arXiv:2206.10394 · Mathematics

Group Actions and Monotone Quantum Metric Tensors

Florio M. Ciaglia, Fabio Di Nocera

Published Jul 26, 2022• 2 citations

Abstract

The interplay between actions of Lie groups and monotone quantum metric tensors on the space of faithful quantum states of a finite-level system observed in recent works is here further developed.

DOI: 10.3390/math10152613

Related papers

Quantum states, groups and monotone metric tensors

Florio M. Ciaglia

Abstract A novel link between monotone metric tensors and actions of suitable extensions of the unitary group on the manifold of faithful quantum states is presented here by means of three illustrative examples related with the Bures–Helstrom metric tensor, the Wigner–Yanase metric tensor, and the Bogoliubov–Kubo–Mori metric tensor.

arXiv:2006.10595The European Physical Journal PlusJun 1, 202011 citations

Group Actions and Monotone Metric Tensors: The Qubit Case

Florio M. Ciaglia, Fabio Di Nocera

arXiv:2105.15067Lecture notes in computer scienceJan 1, 20211 citation

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Universidad Carlos III de Madrid, Spain, Florio M. Ciaglia, Jürgen Jost +1 more

arXiv:2112.09781Symmetry Integrability and Geometry Methods and ApplicationsOct 20, 20232 citations

Unitary representations of lie groups in quantum mechanics

Lochlainn O'Raifeartaigh

Springer eBooksApr 6, 20086 citations

Lie groups and quantum mechanics

Maria Clara Nucci, P. G. L. Leach

Journal of Mathematical Analysis and ApplicationsApr 28, 201310 citations

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

Florio M. Ciaglia, Jürgen Jost, Lorenz Schwachhöfer

arXiv:2005.02023EntropyJun 8, 202020 citations

arXiv:2206.10394 · Mathematics

Group Actions and Monotone Quantum Metric Tensors

Florio M. Ciaglia, Fabio Di Nocera

Published Jul 26, 2022• 2 citations

arXiv HTML PDF

Abstract

The interplay between actions of Lie groups and monotone quantum metric tensors on the space of faithful quantum states of a finite-level system observed in recent works is here further developed.

DOI: 10.3390/math10152613

Related papers

Quantum states, groups and monotone metric tensors

Florio M. Ciaglia

arXiv:2006.10595The European Physical Journal PlusJun 1, 202011 citations

Group Actions and Monotone Metric Tensors: The Qubit Case

Florio M. Ciaglia, Fabio Di Nocera

arXiv:2105.15067Lecture notes in computer scienceJan 1, 20211 citation

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Universidad Carlos III de Madrid, Spain, Florio M. Ciaglia, Jürgen Jost +1 more

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra ${\mathcal J}$ , we exploit the generalized distribution determined by the Jordan product on the dual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of ${\mathcal J}$ , in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on ${\mathcal J}$ , the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of ${\mathcal J}$ , it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra ${\mathcal J}$, we exploit the generalized distribution determined by the Jordan product on the dual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of ${\mathcal J}$, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on ${\mathcal J}$, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of ${\mathcal J}$, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.

arXiv:2112.09781Symmetry Integrability and Geometry Methods and ApplicationsOct 20, 20232 citations

Unitary representations of lie groups in quantum mechanics

Lochlainn O'Raifeartaigh

Springer eBooksApr 6, 20086 citations

Lie groups and quantum mechanics

Maria Clara Nucci, P. G. L. Leach

Journal of Mathematical Analysis and ApplicationsApr 28, 201310 citations

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

Florio M. Ciaglia, Jürgen Jost, Lorenz Schwachhöfer

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.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.

arXiv:2005.02023EntropyJun 8, 202020 citations

{\mathcal J}

, we exploit the generalized distribution determined by the Jordan product on the dual

{\mathcal J}^{\star}

to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of

{\mathcal J}

, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in

{\mathcal J}^{*}

lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on

{\mathcal J}

, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of

{\mathcal J}

, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.