We define the quantum Wasserstein distance such that the optimization is
carried out over bipartite separable states rather than bipartite quantum
states in general, and examine its properties. Surprisingly, we find that its
self-distance is related to the quantum Fisher information. We discuss how the
quantum Wasserstein distance introduced is connected to criteria detecting
quantum entanglement. We define variance-like quantities that can be obtained
from the quantum Wasserstein distance by replacing the minimization over
quantum states by a maximization. We extend our results to a family of
generalized quantum Fisher information.

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