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：考虑通过使每个状态受约束来最小化混合态的熵。若每个状态的谱固定，我们预期为了降低混合熵，应在某种意义上使状态间更难区分。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。原始问题中，混合态中的每个状态被约束为：在n个量子比特子集上自由选择的状态与补集中每个量子比特上的固定态Q的张量积。该猜想指出，通过将每个项中的自由选择状态选为Q的固定最大本征向量投影算子的张量积（即最大化混合项之间的“对齐”），可最小化混合熵。我们从多个方面推广该问题：首先，考虑最大化任意酉不变凸函数（如Fan范数、Schatten范数）而非最小化熵。为形式化并推广猜想所需的对齐，定义对齐为通过优超诱导的自伴算子元组上的预序。我们证明了整数阶Schatten范数、自由选择状态受限为经典情形、以及仅两项状态参与混合且Q正比于投影算子情形下的广义猜想。最后一种情形可归入更一般的分析框架，并给出最大对齐的显式条件。自旋对齐问题具有自然“对偶”形式，其变体可进一步推广，相关推广亦在文中引入。