Tensorial dynamics on the space of quantum states

Jost

Schwachhöfer

2020

Entropy

The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on B ( H ) , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.

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

, are associated with the antisymmetric part (the Lie product) and the symmetric part (the Jordan product) of the associative product in

, respectively.…”

Section: Introductionmentioning

confidence: 99%

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

Jost

Schwachhöfer

2020

Entropy

“…[8,12,13] By using only Hamiltonian and gradient vector fields it is not possible to write Markovian dynamics, and one has to introduce Kraus vector fields. [10,8,3] This will not be the case for dynamics on the simplex, marking an important difference between "classical" and "quantum" evolutions.…”

Section: Discussionmentioning

confidence: 99%

“…The associativity of the product follows from the fact that [[f a , ·]] is (trivially) a derivation of the Jordan product ⊙. The structure constants of this product on the basis {e j } j=1,2,3 introduced before are: e 1 ⋆ e 1 = e 2 ⋆ e 2 = e 3 , e 1 ⋆ e 2 = −e 3 + ıe 3 2 , e 2 ⋆ e 1 = − e 3 + ıe 3 2 , e 3 ⋆ e j = e j ⋆ e 3 = 0 ∀j = 1, 2, 3 .…”

Section: Remark 2 It Would Be Possible To Define a Quadratic Poissonmentioning

confidence: 99%

Geometrical structures for classical and quantum probability spaces

Int. J. Quantum Inform.

Ibort

Marmo

2017

On the affine space containing the space S of quantum states of finitedimensional systems there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding relevant geometrical properties of S. Guided by Dirac's analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyse their geometrical properties. Most of our considerations will be dealt with for the simple example of a threelevel system.

“…is a contraction of the quantum Lie algebra ( [1,7,10]), and we may say that the dynamical evolution is "dissipating" the Poisson tensor Λ. This brings in the possibility of characterizing dissipation in terms of tensor fields rather than functions.…”

Section: Geometrical Description Of the Gkls Equation And Dissipationmentioning

confidence: 99%

Contact manifolds and dissipation, classical and quantum

Cruz

Marmo³

2018

Annals of Physics

Motivated by a geometric decomposition of the vector field associated with the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact Hamiltonian systems for the description of dissipative-like dynamical systems in the context of (non-necessarily exact) contact manifolds. In particular, we show how this class of dynamical systems naturally emerges in the context of Lagrangian Mechanics and in the case of nonlinear evolutions on the space of pure states of a finite-level quantum system. arXiv:1808.06822v2 [math-ph]