Quantum Wasserstein distance based on an optimization over separable states

Géza Tóth1,2,3,4,5 and József Pitrik5,6,7

1Theoretical Physics, University of the Basque Country UPV/EHU, ES-48080 Bilbao, Spain
2EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, ES-48940 Leioa, Biscay, Spain
3Donostia International Physics Center (DIPC), ES-20080 San Sebastián, Spain
4IKERBASQUE, Basque Foundation for Science, ES-48011 Bilbao, Spain
5Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, HU-1525 Budapest, Hungary
6Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15., HU-1053 Budapest, Hungary
7Department of Analysis and Operations Research, Institute of Mathematics, Budapest University of Technology and Economics, Müegyetem rkp. 3., HU-1111 Budapest, Hungary

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We define the quantum Wasserstein distance such that the optimization of the coupling is carried out over bipartite separable states rather than bipartite quantum states in general, and examine its properties. Surprisingly, we find that the self-distance is related to the quantum Fisher information. We present a transport map corresponding to an optimal bipartite separable state. 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 quantities.

In the everyday life, the distance of two cities tells us how many kilometers we have to drive from one to the other. It is also possible to characterize how easily we can get from one city to the other is to measure the fuel consumption during our journey. The latter is more informative in the sense that it reflects the cost of the travel related to the topography of the road, i. e., it is sensitive to the underlying metric. Next, let us imagine that we need to move a heap of sand from one place to another one and the new heap might have a different shape. In this case, again, we can characterize the effort of moving the sand by the cost of the transport.

Distances play a central role in mathematics, physics and engineering. A fundamental problem in probability and statistics is to come up with useful measures of distance between two probability distributions. Unfortunately, many notions of distance between probability distributions, say p(x) and q(x), are maximal if they do not overlap with each other, i. e., one is always zero when the other is non-zero. This is impractical for many applications. For instance, returning to the sand analogy, two non-overlapping piles of sand seem to be equally far from each other, regardless whether their distance is 10km or 100km. Optimal transport theory is a way to construct an alternative notion of distance between probability distributions, the so-called Wasserstein distance. It can be non-maximal even if the distributions do not overlap with each other, it is sensitive to the underlying metric (i.e., the cost of the transport), and essentially, it expresses the effort we need to move one to the other, as if they were sand piles.

Recently, the quantum Wasserstein distance has been defined generalizing the classical Wasserstein distance. It is based on the minimization of a cost function over the quantum states of a bipartite quantum system. It has the property analogous to the one mentioned above in the quantum world. It can be non-maximal for orthogonal states, which is useful, for instance, when we need to teach quantum data to an algorithm.

As we can expect, quantum Wasserstein distance also has properties that are very different from those of its classical counterpart. For instance, when we measure the distance of a quantum state from itself, it can be nonzero. While this is already puzzling, it has also been found that the self-distance is related to the Wigner-Yanase skew information, introduced in 1963 by the Nobel laureate E. P. Wigner, who has vital contributions to the foundations of quantum physics and M. M. Yanase.

In our paper, we look at this mysterious finding from yet another direction. We restrict the minimization mentioned above to so-called separable states. These are the quantum states that do not contain entanglement. We find that the self-distance becomes the quantum Fisher information, a quantity central in quantum metrology and quantum estimation theory, and appearing for instance in the famous Cramer-Rao bound. By examining the properties of such a Wasserstein distance, our work paves the way to connect the theory of quantum Wasserstein distance to the theory of quantum entanglement.

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