论文标题
量子1的量子瓦斯汀距离1
The Quantum Wasserstein Distance of Order 1
论文作者
论文摘要
我们建议将订单1的Wasserstein距离与$ n $ qudits的量子状态进行概括。该提案为规范基础的向量恢复了锤距，更通常，在规范的基础上，量子状态的经典瓦斯汀距离。相对于作用于一个Qudit的Qudits和单一操作的排列，所提出的距离是不变的，并且相对于张量产品是添加的。我们的主要结果是相对于所提出的距离，冯·诺伊曼熵的连续性结合，这显着增强了相对于痕量距离的最佳连续性。我们还提出了Lipschitz常数对量子可观察物的概括。量子Lipschitz常数的概念使我们能够通过半限定程序计算提出的距离。我们证明了Marton的运输不平等的量子版本和量子Lipschitz可观察到的量子的量子高斯浓度不平等。此外，我们在浅量子电路的收缩系数以及相对于所提出的距离方面的张量产物的收缩系数和张量。我们讨论了量子机学习，量子香农理论和量子多体系统中的其他可能应用。
We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.
