AI 中文总结
本文提出有限可逆马尔可夫链的伪量子表示方法，将其编码为复正交流，以对称埃伦费斯特瓮为例验证，预告了其他随机模型的相关结果。
AI 中文摘要
伪量子表示将具有$M$个状态的有限不可约可逆连续时间马尔可夫链重新编码为维度为$2M$的加倍空间上的复正交流。经过均匀化、平方根规范、状态空间加倍和对角幺正扭转后，该链生成一个含复参数$z$的单参数群$W_z = e^{zK}$，本文的研究对象正是该单参数群。当$z$取实数值时，其实切片通过显式解码精确重现随机半群，该解码的概率意义是均匀化链的节拍数的奇偶性；当$z = i\theta$（$\theta$为实数）时，其虚切片是有限维幺正量子系统。因此，该链与量子系统并非两个类似模型，而是同一完整表示的两种限制。该框架还无需额外输入即可导出伪薛定谔方程、针对复对称伪密度的双线性冯·诺依曼方程，以及非紧二次曲面上的伪布洛赫矢量方程。我们从第一性原理出发推导该构造，并完整计算了一个模型——通常的对称埃伦费斯特瓮模型。其规范生成元等于$(2/n) J_x$（自旋$n/2$算符），弛豫模式为洛伦兹推进，其虚切片是深度为1的量子电路，该电路的玻恩分布对应时钟$\theta(t) = \text{arccos}(e^{-2t})$下的经典瓮定律。同一时钟将瓮轨迹的费希尔-饶提升映射为刚性自旋相干态轨道。本文是一篇更长论文的初步简化版本，仅处理有限状态空间和对称埃伦费斯特瓮，且无证明地预告了非对称瓮、两个队列、对称简单排斥过程及随机伊辛模型。附录中的入门材料使量子、几何和信息论语言自成体系。
英文摘要
The pseudo-quantum representation re-encodes a finite, irreducible, reversible continuous-time Markov chain with $M$ states as a complex-orthogonal flow on a doubled space of dimension $2M$. After uniformisation, a square-root gauge, a doubling of the state space and a diagonal unitary twist, the chain generates an entire one-parameter group $W_z = e^{zK}$ with $z$ complex, and this single group is the object of the paper. Its real slice, $z$ real, reproduces the stochastic semigroup exactly through an explicit decoding whose probabilistic meaning is the parity of the number of ticks of the uniformised chain. Its imaginary slice, $z = iθ$, is a finite-dimensional unitary quantum system. The chain and the quantum system are therefore not two analogous models but two restrictions of one entire representation. The setup also produces, with no further input, a pseudo-Schroedinger equation, a bilinear von Neumann equation for a complex-symmetric pseudo-density, and a pseudo-Bloch vector equation on a non-compact quadric. We develop the construction from first principles and work out one model completely, the usual symmetric Ehrenfest urn. Its gauged generator equals $(2/n) J_x$, the spin-$n/2$ operator, its relaxation modes are Lorentz boosts, and its imaginary slice is a depth-one quantum circuit whose Born distribution is the classical urn law under the clock $θ(t) = \arccos(e^{-2t})$. The same clock identifies the Fisher-Rao lift of the urn trajectory with a rigid spin-coherent-state orbit. This is a preliminary simplified version of a longer paper. It treats only finite state spaces and only the symmetric Ehrenfest urn, and it previews without proofs the asymmetric urn, two queues, the symmetric simple exclusion process and the stochastic Ising model. Appendix primers make the quantum, geometric and information-theoretic language self-contained.
