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- W4384392912 abstract "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." @default.
- W4384392912 created "2023-07-15" @default.
- W4384392912 creator A5036309493 @default.
- W4384392912 creator A5041610548 @default.
- W4384392912 date "2023-07-13" @default.
- W4384392912 modified "2023-09-23" @default.
- W4384392912 title "Towards a resolution of the spin alignment problem" @default.
- W4384392912 doi "https://doi.org/10.48550/arxiv.2307.06894" @default.
- W4384392912 hasPublicationYear "2023" @default.
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