The quantum Wasserstein distance of order 1

The quantum Wasserstein distance of order 1

Seminar author:Giacomo De Palma

Event date and time:02/25/2021 04:00:pm

Event location:https://us02web.zoom.us/j/89335132125?pwd=bStvUFhiZ2dUZmxZTzFzTjg1NXFZQT09

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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. In particular, we focus on the application of the proposed distance as cost function for the quantum counterpart of Generative Adversarial Networks (GANs), which provide an algorithm to learn a quantum state. Contrarily to the previous proposals, our quantum GAN does not suffer from the issue of exponentially vanishing gradients and is capable to learn a broad family of quantum states.