Geometric Quantum Mechanics

Abstract

The manifold of pure quantum states is a complex projective space endowed with the unitary-invariant geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold. Here we construct a number of examples of such geometrical features as they arise in the state spaces for spin-1/2, spin-1, and spin-3/2 systems, and for pairs of spin-1/2 systems. A study is undertaken on the geometry of entangled states, and a natural measure is assigned to the degree of entanglement of a given state for a general multi-particle system. The properties of this measure are analysed for the entangled states of a pair of spin-1/2 particles. With the specification of a quantum Hamiltonian, the resulting Schrodinger trajectory induces a Killing field, which is quasiergodic on a toroidal subspace of the energy surface. When the dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of a general state is in its applicability in various attempts to go beyond the standard quantum theory.

27 pages. Extended with additional material

References in corpus (4)

• Limits for entanglement measures
• Classical and Quantum Probability
• Structure and Properties of Hughston's Stochastic Extension of the Schrödinger Equation
• Information Content for Quantum States

Cited by in corpus (229)

• Biorthogonal Quantum Mechanics
• Feedback control of quantum state reduction
• Classification of multipartite entangled states by multidimensional determinants
• Contact Hamiltonian Mechanics
• Quantum Classical Correspondence for a non-Hermitian Bose-Hubbard Dimer
• Phase transitions and configuration space topology
• The maximally entangled symmetric state in terms of the geometric measure
• Experimental measurement of the quantum geometric tensor using coupled qubits in diamond
• Quantum Machine Learning for Particle Physics using a Variational Quantum Classifier
• Elementary Derivation for Passage Times
• Geometry of quantum systems: density states and entanglement
• Monge Metric on the Sphere and Geometry of Quantum States
• On optimum Hamiltonians for state transformations
• Geometry of Matrix Product States: metric, parallel transport and curvature
• Information geometry in vapour-liquid equilibrium
• The geometry of entanglement: metrics, connections and the geometric phase
• An elementary formula for entanglement entropies of fermionic systems
• A generalized skew information and uncertainty relation
• Geometrization of Quantum Mechanics
• Classical limit of non-Hermitian quantum dynamics - a generalised canonical structure
• Parity-time symmetry breaking in magnetic systems
• On the geometry of four qubit invariants
• The twistor geometry of three-qubit entanglement
• Properties of the general NHDM. I. The orbit space
• Time-optimal navigation through quantum wind
• A General Theory of Quantum Relativity
• Geometric Strategy for the Optimal Quantum Search
• Barycentric measure of quantum entanglement
• CP^n, or, entanglement illustrated
• Time and M-theory
• Background Independent Quantum Mechanics and Gravity
• Generalized Geometric Measure of Entanglement for Multiparty Mixed States
• Entanglement minimization in hadronic scattering with pions
• An Introduction to Quantum Entanglement: a Geometric Approach
• The geometric measure of multipartite entanglement and the singular values of a hypermatrix
• A machine learning approach to dynamical properties of quantum many-body systems
• A Note on Fully Entangled Fraction
• Spread complexity as classical dilaton solutions
• Nambu Quantum Mechanics: A Nonlinear Generalization of Geometric Quantum Mechanics
• Classification of Entanglement in Symmetric States
• From SICs and MUBs to Eddington
• Hamiltonian Quantum Dynamics With Separability Constraints
• Correlation experiments in nonlinear quantum mechanics
• On the geometry of a class of N-qubit entanglement monotones
• Alternative Structures and Bihamiltonian Systems
• Time-Dependent Hilbert Spaces, Geometric Phases, and General Covariance in Quantum Mechanics
• Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
• A comparison of old and new definitions of the geometric measure of entanglement
• Statistical ensembles in Hamiltonian formulation of hybrid quantum-classical systems
• Statistics and Nosé formalism for Ehrenfest dynamics
• Geometry of spin coherent states
• Geometrical Description of Quantum Mechanics - Transformations and Dynamics
• Quantum phase transitions without thermodynamic limits
• Geometric Entanglement in Topologically Ordered States
• Ehrenfest dynamics is purity non-preserving: a necessary ingredient for decoherence
• Differential geometry of density states
• Geometric descriptions of entangled states by auxiliaries varieties
• The quantum brachistochrone problem for an arbitrary spin in a magnetic field
• On the geometry of quantum constrained systems
• Classical Tensors and Quantum Entanglement I: Pure States
• Quantifying geometric measure of entanglement by mean value of spin and spin correlations for pure and mixed states
• Geometry and symmetry of quantum and classical-quantum variational principles
• Geometric characteristics of quantum evolution: curvature and torsion
• Why there is something so close to nothing: towards a fundamental theory of the cosmological constant
• Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension
• Visualizing Entanglement in multi-Qubit Systems
• From Classical to Quantum Information Geometry: A Guide for Physicists
• Collectibility for Mixed Quantum States
• Exposing the quantum geometry of spin-orbit coupled Fermi superfluids
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Symmetric logarithmic derivative for general n-level systems and the quantum Fisher information tensor for three-level systems
• Unitarity, ergodicity, and quantum thermodynamics
• The Structure of Classical Extensions of Quantum Probability Theory
• Beyond Density Matrices: Geometric Quantum States
• Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer
• Computable genuine multimode entanglement measure: Gaussian vs. non-Gaussian
• Entanglement of three-qubit geometry
• Existence of Universal Entangler
• The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors
• Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians
• Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation
• Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states
• Local geometry and quantum geometric tensor of mixed states
• Stochastic Reduction in Nonlinear Quantum Mechanics
• Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry
• Coherent states and rational surfaces
• Thermodynamics of Quantum Heat Bath
• Symplectic approach to quantum constraints
• Emergence of equilibrium thermodynamic properties in quantum pure states. I. Theory
• Emergence of classical behavior from the quantum spin
• Quantum computing with classical bits
• General pure multipartite entangled states and the Segre variety
• Unveiling the geometric meaning of quantum entanglement: discrete and continuous variable systems
• Entanglement in permutation symmetric states, fractal dimensions, and geometric quantum mechanics
• Exponential families, Kahler geometry and quantum mechanics
• Entanglement Classification with Algebraic Geometry
• On the geometry of mixed states and the Fisher information tensor
• Geometry of a two-spin quantum state in evolution
• Entanglement and quantum state geometry of spin system with all-range Ising-type interaction
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Branching States as The Emergent Structure of a Quantum Universe
• Accurate calculation of the geometric measure of entanglement for multipartite quantum states
• Quantum Information Dimension and Geometric Entropy
• Finslerian geometrization of quantum mechanics in the hydrodynamical representation
• On quantum microcanonical equilibrium
• Singularity of type $D_4$ arising from four qubit systems
• Quantum Control Landscapes Are Almost Always Trap Free
• Quantum state geometry and entanglement of two spins with anisotropic interaction in evolution
• An Algebraic-Geometric Characterization of Tripartite Entanglement
• Geometrical, topological and dynamical description of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model and their interplay with quantum entanglement
• Unifying Geometric Entanglement and Geometric Phase in a Quantum Phase Transition
• The Two-fold Role of Observables in Classical and Quantum Kinematics
• Quantum splines
• Four remarks on spin coherent states
• Ehrenfest statistical dynamics in chemistry: study of decoherence effects
• Convex politopes and quantum separability
• Entanglement monotones for multi-qubit states based on geometric invariant theory
• Digital finite quantum Riemannian geometries
• Constructive entanglement test from triangle inequality
• Metric approach to quantum constraints
• Geometry and speed of evolution for a spin-s system with long-range zz-type Ising interaction
• Generalized Weyl-Heisenberg algebra, qudit systems and entanglement measure of symmetric states via spin coherent states
• Duality, Quantum Mechanics and (Almost) Complex Manifolds
• Sj$\ddot{\text{o}}$qvist quantum geometric tensor of finite-temperature mixed states
• Classical Tensors and Quantum Entanglement II: Mixed States
• Electric field induced second-order anomalous Hall transport in unconventional Rashba systems
• The space of density states in geometrical quantum mechanics
• Geometrical entanglement of highly symmetric multipartite states and the Schmidt decomposition
• Measure of the density of quantum states in information geometry and its application in the quantum multi-parameter estimation
• Exact scaling of geometric phase and fidelity susceptibility and their breakdown across the critical points
• Geometrical modification of quaternionic quantum mechanics
• Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces
• Twistor geometry of the Flag manifold
• Complementarity between quantum entanglement, geometrical and dynamical appearances in N spin-$1/2$ system under all-range Ising model
• Geometric properties of evolutionary graph states and their detection on a quantum computer
• Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations
• Maximum Geometric Quantum Entropy
• Jensen-Shannon Divergence and Non-linear Quantum Dynamics
• Gaussian distributions, Jacobi group and Siegel-Jacobi space
• Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics
• Framework for Learning and Control in the Classical and Quantum Domains
• Minimum Entangling Power is Close to Its Maximum
• Duality constructions from quantum state manifolds
• The Exclusivity Principle and the Set of Quantum Correlations
• Loop Quantum Cosmology: Physics of Singularity Resolution and its Implications
• Microcanonical distributions for quantum systems
• Efficiency of neural quantum states in light of the quantum geometric tensor
• Deformation Quantization of Geometric Quantum Mechanics
• Theory of Quantum Space-Time
• Semiquantisation Functor and Poisson-Riemannian Geometry, I
• On some hydrodynamical aspects of quantum mechanics
• Hermitian - non-Hermitian interfaces in quantum theory
• Geometry of the adiabatic theorem
• From Geometric Quantum Mechanics to Quantum Information
• Introduction to Quantum Mechanics and the Quantum-Classical transition
• Frame functions in finite-dimensional Quantum Mechanics and its Hamiltonian formulation on complex projective spaces
• Experimental Entanglement Quantification for Unknown Quantum States in a Semi-Device-Independent Manner
• A general framework for quantum splines
• Compactifications of Deformed Conifolds, Branes and the Geometry of Qubits
• On the Relativity of Quantumness as Implied by Relativity of Arithmetic and Probability
• Phase space hybrid theory of quantum measurement with nonlinear and stochastic dynamics
• Segre embedding and related maps and immersions in differential geometry
• Quantum geometry in many-body systems with precursors of criticality
• Orbits of hybrid systems as qualitative indicators of quantum dynamics
• Relations between different notions of degrees of freedom of a quantum system and its classical model
• A Geometrical Derivation of a Family of Quantum Speed Limit Results
• The Quantum Eigenvalue Problem and Z-Eigenvalues of Tensors
• On Arguments for Linear Quantum Dynamics
• Quantum Superposition Principle and Geometry
• Non-Markovianity of geometrical qudit decoherence
• Derivation of the robustness from the concurrence
• Geometric analysis of entangled qubit pairs
• Geometry of quantum state manifolds generated by the Lie algebra operators
• Decoherence of a qubit as a diffusion on the Bloch sphere
• Majorana stellar representation for mixed-spin $(s,\frac{1}{2})$ systems
• Entanglement bound for multipartite pure states based on local measurements
• Efficient criteria of quantumness for a large system of qubits
• Local quantum information dynamics
• Quantifying Multipartite Quantum Entanglement in a Semi-Device-Independent Manner
• Geometric Structures Induced by Deformations of the Legendre Transform
• Dynamics and Geometry of Entanglement in Many-Body Quantum Systems
• Is there a Jordan geometry underlying quantum physics?
• $SU(4)$ description of bilayer skyrmion-antiskyrmion pairs
• Classifying Entanglement by Algebraic Geometry
• Topologies and Measurable Structures on the Projective Hilbert Space
• Geometric measure of entanglement of multipartite mixed states
• A geometric approach to quantum control in projective Hilbert spaces
• Surveying points in the complex projective plane
• Remarks on the GNS Representation and the Geometry of Quantum States
• Fragile entanglement statistics
• Universal Quantum Measurements
• The uncertainty principle and the energy identity for holomorphic maps in geometric quantum mechanics
• An Exact Five-Step Method for Classicalizing N-level Quantum Systems: Application to Quantum Entanglement Dynamics
• Lie systems and Schrödinger equations
• Nonlinear evolution of coarse-grained quantum systems with generalized purity constraints
• Particles as superoscillations of spacetime with a nonlocal metric?
• Cloning in nonlinear Hamiltonian quantum and hybrid mechanics
• Poincaré polynomials for Abelian symplectic quotients of pure $r$-qubits via wall-crossings
• Constrained quantum dynamics
• Tensorial characterization and quantum estimation of weakly entangled qubits
• Classical tensors from quantum states
• Visualizing Quantum States: A Pilot Study on Problem Solving in Quantum Information Science Education
• Constraints on a Generalization of Geometric Quantum Mechanics from Neutrino and $B^0$-$\overline{B^0}$ Oscillations
• Shapes of Quantum States
• The Geometry of Stochastic Reduction of an Entangled System
• Probabilistic Simulation of Quantum Computation
• Complex extension of Wigner's theorem
• Space of events and the time observable
• A Quantum is a Complex Structure on Classical Phase Space
• A kinetic theory for quantum information transport
• Cohomological invariants of Abelian symplectic quotients of pure $r$-qubits
• Geometry and Dynamics of Quantum State Diffusion
• A detailed analysis of mathematics of entanglement in Non-Hermitian systems in real eigenvalue regime
• Topological phases and entanglement in real space for 1D SSH topological insulator: effects of first and second neighbor-hoppings
• Evolution of quantum geometric tensor of 1D periodic systems after a quench
• Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement
• Qubit Geometry through Holomorphic Quantization
• Geometric viewpoint on the quantization of a fuzzy logic
• Quantum Mechanics Relative to a Quantum Reference System: a Relative State Approach
• Geometric representation of higher-order optical modes
• Geometry and quantum brachistochrone analysis of multiple entangled spin-1/2 particles under all-range Ising interaction
• PT symmetry and the evolution speed in open quantum systems
• Quantum Measurement of Space-Time Events
• Geometrical Properties of Feynman Path Integrals
• Quantum Computation
• Quantum states and space-time causality
• Measuring distance between quantum states on a quantum computer
• Dequantisation of the Dirac Monopole
• Kähler Representation Theory