论文标题
Kraus样分解
Kraus-Like Decompositions
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
We introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like (operator) decompositions. We motivate this decomposition with a general nonexistence result for Kraus operator decompositions in this setting. Given a length function which is a class function on a finite group, we construct a corresponding Kraus-like decomposition. We prove that this Kraus-like decomposition is \textit{convex} (meaning its coefficients are nonnegative and satisfy a sum rule) if and only if the length is conditionally negative definite. For a general finite group, we prove a stability condition which shows that the existence of a convex Kraus-like decomposition for all $t>0$ small enough necessarily implies existence for all time $t>0$. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. Our result implies that in the group algebra setting, a semigroup $P_t$ induced by a length function which is a class function is a quantum channel for all $t\geq 0$ if and only if it possesses a convex Kraus-like decomposition for all $t>0$.
