Classical Tensors and Quantum Entanglement I: Pure States

Abstract

The geometrical description of a Hilbert space asociated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

16 pages, 1 figure, to appear in Int. J. Geom. Meth. Mod. Phys

References in corpus (10)

• Quantum entanglement
• An introduction to the tomographic picture of quantum mechanics
• Unifying several separability conditions using the covariance matrix criterion
• Geometrical aspects of entanglement
• Geometry of quantum systems: density states and entanglement
• Geometrization of Quantum Mechanics
• Separability criteria based on the Bloch representation of density matrices
• Classical Tensors and Quantum Entanglement I: Pure States
• Geometrization of some quantum mechanics formalism
• A Curious Geometrical Fact About Entanglement

Cited by in corpus (16)

• Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics
• Optimal Generators for Quantum Sensing
• Geometrical Description of Quantum Mechanics - Transformations and Dynamics
• Classical Tensors and Quantum Entanglement I: Pure States
• Fisher Metric, Geometric Entanglement and Spin Networks
• Tensorial dynamics on the space of quantum states
• A geometric Hamiltonian description of composite quantum systems and quantum entanglement
• Geometry of the parameter space of a quantum system: Classical point of view
• Stratified Manifold of Quantum States, actions of the complex special linear group
• Invariant measures on multimode quantum Gaussian states
• Convex politopes and quantum separability
• Classical Tensors and Quantum Entanglement II: Mixed States
• From Geometric Quantum Mechanics to Quantum Information
• Tensorial description of quantum mechanics
• Tensorial characterization and quantum estimation of weakly entangled qubits
• Entanglement of extremal density matrices of 2-qubit Hamiltonian with Kramers degeneracy