论文标题
从约旦产品到经典和量子状态的Riemannian几何形状
From the Jordan product to Riemannian geometries on classical and quantum states
论文作者
论文摘要
约旦在有限维$ c^{*} $ - 代数$ \ mathscr {a} $的自相关部分中的产品显示出在适当的国家对$ \ \ \ mathscr {a} $的riemannian度量张量产生公制张量是明确计算的。特别是，证明在Abelian案件中恢复了Fisher-raO公制张量，当我们考虑在代数$ \ Mathcal {b}上纯净的状态时，将恢复fubini-study公制张量。 BURES-当我们在$ \ Mathcal {b}上考虑忠实的状态（\ Mathcal {H}）$时，将恢复Helstrom度量张量。此外，提出了这些riemannian度量张量的替代推导，就与国家相关的GNS结构而言。如果在$ \ MATHCAL {B}（\ MATHCAL {H}）$上纯粹和忠实的状态，此替代的几何描述阐明了Fubini-and-dudy-study和Bures-Helstrom Metror Metror Metric Tensor之间的类比。
The Jordan product on the self-adjoint part of a finite-dimensional $C^{*}$-algebra $\mathscr{A}$ is shown to give rise to Riemannian metric tensors on suitable manifolds of states on $\mathscr{A}$, 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 $\mathcal{B}(\mathcal{H})$ of linear operators on a finite-dimensional Hilbert space $\mathcal{H}$, and that the Bures--Helstrom metric tensors is recovered when we consider faithful states on $\mathcal{B}(\mathcal{H})$. 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 $\mathcal{B}(\mathcal{H})$, this alternative geometrical description clarifies the analogy between the Fubini--Study and the Bures--Helstrom metric tensor.
