The problem of an optimal mapping between Hilbert spaces $IN$ of $\left|\psi\right\rangle$ and $OUT$ of $\left|\phi\right\rangle$ based on a set of wavefunction measurements (within a phase) $\psi_l \to \phi_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\sum_{l=1}^{M} \omega^{(l)} \left|\langle\phi_l|\mathcal{U}|\psi_l\rangle\right|^2$ subject to probability preservation constraints on $\mathcal{U}$ (partial unitarity). Constructed operator $\mathcal{U}$ can be considered as a $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix of the dimension $\dim(OUT) \times \dim(IN)$ transforming operators as $A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$. An iteration algorithm finding the global maximum of this optimization problem is developed and it's application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.

翻译：基于一组波函数测量（相位内）$\psi_l \to \phi_l$（$l=1\dots M$），寻找希尔伯特空间$IN$（对应$\left|\psi\right\rangle$）与$OUT$（对应$\left|\phi\right\rangle$）间最优映射的问题，被建模为在$\mathcal{U}$的概率保持约束（部分酉性）下最大化总保真度$\sum_{l=1}^{M} \omega^{(l)} \left|\langle\phi_l|\mathcal{U}|\psi_l\rangle\right|^2$的优化问题。所构造的算子$\mathcal{U}$可视为从$IN$到$OUT$的量子通道；它是一个维度为$\dim(OUT) \times \dim(IN)$的部分酉矩形矩阵，通过$A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$变换算子。本文开发了一种能全局求解该优化问题的迭代算法，并展示了其在多个问题上的应用。实现该算法的软件产品可从作者处获取。