Differential Geometry of Quantum States, Observables and Evolution

Abstract

The geometrical description of Quantum Mechanics is reviewed and proposed as an alternative picture to the standard ones. The basic notions of observables, states, evolution and composition of systems are analised from this perspective, the relevant geometrical structures and their associated algebraic properties are highlighted, and the Qubit example is thoroughly discussed. Note. At the time of the creation and submission of this work to the Lecture Notes of theUnione Matematica Italiana 25, F. M. Ciaglia was not yet affiliated with the Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig with which he is affiliated at the time of the submission of this work to arXiv.

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

“…From a purely theoretical point of view, there is no need to restrict our attention to geometrical structures on quantum states associated with covariant tensor fields as in the case of metric tensors discussed above. Indeed, in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ], the associative product of the algebra of linear operators on the finite-dimensional Hilbert space associated with a quantum system has been suitably exploited to define two contravariant tensor fields on the space of self-adjoint operators on , and these tensor fields have been used to give a geometrical description of the Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation describing the dynamical evolution of open quantum systems (see [ 19 , 21 , 22 , 24 , 26 , 27 , 28 ]). These two tensor fields, named and , are associated with the antisymmetric part (the Lie product) and the symmetric part (the Jordan product) of the associative product in , respectively.…”

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

“…In addition to the action of the special unitary group SU (H), it has recently been shown that there is an action of the special linear group SL(H) on the space S (H) of quantum states 1 [16,21,29,30,31]. This action does not preserve the convex structure of S (H), and it allows us to partition it into the disjoint union of smooth manifolds labelled by the rank of the quantum states (density operators) they contain.…”

“…With a little effort [16,21,29,30,31] it is possible to prove that the orbits of SL(H) on S (H) by means of the action given in equation (32) are labelled by the rank of the quantum states they contain, and that they are differential manifolds. We will denote these orbits as S k (H), where k denotes the rank of the quantum states in S k (H).…”

“…It has been argued elsewhere that even though the action of the SL(H) group does not preserve the metric structure, it preserves Hermiticity, positivity and rank of the states. Recently, it has been proved that there is a (nonlinear) action of the special linear group SL(H) on the whole S (H) [16,21,29]. This Lie group is the complexification of the special unitary group and its action on S (H) allows us to build the submanifolds of quantum states with fixed rank.…”

From the Jordan product to Riemannian geometries on classical and quantum states

“…From a purely theoretical point of view, there is no need to restrict our attention to geometrical structures on quantum states associated with covariant tensor fields as in the case of metric tensors discussed above. Indeed, in [19,21,23,24,25,28,31], the associative product of the algebra B(H) of linear operators on the finite-dimensional Hilbert space H associated with a quantum system has been suitably exploited to define two contravariant tensor fields on the space of self-adjoint operators on H, and these tensor fields have been used to give a geometrical description of the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation describing the dynamical evolution of open quantum systems (see [19,23,24,28,33,49,60]). These two tensor fields, named Λ and R, are associated with the antisymmetric part (the Lie product) and the symmetric part (the Jordan product) of the associative product in B(H), respectively.…”