Qubit Geodesics on the Bloch Sphere from Optimal-Speed Hamiltonian Evolutions
arXiv:2210.09142 · doi:10.1088/1361-6382/acce1a
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.
arXiv comment19 pages, 1 figure, 2 tables
References in corpus (37)
- Quantum speed limit for physical processes
- Quantum speed limit for non-Markovian dynamics
- Quantum speed limits in open system dynamics
- Faster than Hermitian Quantum Mechanics
- Bloch vectors for qudits
- Quantum Brachistochrone
- Wigner-Yanase information on quantum state space:the geometric approach
- Geometry of the generalized Bloch sphere for qutrits
- Elementary Derivation for Passage Times
- On optimum Hamiltonians for state transformations
- Action quantum speed limits
- The Parity Operator: applications in quantum metrology
- Quantum Brachistochrone for Mixed States
- Energy-Sensitive and "Classical-like" Distances Between Quantum States
- Jacobi Fields on Statistical Manifolds of Negative Curvature
- Bargmann Invariants and Geometric Phases - a Generalised Connection
- Three-dimensional visualization of a qutrit
- Geometry of the set of mixed quantum states: An apophatic approach
- On Grover's Search Algorithm from a Quantum Information Geometry Viewpoint
- On Hamiltonians Generating Optimal-Speed Evolutions
- Extensions of the Mandelstam-Tamm quantum speed limit to systems in mixed states
- Evolution speed of open quantum dynamics
- Geometric Algebra and Information Geometry for Quantum Computational Software
- Quantum gravity on a square graph
- Quantifying The Complexity Of Geodesic Paths On Curved Statistical Manifolds Through Information Geometric Entropies and Jacobi Fields
- Geometric characteristics of quantum evolution: curvature and torsion
- Extending the Fisher metric to density matrices
- An algorithm for solving unconstrained unitary quantum brachistochrone problems
- Geometry along Evolution of Mixed Quantum States
- Geometric aspects of analog quantum search evolutions
- Complexity of Pure and Mixed Qubit Geodesic Paths on Curved Manifolds
- Quantum geodesic flows and curvature
- Theoretical analysis of a nearly optimal analog quantum search
- Transition Probabilities in Generalized Quantum Search Hamiltonian Evolutions
- Null Phase Curves and Manifolds in Geometric Phase Theory
- Geometric decomposition of geodesics and null phase curves using Majorana star representation
- Optimal-speed unitary quantum time evolutions and propagation of light with maximal degree of coherence