Ciaglia Florio M, Jost Jürgen, Schwachhöfer Lorenz
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany.
Faculty for Mathematics, TU Dortmund University, 44221 Dortmund, Germany.
Entropy (Basel). 2020 Jun 8;22(6):637. doi: 10.3390/e22060637.
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.
有限维C* -代数A的自伴部分上的约当积被证明能在A上合适的态流形上产生黎曼度量张量，并且明确计算了所有这些度量张量的协变导数、测地线、黎曼张量和截面曲率。特别地，证明了在阿贝尔情形下可恢复费希尔 - 拉奥度量张量，当考虑有限维希尔伯特空间H上线性算子的代数B(H)上的纯态时可恢复富比尼 - 施图迪度量张量，以及当考虑B(H)上的忠实态时可恢复布雷斯 - 赫尔斯托姆度量张量。此外，给出了根据与一个态相关的GNS构造对这些黎曼度量张量的另一种推导。在B(H)上的纯态和忠实态的情形下，这种替代的几何描述阐明了富比尼 - 施图迪度量张量与布雷斯 - 赫尔斯托姆度量张量之间的类比。
