Consider minimizing the entropy of a mixture of states by choosing each state subject to constraints. If the spectrum of each state is fixed, we expect that in order to reduce the entropy of the mixture, we should make the states less distinguishable in some sense. Here, we study a class of optimization problems that are inspired by this situation and shed light on the relevant notions of distinguishability. The motivation for our study is the recently introduced spin alignment conjecture. In the original version of the underlying problem, each state in the mixture is constrained to be a freely chosen state on a subset of $n$ qubits tensored with a fixed state $Q$ on each of the qubits in the complement. According to the conjecture, the entropy of the mixture is minimized by choosing the freely chosen state in each term to be a tensor product of projectors onto a fixed maximal eigenvector of $Q$, which maximally "aligns" the terms in the mixture. We generalize this problem in several ways. First, instead of minimizing entropy, we consider maximizing arbitrary unitarily invariant convex functions such as Fan norms and Schatten norms. To formalize and generalize the conjectured required alignment, we define alignment as a preorder on tuples of self-adjoint operators that is induced by majorization. We prove the generalized conjecture for Schatten norms of integer order, for the case where the freely chosen states are constrained to be classical, and for the case where only two states contribute to the mixture and $Q$ is proportional to a projector. The last case fits into a more general situation where we give explicit conditions for maximal alignment. The spin alignment problem has a natural "dual" formulation, versions of which have further generalizations that we introduce.

翻译：考虑通过选择满足约束的每个状态来最小化混合状态的熵。若每个状态的谱固定，我们预期为减少混合熵，应在某种意义上使状态间更难以区分。本文研究受此情境启发的一类优化问题，并阐明相关可区分性概念。研究动机源于近期提出的自旋对齐猜想。在原始问题版本中，混合中的每个状态被约束为自由选择的、作用于$n$个量子比特子集上的状态，与互补量子比特上每个固定状态$Q$的张量积。根据猜想，通过将每项自由选择的状态选为向$Q$的固定最大特征向量投影算子的张量积（即最大化混合中各项的"对齐"），可使混合熵最小化。我们从多个方面推广该问题：首先，将熵最小化推广为最大化任意酉不变凸函数（如Fan范数和Schatten范数）；其次，为形式化并推广猜想所要求的对齐，定义由优超诱导的自伴算子元组的对齐预序。我们证明该广义猜想在以下情形成立：整数阶Schatten范数、自由选择状态被约束为经典态的情况，以及仅两个状态参与混合且$Q$正比于投影算子的情况。最后一种情况可纳入更一般框架，其中我们给出最大对齐的显式条件。自旋对齐问题存在自然的"对偶"表述，其变体可进一步推广，这些推广也将在本文中引入。