Tensors with unit Frobenius norm are fundamental objects in many fields, including scientific computing and quantum physics, which are able to represent normalized eigenvectors and pure quantum states. While the tensor train decomposition provides a powerful low-rank format for tackling high-dimensional problems, it does not intrinsically enforce the unit-norm constraint. To address this, we introduce the normalized tensor train (NTT) decomposition, which aims to approximate a tensor by unit-norm tensors in tensor train format. The low-rank structure of NTT decomposition not only saves storage and computational cost but also preserves the underlying unit-norm structure. We prove that the set of fixed-rank NTT tensors forms a smooth manifold, and the corresponding Riemannian geometry is derived, paving the way for geometric methods. We propose NTT-based methods for low-rank tensor recovery, high-dimensional eigenvalue problem, estimation of stabilizer rank, and calculation of the minimum output R\'enyi 2-entropy of quantum channels. Numerical experiments demonstrate the superior efficiency and scalability of the proposed NTT-based methods.

翻译：具有单位Frobenius范数的张量是许多领域的基本对象，包括科学计算和量子物理，它们能够表示归一化特征向量和纯量子态。虽然张量列车分解为解决高维问题提供了强大的低秩格式，但其本身并不强制执行单位范数约束。为此，我们引入了归一化张量列车（NTT）分解，旨在通过张量列车格式中的单位范数张量来逼近一个张量。NTT分解的低秩结构不仅节省了存储和计算成本，还保留了基础的单位范数结构。我们证明了固定秩NTT张量的集合形成了一个光滑流形，并推导了相应的黎曼几何，为几何方法铺平了道路。我们提出了基于NTT的方法，用于低秩张量恢复、高维特征值问题、稳定子秩的估计以及量子信道的最小输出Rényi 2-熵的计算。数值实验证明了所提出的基于NTT的方法具有卓越的效率和可扩展性。