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 spin alignment conjecture introduced recently in Ref.~\cite{Leditzky2022a}. 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 \textit{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：考虑通过选择每个满足约束的状态来最小化混合态的熵。若每个态的谱固定，我们预期为降低混合熵，应在某种意义上使各态更难以区分。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。研究动机源于近期文献[1]提出的自旋对齐猜想。原始问题中，混合态中的每个态被限制为自由选择|n个量子比特子集上的态，与互补比特上固定态Q的张量积。该猜想指出，若在每个项中选择自由态为Q的固定最大本征矢的投影算符张量积，则混合熵最小化——这使混合项达到最大程度“对齐”。我们以多种方式推广该问题：首先，将熵最小化替换为任意酉不变凸函数（如Fan范数、Schatten范数）的最大化。为形式化并推广猜想所需的“对齐”，我们定义由majorization诱导的自伴算子元组上的预序关系——对齐。我们证明了以下情形下猜想的推广形式：整数阶Schatten范数；自由态受经典约束；仅有两态构成混合且Q正比于投影算符。最后一例可纳入更一般情况——我们给出最大对齐的显式条件。自旋对齐问题具有自然“对偶”表述，其变体引入的进一步推广将在本文中展开。