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

Chruściński, Dariusz; Ciaglia, Florio M.; Ibort, Alberto; Marmo, Giuseppe; Ventriglia, F.

doi:10.1016/j.aop.2018.11.015

Cited by 14 publications

(26 citation statements)

References 53 publications

(110 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will now exploit the Jordan–Lie-algebra structure of

introduced above to obtain geometric tensor fields on

, specifically, we obtain a symmetric, contravariant bivector field

associated with the Jordan product

, and a Poisson bivector field

associated with the Lie product

. This is the generalization to a generic (finite-dimensional)

-algebra

of what is done in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ] for the specific case

for a finite-dimensional Hilbert space

. Then, we will show how the manifolds of positive linear functionals introduced in the previous section may be interpreted as a sort of analogs of symplectic leaves for the symmetric tensor

in a sense that will be specified later.…”

Section: From the Jordan Product To Riemannian Geometriesmentioning

confidence: 99%

“…If

for some finite-dimensional Hilbert space

, it is a matter of direct inspection to show that tensor fields

and

defined above coincide with those introduced in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ].…”

Section: From the Jordan Product To Riemannian Geometriesmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…The manifold of quantum states with maximal rank thus obtained coincides with the manifold of invertible quantum states on which Petz’s metric tensors are defined. Furthermore, it is possible to link the tensor fields

and

with the action of

by showing that the vector fields associated with linear functions by means of

and

provide the representation of the Lie algebra of

integrating to the action of

described above (see [ 20 , 22 ]).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Ciaglia

Jost

Schwachhöfer

2020

Entropy

View full text Add to dashboard Cite

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.

show abstract

“…We will now exploit the Jordan–Lie-algebra structure of

introduced above to obtain geometric tensor fields on

, specifically, we obtain a symmetric, contravariant bivector field

associated with the Jordan product

, and a Poisson bivector field

associated with the Lie product

. This is the generalization to a generic (finite-dimensional)

-algebra

of what is done in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ] for the specific case

for a finite-dimensional Hilbert space

. Then, we will show how the manifolds of positive linear functionals introduced in the previous section may be interpreted as a sort of analogs of symplectic leaves for the symmetric tensor

in a sense that will be specified later.…”

Section: From the Jordan Product To Riemannian Geometriesmentioning

confidence: 99%

“…If

for some finite-dimensional Hilbert space

, it is a matter of direct inspection to show that tensor fields

and

defined above coincide with those introduced in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ].…”

Section: From the Jordan Product To Riemannian Geometriesmentioning

confidence: 99%

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%

and

with the action of

by showing that the vector fields associated with linear functions by means of

and

provide the representation of the Lie algebra of

integrating to the action of

described above (see [ 20 , 22 ]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Ciaglia

Jost

Schwachhöfer

2020

Entropy

View full text Add to dashboard Cite

show abstract

“…where the ε jk l are the structure constants of SU(H) and we raise and lower indices with the Euclidean metric in R 3 . Note that, according to the work in [9,11], the Hamiltonian vector field may be written as X A = Λ(df A , ·) where f A = a j u j and Λ is the Poisson tensor Λ = ε jkl x j ∂ ∂x k ∧ ∂ ∂x l on T 1 (H), while the Gradient vector field may be written as…”

Section: Geometrical Description Of the Gkls Equation And Dissipationmentioning

confidence: 99%

Contact manifolds and dissipation, classical and quantum

Ciaglia

Cruz

Marmo³

2018

Annals of Physics

Self Cite

View full text Add to dashboard Cite

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]

show abstract