Tensor Product Structure and Non-associative Entanglement Degree非结合代数的张量积结构与非结合纠缠度

Abstract

Abstract This paper studies the associator operator structure on the two-body state space of the three-dimensional dissipative algebra (TDDA), defined by three axioms , , . Three core results are obtained. First, the decomposition theorem shows that the associator operator of pure tensors can be split into two tensor-product terms with fully controllable rank structures. Second, the direction-locking theorem indicates any single-body associator always lies on a fixed one-dimensional ray uniquely determined by the intermediate element , and its amplitude is governed by the area determinant of the non-classical projection plane. Third, symbolic calculation and massive random-matrix experiments lead to a dimension experimental law: the maximum rank of for generic entangled two-body states is 7, strictly smaller than the dimension 8 of the constraint shell . The rank degenerates to 6 or 5 when one or both sides have collinear distributions. Based on the hierarchical rank classification, we propose the non-associative entanglement degree taking values , as an intrinsic algebraic metric exclusive to non-associative algebras. All algebraic identities and numerical results are double-validated via symbolic programs and random matrix samplin 摘要 本文基于三条核心公理 定义三维耗散代数 TDDA,系统研究其两体态空间 内结合算子的完整结构。全文得出三项核心结论:第一,纯张量形式的结合子存在标准张量分解式,算子秩可通过两项张量积结构精确管控;第二,单体结合子满足方向锁定规则,输出矢量方向仅由中间代数元唯一确定,耦合强度由平面行列式面积因子决定;第三,大量符号与随机数值实验证明,两体结合子算子秩存在刚性上界 7,该空间被 8 维约束壳完整包裹,内部存在固有线性约束造成维度损失;当体系分量发生几何退化时,算子秩会同步降至 6、5 两档。依托秩分层规律,本文原创定义非结合纠缠度 ,取值仅为 0、1、2 三档,作为非结合代数专属关联度量。全部代数恒等式、维度实验均经过符号程序与大规模随机矩阵双重核验。