On quantum versions of the classical Wasserstein distance

Abstract

We investigate a definition of quantum Wasserstein distance of two states based on their couplings on the product algebra as in the classical case. A detailed analysis of the two qubit model leads to a formal definition fulfilling some minimal requirements. It also shows that a clear-cut definition, by direct generalization of the classical one, is not achievable.

Keywords:

• Wasserstein distance
• quantum probability
• qbit state

Acknowledgements

Part of this work was done while the authors were visiting the ANESTOC Centre in September 2015. We wish to thank the director R. Rebolledo for valuable discussions and comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

JA and Universidad Nacional de Colombia want to highlight the financial support from ‘Patrimonio autónomo fondo nacional de financiamiento para la ciencia, tecnología y la innovación, Francisco Jose de Caldas’ and ‘Universidad Nacional de Colombia (UN)’.