关于量子Wasserstein散度的度量性质
On the metric property of quantum Wasserstein divergences
AI总结：
本文证明了在纯态和有限能量假设下，量子Wasserstein散度在可分Hilbert空间中满足三角不等式，并提供了该不等式对任意状态普遍成立的数值证据。
摘要：
量子Wasserstein散度是由信道定义的量子Wasserstein距离的修改版本，De Palma和Trevisan猜想它们是量子态空间上的真正度量。我们证明了在涉及的一个特定状态是纯态且所有状态具有有限能量的假设下，对于由可分Hilbert空间描述的每个量子系统和任何二次成本算子，量子Wasserstein散度满足三角不等式。我们还提供了强有力的数值证据，表明三角不等式在一般情况下，对于任意选择的状态都成立。
英文摘要：
Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels, and they are conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure, and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general, for an arbitrary choice of states.
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