Classical Tensors and Quantum Entanglement I: Pure States

Abstract

The geometrical description of a Hilbert space associated 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.

Stratified manifold of quantum states, actions of the complex special linear group

“…Composite quantum systems and quantum entanglement manipulation are of fundamental importance in the context of quantum information theory and the geometry of quantum entanglement is a fascinating and very complex subject [3,6,11,29,30,32,33,34,37,46,47,48,49,50,51,55]. Here we want to present some possible connections between the geometry of quantum entanglement and the geometrical tools introduced in the previous sections.…”

“…Our understanding of the geometry of the space of quantum states is in constant evolution and there are different fields of application in which it is possible to use the knowledge we gain. For instance, geometrical ideas have been successfully exploited when addressing the foundations of quantum mechanics [4,7,8,10,13,20,22,23,25,31,35,40], quantum information theory [5,15,19,27,36,39,43,45,54], quantum dynamics [9,12,14,16,17,18,24], entanglement theory [3,6,11,29,30,34,48,49].…”

Fisher metric, geometric entanglement, and spin networks

“…dρ 0 = 0 being ρ 0 our fixed fiducial state. The pull-back of the Hermitian tensor (4) on the orbit submanifold embedded in D 1 (H) ∼ = R(H) then yields [29,55]…”

Tensorial description of quantum mechanics

“…The space of pure states of a quantum system (usually identified with rays of a Hilbert space H or with rank-one projectors defined on H) is not a linear space, rather it is a Hilbert manifold (i.e., a differential manifold whose tangent spaces (at each point) are endowed with a Hilbert space structure). Additional instances of manifolds of states, instead of a representation as elements of linear spaces, are provided by coherent states and generalized coherent states (see [1,2]). Further, the space of entangled states, in composite systems, does not carry a linear structure and therefore a manifold point of view may help in studying their properties.…”