单向量子自动机中对称下的行为记忆
Behavioral Memory under Symmetry in One-Way Quantum Automata
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中文总结 AI 辅助
该研究针对单向量子自动机，通过算子代数理论分析对称下的行为记忆，揭示非交换性的最坏情况经典代价为一个状态，且对称可改变最坏记忆尺度。
中文摘要 AI 辅助
在紧致对称下，可观测行为简化为一个不变算子代数，但其维度并非经典记忆：部分坐标被动态冻结，部分对阈值测试不可见，部分已属经典范畴。我们构建了一种算子代数理论，通过三个过滤器分离这些效应。对于一个自动机，行为是前缀可达态与后缀可观测效应之间的希尔伯特-施密特配对，其秩等于无可控性或可观性假设下的实汉克尔秩。在对称约束动力学类上最大化该不变量，得到由对称交换子控制的结构容量：其中心存储由可逆动力学冻结的同型种群，其无迹重数块携带可移动非交换坐标，耗散消除这些块内部的一元谱损失，而协变移动性释放受组分守恒约束的相对种群。随后，操作实现确定哪些存活坐标会强制概率态。对于固定的非平凡不变读出，完全移动性给出最坏情况状态代价的精确二分法：交换不变代数的代价恰好为其维度，而非交换重数块则使无约束代价恰好提高一个状态。因此，非交换性具有一个状态的最坏情况经典代价。平凡对称下已知的四字母二次加一法则是该原理的完全移动端点。舒尔-外尔对偶进一步表明，同一张量幂希尔伯特空间上不同的保留对称可将最坏记忆尺度从多项式变为指数，而固定权重模在半填充时给出精确的卡特兰法则，结构容量等于卡特兰数减去其中心扇区修正项。
英文摘要
Under compact symmetry, observable behavior reduces to an invariant operator algebra, but its dimension is not yet classical memory: some coordinates are dynamically frozen, some invisible to threshold tests, and some already classical. We develop an operator-algebraic theory that separates these effects through three filters. For one automaton, behavior is the Hilbert--Schmidt pairing between prefix-reachable states and suffix-observable effects, whose rank equals the real Hankel rank without controllability or observability assumptions. Maximizing this invariant over a symmetry-constrained dynamical class gives a structural capacity controlled by the symmetry commutant: its center stores isotypic populations frozen by reversible dynamics, its traceless multiplicity blocks carry movable noncommutative coordinates, dissipation removes the unary spectral loss inside those blocks, and covariant mobility releases relative populations subject to component conservation. Operational realization then determines which surviving coordinates force probabilistic states. For a fixed nontrivial invariant readout, full mobility gives an exact dichotomy in worst-case state cost: a commutative invariant algebra costs exactly its dimension, whereas a noncommutative multiplicity block raises the unrestricted cost by exactly one state. Thus noncommutativity has a one-state worst-case classical price. The known four-letter quadratic-plus-one law at trivial symmetry is the fully mobile endpoint of this principle. Schur--Weyl duality further shows that different preserved symmetries on the same tensor-power Hilbert space can change the worst memory scale from polynomial to exponential, while fixed-weight modules give an exact Catalan law at half filling, with structural capacity equal to the Catalan count minus its central-sector correction.
