Complexity of Pure and Mixed Qubit Geodesic Paths on Curved Manifolds
arXiv:2209.10661 · doi:10.1103/PhysRevD.106.096004
Abstract
It is known that mixed quantum states are highly entropic states of imperfect knowledge (i.e., incomplete information) about a quantum system, while pure quantum states are states of perfect knowledge (i.e., complete information) with vanishing von Neumann entropy. In this paper, we propose an information geometric theoretical construct to describe and, to a certain extent, understand the complex behavior of evolutions of quantum systems in pure and mixed states. The comparative analysis is probabilistic in nature, it uses a complexity measure that relies on a temporal averaging procedure along with a long-time limit, and is limited to analyzing expected geodesic evolutions on the underlying manifolds. More specifically, we study the complexity of geodesic paths on the manifolds of single-qubit pure and mixed quantum states equipped with the Fubini-Study metric and the Sjoqvist metric, respectively. We analytically show that the evolution of mixed quantum states in the Bloch ball is more complex than the evolution of pure states on the Bloch sphere. We also verify that the ranking based on our proposed measure of complexity, a quantity that represents the asymptotic temporal behavior of an averaged volume of the region explored on the manifold during the evolution of the systems, agrees with the geodesic length-based ranking. Finally, focusing on geodesic lengths and curvature properties in manifolds of mixed quantum states, we observed a softening of the complexity on the Bures manifold compared to the Sjoqvist manifold.
arXiv comment34 pages, 3 tables
References in corpus (51)
- On the volume of the set of mixed entangled states
- Complexity Equals Action
- Complexity and Shock Wave Geometries
- Localized shocks
- Complexity, action, and black holes
- Quantum Computation as Geometry
- Quantum speed limit for physical processes
- Quantum speed limit for non-Markovian dynamics
- Quantum speed limits in open system dynamics
- Measuring thermodynamic length
- Towards Complexity for Quantum Field Theory States
- Geometric approach to Hamiltonian dynamics and statistical mechanics
- Quantum Speed Limits Across the Quantum-to-Classical Transition
- Quantum Complexity and Negative Curvature
- Bures volume of the set of mixed quantum states
- Quantum Speed Limit is Not Quantum
- Bures metric over thermal state manifolds and quantum criticality
- The Complexity Geometry of a Single Qubit
- Renormalized Circuit Complexity
- Action quantum speed limits
- Time-dependent Hamiltonians with 100% evolution speed efficiency
- Quantum Brachistochrone for Mixed States
- Jacobi Fields on Statistical Manifolds of Negative Curvature
- Information geometric methods for complexity
- Works on an information geometrodynamical approach to chaos
- How difficult is it to prepare a quantum state?
- Extensions of the Mandelstam-Tamm quantum speed limit to systems in mixed states
- Evolution speed of open quantum dynamics
- Taking the temperature of a pure quantum state
- The Effect Of Microscopic Correlations On The Information Geometric Complexity Of Gaussian Statistical Models
- Quantifying The Complexity Of Geodesic Paths On Curved Statistical Manifolds Through Information Geometric Entropies and Jacobi Fields
- Information Geometry Aspects of Minimum Entropy Production Paths from Quantum Mechanical Evolutions
- Information geometry for Fermi-Dirac and Bose-Einstein quantum statistics
- An algorithm for solving unconstrained unitary quantum brachistochrone problems
- Geometry along Evolution of Mixed Quantum States
- Information-Geometric Indicators of Chaos in Gaussian Models on Statistical Manifolds of Negative Ricci Curvature
- Information Geometry, Inference Methods and Chaotic Energy Levels Statistics
- Information Geometric Modeling of Scattering Induced Quantum Entanglement
- Complexity and efficiency of minimum entropy production probability paths from quantum dynamical evolutions
- Geometric aspects of analog quantum search evolutions
- Quantum Chaos at Finite Temperature
- Information Geometry of Quantum Entangled Gaussian Wave-Packets
- Theoretical investigations of an information geometric approach to complexity
- Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions
- Theoretical analysis of a nearly optimal analog quantum search
- Classical and quantum speed limits
- On The Complexity Of Statistical Models Admitting Correlations
- Dynamics of Bloch vector in thermal Jaynes-Cummings model
- Bures Metrics for Certain High-Dimensional Quantum Systems
- Is Quantum Chaos Weaker Than Classical Chaos?
- Quantum sensing of curvature