The infinitesimal transition probability operator for a continuous-time discrete-state Markov process, $\mathcal{Q}$, can be decomposed into a symmetric and a skew-symmetric parts. As recently shown for the case of diffusion processes, while the symmetric part corresponding to a gradient system stands for a reversible Markov process, the skew-symmetric part, $\frac{d}{dt}u(t)=\mcA u$, is mathematically equivalent to a linear Hamiltonian dynamics with Hamiltonian $H=1/2u^T\big(\mcA^T\mcA)^{1/2}u$. It can also be transformed into a Schrödinger-like equation $\frac{d}{dt}u=i\mathcal{H}u$ where the "Hamiltonian" operator $\mathcal{H}=-i\mcA$ is Hermitian. In fact, these two representations of a skew-symmetric dynamics emerge natually through singular-value and eigen-value decompositions, respectively. The stationary probability of the Markov process can be expressed as $\|u^s_i\|^2$. The motion can be viewed as "harmonic" since $\frac{d}{dt}\|u(t)-\vec{c}\|^2=0$ where $\vec{c}=(c,c,...,c)$ with $c$ being a constant. More interestingly, we discover that $\textrm{Tr}(\mcA^T\mcA)=\sum_{j,\ell=1}^n \frac{(q_{j\ell}π_\ell-q_{\ell j}π_j)^2}{π_jπ_{\ell}}$, whose right-hand-side is intimately related to the entropy production rate of the Markov process in a nonequilibrium steady state with stationary distribution $\{π_j\}$. The physical implication of this intriguing connection between conservative Hamiltonian dynamics and dissipative entropy production remains to be further explored.
Entropy non-conservation and boundary conditions for Hamiltonian dynamical systems
将厄米算符的自伴扩张理论应用于Koopman von Neumann经典力学，发现了一组最一般的概率分布，其熵由哈密顿演化守恒。一个新的动力学阶段与这样的建设确定。通过选择不属于这一类的分布，我们产生明确的例子，自由粒子和谐波系统在有界相空间中的熵是不守恒的。虽然这些非守恒态在经典上是被禁止的，但它们可以被解释为量子系统隧穿势垒边界的状态。在这种情况下，允许的边界条件是经典系统和量子系统之间的唯一区别。我们证明了隧穿量子系统的边界条件成为经典极限下熵守恒的判据。这些发现突出了边界效应如何彻底改变系统的性质。
Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
我们证明了两体玻色量子系统的任何纯初态的纠缠熵都随时间线性增长，并且与由不稳定的二次哈密顿量引起的动力学有关。增长率不依赖于初始状态，等于相应经典动力学的某些李雅普诺夫指数之和。本文概括了[Bianchi等人，JHEP 2018，25（2018）]，在高斯初始状态的特殊情况下证明了相同的结果。我们的证明是基于最近对玻色子量子系统的冯诺伊曼熵的强次可加性的推广[De Palma et al.，arXiv：2105.05627]。这种技术使我们能够将我们的结果扩展到一般的混合初态，压缩纠缠提供了正确的概括的纠缠熵。我们讨论了我们的结果的几个应用程序的物理系统（弱）相互作用的哈密顿和周期性驱动的量子系统，包括某些量子场论模型。
