Quantum tensor product structures are observable-induced

Tensor-product vs. geometric-product coding

Quantum computation is based on tensor products and entangled states. We discuss an alternative to the quantum framework where tensor products are replaced by geometric products and entangled states by multivectors. The resulting theory is analogous to quantum computation but does not involve quantum mechanics. We discuss in detail similarities and differences between the two approaches and illustrate the formulas by explicit geometric objects where multivector versions of the Bell-basis, GHZ, and Hadamard states are visualized by means of colored oriented polylines.

Light cone tensor network and time evolution

The transverse folding algorithm [Phys. Rev. Lett. 102, 240603] is a tensor network method to compute time-dependent local observables in out-of-equilibrium quantum spin chains that can sometimes overcome the limitations of matrix product states. We present a contraction strategy that makes use of the exact light cone structure of the tensor network representing the observables. The strategy can be combined with the hybrid truncation proposed for global quenches in [Phys. Rev. A 91, 032306], which significantly improves the efficiency of the method. We demonstrate the performance of this trans…

Tensor network techniques for the computation of dynamical observables in 1D quantum spin systems

We analyze the recently developed folding algorithm [Phys. Rev. Lett. 102, 240603 (2009)] to simulate the dynamics of infinite quantum spin chains, and relate its performance to the kind of entanglement produced under the evolution of product states. We benchmark the accomplishments of this technique with respect to alternative strategies using Ising Hamiltonians with transverse and parallel fields, as well as XY models. Additionally, we evaluate its ability to find ground and thermal equilibrium states.

Capturing exponential variance using polynomial resources: applying tensor networks to non-equilibrium stochastic processes

Estimating the expected value of an observable appearing in a non-equilibrium stochastic process usually involves sampling. If the observable's variance is high, many samples are required. In contrast, we show that performing the same task without sampling, using tensor network compression, efficiently captures high variances in systems of various geometries and dimensions. We provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size. In particular, the high variance observable $\mathrm{e}^{-βW}$, motivated by Ja…

TENSKEL：用于结构化测量与重建的组合可观测张量

TENSKEL: A Combinatorial Observable Tensor for Structured Measurement and Reconstruction

TENSKEL是一种基于离散域张量表示的组合可观测框架，可明确测量诱导的潜构型组织，为结构化测量与重建提供数学基础，在多领域具有潜在应用

The tensor rank of tensor product of two three-qubit W states is eight

We show that the tensor rank of tensor product of two three-qubit W states is not less than eight. Combining this result with the recent result of M. Christandl, A. K. Jensen, and J. Zuiddam that the tensor rank of tensor product of two three-qubit W states is at most eight, we deduce that the tensor rank of tensor product of two three-qubit W states is eight. We also construct the upper bound of the tensor rank of tensor product of many three-qubit W states.

Tensor rank is not multiplicative under the tensor product

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not …

高斯普适性及其在张量网络机器学习中的破缺

Gaussian Universality and Its Breakdown in Tensor-Network Machine Learning

本文解析研究了张量网络机器学习中高斯普适性的出现与破缺条件，证明大键维下模型收敛于高斯过程，而大物理维下全局可观测量出现非高斯修正，揭示其受参数数、架构缩放、可观测量局域性和谱性质共同控制。

Interaction of Multiple Tensor Product Operators of the Same Type: an Introduction

Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly diffe…

关于一般协变量子系统的张量结构

On the tensorial structure of general covariant quantum systems

本文探讨了量子系统中可观测量代数的结构与其希尔伯特空间张量分解的关系，指出哈密顿量可能影响这种分解，但需谨慎处理背景无关情况下的哈密顿约束问题，强调定义可观测量及其与动力学的相互作用对量子理论的重要性。