基于 GNS 构造的 \(C^*\) -代数上光滑参数统计模型的几何张量
Metric tensors and two-forms in information geometry from the GNS construction
中文总结 AI 辅助
研究在 \(C^*\) -代数上光滑参数统计模型的几何张量，基于 GNS 构造，通过拉回对偶 GNS 厄米积产生厄米张量 \(K\)，进而定义度量张量 \(G\) 和二形式 \(\Omega\)，恢复多种度量，还研究了其闭性等性质。
中文摘要 AI 辅助
我们在 \(C^*\) -代数上的光滑参数统计模型上，基于 GNS 构造开发了几何张量。由于 \(C^*\) -代数的态空间通常不是光滑流形，该构造不依赖于从周围态流形拉回张量。GNS 希尔伯特空间及其对偶被组织成态空间上的非局部平凡希尔伯特纤维丛。对于满足将期望值导数表示为实化 GNS 纤维上连续泛函的相容性条件的模型，每个切向量都有一个规范的对偶 GNS 代表。沿着相应的规范提升拉回对偶 GNS 厄米积，在模型的复化切丛上产生一个厄米张量 \(K\)，在适当的正则性假设下，其实部和虚部分别定义一个光滑的弱黎曼度量张量 \(G\) 和一个光滑的二形式 \(\Omega\)。在有限维参数流形中，度量当然是强的。该构造在交换主导情况下恢复了 Fisher - Rao 度量，在对偶 GNS 配对施加的归一化和符号约定下恢复了纯态的 Fubini - Study 几何，以及忠实量子态的 SLD 度量。在有限维忠实模型中，二形式 \(\Omega\) 与 SLD 代表的期望对易子成比例，等价于平均 Uhlmann 曲率。通过忠实量子比特和位移热态表明 \(\Omega\) 不一定是闭的。对于丛正则模型，实对偶 GNS 丛上相关的纤维辛形式允许依赖于联络的闭扩展到全空间，而 \(\Omega\) 在参数流形上的闭性由规范实对偶 GNS 提升的协变外导数控制。
英文摘要
We develop a GNS-based construction of geometric tensors on smooth parametric statistical models over $C^*$-algebras. Since the state space of a $C^*$-algebra is generally not a smooth manifold, the construction does not rely on pulling back tensors from an ambient state manifold. Instead, the GNS Hilbert spaces and their duals are organized into non-locally-trivial Hilbert fibrations over the state space. For models satisfying a compatibility condition expressing derivatives of expectation values as continuous functionals on the realified GNS fibers, each tangent vector admits a canonical dual GNS representative. Pulling back the dual GNS Hermitian product along the corresponding canonical lift produces a Hermitian tensor $K$ on the complexified tangent bundle of the model, whose real and imaginary parts define, under suitable regularity assumptions, a smooth weak Riemannian metric tensor $G$ and a smooth two-form $Ω$. In finite-dimensional parameter manifolds the metric is, of course, strong. The construction recovers the Fisher--Rao metric in the commutative dominated case, the Fubini--Study geometry for pure states up to the normalization and sign convention imposed by the dual GNS pairing, and the SLD metric for faithful quantum states. In finite-dimensional faithful models, the two-form $Ω$ is proportional, up to convention, to the expected commutator of the SLD representatives, equivalently to the mean Uhlmann curvature. We show through faithful qubits and displaced thermal states that $Ω$ need not be closed. For bundle-regular models, the associated fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, while closedness of $Ω$ on the parameter manifold is controlled by the covariant exterior derivative of the canonical real dual GNS lift.
