发表机构
ÉTS Montréal, Université du Québec（魁北克大学蒙特利尔工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究量子理论中概率定律无法一致拼接的谱系悖论，分类例外情况，提出Born–Chapman–Kolmogorov流度量，并证明音乐转换预测为PromiseBQP完全问题。
AI 中文摘要
量子理论通常不允许将通过Born规则从单个幺正步骤获得的概率定律拼接成一致的谱系；我们对例外情况进行分类，并表明忠实组合可以从初始边界成立，但在内部重启后却失败。对于有限维、组合封闭的幺正族，普遍组合恰好对幺正单项式成立，即其Born核实现可逆确定性状态机的相位修饰置换。预设序列规避了这一障碍。我们对所有量子比特幺正步骤对进行分类，给出了量子三态幺正步骤对的充要判据，并证明了在$d$维Hilbert空间上的边界稳定幺正序列至多包含$d$个完全混合步骤，且在每一个素数维度上等号成立。我们还构造了任意长的真正混合序列，这些序列从初始边界组合成立，但在内部重启后失败，以及一个具有主动干涉的量子三态控制的双量子比特实现。我们定义了一个Born–Chapman–Kolmogorov流，当相干且逐步检验的端点定律一致时，该流恰好消失，并附带一个度量，用于衡量端点揭示中间检查时间表的比特数。这种状态机联系为量子音乐理论提供了基础，其中幺正操作是音符，时间边界是提示。音乐转换预测被证明是PromiseBQP完全的，而保持谱系的逐步模式规定了节奏规则，而从内部提示出发的谱系的异常失败则是量子音乐悖论。
英文摘要
Quantum theory does not generally permit the probability laws obtained from individual unitary steps by the Born rule to be sewn into a consistent genealogy; we classify the exceptions and show that faithful composition can hold from an initial boundary yet fail after an internal restart. For finite-dimensional, composition-closed unitary families, universal composition holds exactly for unitary monomials, the phase-dressed permutations whose Born kernels realize reversible deterministic state machines. Prescribed sequences evade this obstruction. We classify all pairs of qubit unitary steps, give a necessary-and-sufficient criterion for pairs of qutrit unitary steps, and prove that a boundary-stable unitary sequence on a $d$-dimensional Hilbert space contains at most $d$ fully mixing steps, with equality in every prime dimension. We also construct arbitrarily long genuinely mixing sequences that compose from their initial boundary but fail after an internal restart, and a qutrit-controlled two-qubit realization with active interference. We define a Born--Chapman--Kolmogorov current that vanishes exactly when coherent and stepwise-checked endpoint laws agree, together with an associated measure of how many bits the endpoint reveals about the intermediate checking schedule. This state-machine connection provides a foundation for a quantum theory of music, in which unitary operations are notes and temporal boundaries are cues. Musical-transition prediction is proved PromiseBQP-complete, and the stepwise patterns that preserve a genealogy specify rules for rhythm, whereas the exceptional failure of that genealogy from an internal cue is the quantum music paradox.
Comments66 pages, 8 figures. Interactive project page: https://institut-kets.github.io/quantum-composition
