To what extent does Noether’s principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law and show that this relates to physically impossible transformations in quantum theory, such as time reversal and spin inversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group $G$ becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel $\mathcal{E}$. In particular, we investigate in detail the U(1) and SU(2) symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin-$j_A$ system can be used to polarize a larger spin-$j_B$ system, and on how much one can invert spin polarization using a rotationally symmetric operation. Finally, we also establish novel links between unitarity, complementary channels, and purity that are of independent interest.

中文翻译：

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Noether原理的鲁棒性：量子理论中守恒律和对称性之间的最大联系

Noether原理在多大程度上适用于量子通道？在这里，我们量化了在量子通道上施加对称约束的隐含守恒律的程度，并表明这与量子理论中物理上不可能的转换有关，例如时间反转和自旋反转。在此分析中，一组量子通道的凸结构和极点在Lie基团的作用下对称$G$变得至关重要。它允许我们根据任何对称量子通道得出的与封闭定律的偏差，得出与守恒律的偏差的界限，该偏差由通道的均匀性来衡量$\mathcal{Ë}$。特别是，我们详细研究了与能量和角动量守恒定律有关的U（1）和SU（2）对称性。在后一种情况下，我们对自旋${Ĵ}_{\mathrm{一种}}$ 系统可以用来极化更大的自旋${Ĵ}_{乙}$系统，以及有多少人可以使用旋转对称操作来反转自旋极化。最后，我们还建立了具有独立利益的统一性，互补渠道和纯度之间的新颖联系。