The Quantum Wasserstein Distance of Order 1

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of n qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The pro...

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Bibliographic Details
Published in:IEEE transactions on information theory 2021-10, Vol.67 (10), p.6627-6643
Main Authors: De Palma, Giacomo, Marvian, Milad, Trevisan, Dario, Lloyd, Seth
Format: Article
Language:English
Subjects:
concentration inequalities
Continuity
Entropy
Hamming distance
Lipschitz constant
Machine learning
Mathematical analysis
Permutations
Probability distribution
Quantum mechanics
Quantum optimal mass transport
Quantum state
qudits
Tensors
Training
von Neumann entropy
Wasserstein distance
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Summary:We propose a generalization of the Wasserstein distance of order 1 to the quantum states of n qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2021.3076442