The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $\rho^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} \omega^{(l)} F\left(\varrho^{(l)},\sum_s B_s \rho^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,\sigma)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed to find the global maximum. The result comprises in $N_s$ operators $B_s$ that collectively form an $IN$ to $OUT$ quantum channel $A^{OUT}=\sum_s B_s A^{IN} B_s^{\dagger}$. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a general quantum channel. This marks a crucial advancement from the commonly studied unitary mapping of pure states $\phi_l=\mathcal{U} \psi_l$ to a general quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} \rho^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{\rho^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a general quantum channel of Kraus rank $N_s$, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then built as a hierarchy of unitary mappings. The approach can be applied to study decoherence effects, spontaneous coherence, synchronizing, etc.

翻译：基于一系列密度矩阵映射测量 $\rho^{(l)} \to \varrho^{(l)}$（$l=1\dots M$）的希尔伯特空间 $IN$ 与 $OUT$ 间最优映射问题，被表述为在满足 Kraus 算子 $B_s$ 的概率守恒约束下，最大化总保真度 $\mathcal{F}=\sum_{l=1}^{M} \omega^{(l)} F\left(\varrho^{(l)},\sum_s B_s \rho^{(l)} B^{\dagger}_s\right)$ 的优化问题。当 $F(\varrho,\sigma)$ 的形式使得总保真度可表示为超算子的二次型 $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$（精确或近似成立）时，本文提出了一种寻找全局最大值的迭代算法。该算法最终生成 $N_s$ 个算子 $B_s$，它们共同构成从 $IN$ 到 $OUT$ 的量子信道 $A^{OUT}=\sum_s B_s A^{IN} B_s^{\dagger}$。本工作实现了对酉学习的两个重要推广：1. $IN$/$OUT$ 态以密度矩阵表示；2. 映射本身被表述为一般量子信道。这标志着从普遍研究的纯态酉映射 $\phi_l=\mathcal{U} \psi_l$ 到一般量子信道的重大进展，使我们能够区分态的概率混合与其叠加。该方法的应用通过密度矩阵映射的酉学习 $\varrho^{(l)}=\mathcal{U} \rho^{(l)} \mathcal{U}^{\dagger}$ 得到验证——在此情形下，可通过考虑 $\sqrt{\rho^{(l)}} \to \sqrt{\varrho^{(l)}}$ 映射构造关于 $\mathcal{U}$ 的二次保真度；并在 Kraus 秩为 $N_s$ 的一般量子信道上得到验证——此时关于 $B_s$ 的二次保真度为近似形式，量子信道通过酉映射的层级结构构建。该方法可应用于研究退相干效应、自发相干、同步等现象。