Dynamical Vector Fields on the Manifold of Quantum States

Abstract

In this paper we shall consider the stratified manifold of quantum states and the vector fields which act on it. In particular, we show that the infinitesimal generator of the GKLS evolution is composed of a generator of unitary transformations plus a gradient vector field along with a Kraus vector field transversal to the strata defined by the involutive distribution generated by the former ones.Comment: 30 pages, 2 figures, comments are welcom

Differential Geometry of Quantum States, Observables and Evolution

“…For a system with n levels (dim H = n) we would get n different orbits. The one of maximal dimension would be the bulk, while the boundary of the closed convex body S of quantum states would be the union of orbits of dimensions less than n. The geometry of S as developed in [6,7,16] will be exposed in Sect. 2.1.…”

“…We will briefly recall here the results of [6,7] concerning the geometry of the space of all states, pure or mixed for the qubit. Every 2 by 2 Hermitean matrix A may be written in the form:…”

“…It is now possible to construct a Lie-Jordan algebra (see for instance [6,7,14]) with commutative Jordan product • and Lie product {·, ·} on the space of observables (affine functions) out of the tensors R and Λ. Such algebra is defined by:…”

“…The nonlinearity pops up because the two maps are not trace preserving therefore we have to introduce a denominator for the map to transform states into states. The "miracle" of the Kossakowski-Lindblad form of the equation is that the two nonlinearities cancel each other so that the resulting vector field is actually linear [7,6].…”

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

“…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].…”