamics of quantum systems whose evolution is governed by linear generators of fixed unitarily invariant norm. For this broad class of evolutions, we derive a universal time-optimal generator of the dynamics converting one given state into another. It provides a time-optimal non-Hermitian Hamiltonian for pure state conversion. We then use this general result to derive quantum speed limits for mixed-state evolutions governed by Lindbladians and non-Hermitian Hamiltonians. Our findings follow from the geometric properties of the group GL(n,C)GL(n,\mathbb{C})GL(n,C), in particular the geodesic distance. The derived bounds hold for an arbitrary system size nnn, depend only on the norms of the evolution generator and the initial and final states, and do not require performing any time-averaging.
Abstract We present a comparative analysis of two different constructions of optimal-speed quantum Hamiltonian evolutions on the Bloch sphere. In the first approach (Mostafazadeh’s approach), the evolution is specified by a traceless stationary Hermitian Hamiltonian and occurs between two arbitrary qubit states by maximizing the energy uncertainty. In the second approach (Bender’s approach), instead, the evolution is characterized by a stationary Hermitian Hamiltonian which is not traceless and occurs between an initial qubit state on the north pole and an arbitrary final qubit state. In this second approach, the evolution occurs by minimizing the evolution time subject to the constraint that the difference between the largest and the smallest eigenvalues of the Hamiltonian is kept fixed. For both approaches we calculate explicitly the optimal Hamiltonian, the optimal unitary evolution operator and, finally, the optimal magnetic field configuration. Furthermore, we show in a clear way that Mostafazadeh’s and Bender’s approaches are equivalent when we extend Mostafazadeh’s approach to Hamiltonians with nonzero trace and, at the same time, focus on an initial quantum state placed on the north pole of the Bloch sphere. Finally, we demonstrate in both scenarios that the optimal unitary evolution operator is a rotation about an axis that is orthogonal to the unit Bloch vectors that correspond to the initial and final qubit states.
We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric (inner product) dependence of the lower bound on the travel time and the universality (metric independence) of the upper bound on the speed of unitary evolutions.
A fundamental problem in quantum engineering is determining the lowest time required to ensure that all possible unitaries can be generated with the tools available, which is one of a number of possible quantum speed limits. We examine this problem from the perspective of quantum control, where the system of interest is described by a drift Hamiltonian and set of control Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups, and differential geometry and formulates the problem in terms of geodesics on a differentiable manifold. We provide explicit lower bounds on the quantum speed limit for the case of an arbitrary drift, requiring only that the control Hamiltonians generate a topologically closed subgroup of the full unitary group, and formulate criteria as to when our expression for the speed limit is exact and not merely a lower bound. These analytic results are then tested and confirmed using a numerical optimization scheme. Finally, we extend the analysis to find a lower bound on the quantum speed limit in the common case where the system is described by a drift Hamiltonian and a single control Hamiltonian.
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.
The speed of evolution between perfectly distinguishable states is thoroughly analyzed in a closed three-level (qutrit) quantum system. Considering an evolution under an arbitrary time-independent Hamiltonian, we fully characterize the relevant parameters according to whether the corresponding quantum speed limit is given by the Mandelstam-Tamm, the Margolus-Levitin, or the Ness-Alberti-Sagi (dual) bound, thereby elucidating their hierarchy and relative importance. We revisit the necessary and sufficient conditions that guarantee the evolution of the initial state towards an orthogonal one in a finite time, and pay special attention to the full characterization of the speed of evolution, offering a speed map in parameter space that highlights regions associated with faster or slower dynamics. The general analysis is applied to concrete physical settings, particularly a pair of bosons governed by an extended Bose-Hubbard Hamiltonian, and a single particle in a triple-well potential. Our findings provide a framework to explore how the energetic resources and the initial configurations shape the pace of the dynamics in higher-dimensional systems.