论文标题
量子2-wasserstein距离的单调性
Monotonicity of the quantum 2-Wasserstein distance
论文作者
论文摘要
我们研究了2-wasserstein距离的量子类似物，以衡量尺寸$ n $密度矩阵的集合$ω_n$。我们表明，这种（半）距离不会在$ω_n$的切线束上诱导Riemannian指标，并且通常不是统一的不变性。然而，我们证明，对于$ n = 2 $尺寸希尔伯特空间，量子2-wasserstein距离（唯一重新缩放）对于任何单量量子操作都是单调的，而量子传输问题的解决方案本质上是独一无二的。此外，对于任何$ n \ geq 3 $和与投影仪成正比的量子成本矩阵，我们证明了在任意混合统一渠道下的单调性。最后，我们提供了数值证据，使我们能够猜想统一不变的Quantum 2-Wasserstein semi-distance相对于任何维度$ n $中的所有CPTP地图都是单调的。
We study a quantum analogue of the 2-Wasserstein distance as a measure of proximity on the set $Ω_N$ of density matrices of dimension $N$. We show that such (semi-)distances do not induce Riemannian metrics on the tangent bundle of $Ω_N$ and are typically not unitary invariant. Nevertheless, we prove that for $N=2$ dimensional Hilbert space the quantum 2-Wasserstein distance (unique up to rescaling) is monotonous with respect to any single-qubit quantum operation and the solution of the quantum transport problem is essentially unique. Furthermore, for any $N \geq 3$ and the quantum cost matrix proportional to a projector we demonstrate the monotonicity under arbitrary mixed unitary channels. Finally, we provide numerical evidence which allows us to conjecture that the unitary invariant quantum 2-Wasserstein semi-distance is monotonous with respect to all CPTP maps in any dimension $N$.
