论文标题
无限晶格上的量子自旋系统的订单1的Wasserstein距离
The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
论文作者
论文摘要
我们提出了订单1的Wasserstein距离对晶格$ \ mathbb {z}^d $上量子旋转系统的概括，我们称之为特定的量子$ W_1 $距离。该提案基于[de Palma等，IEEE Trans的Qudits的$ W_1 $距离。 inf。理论67，6627（2021）]并恢复了Ornstein的$ \ bar {d} $ - 对于量子状态的距离，其在任何有限数量的旋转的边际状态在规范的基础上都是对角线的。我们还提出了对Lipschitz常数的概括，以在$ \ mathbb {z}^d $上进行量子相互作用，并证明这种量子Lipschitz常数和特定的量子$ W_1 $距离是相互偶发的。我们证明，就量子$ W_1 $距离而言，von Neumann熵是有限的量子旋转的新连续性，我们将其应用于特定的von Neumann熵的连续性，该连续性限制了特定的量子$ W_1 $ $ W_1 $的距离，用于$ \ \ \ \ \ \ \ nathbb {z}^d $。最后，我们证明，在临界温度以上的局部量子通勤相互作用满足运输成本不平等的情况，这意味着其Gibbs状态的独特性。
We propose a generalization of the Wasserstein distance of order 1 to quantum spin systems on the lattice $\mathbb{Z}^d$, which we call specific quantum $W_1$ distance. The proposal is based on the $W_1$ distance for qudits of [De Palma et al., IEEE Trans. Inf. Theory 67, 6627 (2021)] and recovers Ornstein's $\bar{d}$-distance for the quantum states whose marginal states on any finite number of spins are diagonal in the canonical basis. We also propose a generalization of the Lipschitz constant to quantum interactions on $\mathbb{Z}^d$ and prove that such quantum Lipschitz constant and the specific quantum $W_1$ distance are mutually dual. We prove a new continuity bound for the von Neumann entropy for a finite set of quantum spins in terms of the quantum $W_1$ distance, and we apply it to prove a continuity bound for the specific von Neumann entropy in terms of the specific quantum $W_1$ distance for quantum spin systems on $\mathbb{Z}^d$. Finally, we prove that local quantum commuting interactions above a critical temperature satisfy a transportation-cost inequality, which implies the uniqueness of their Gibbs states.
