Establishes Kantorovich duality for linearized non-quadratic quantum optimal transport realized by channels, determines optimal primal-dual solutions for qubits under state restrictions, and proves the triangle inequality for the square of the induced quantum Wasserstein divergences.
Wasserstein distances and divergences of order $p$ by quantum channels
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abstract
We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.
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UNVERDICTED 3representative citing papers
Two quantum Wasserstein distance definitions coincide for qubits with single-operator cost functions, implying the self-distance equals the Wigner-Yanase skew information.
Proves equalities among quantum Wasserstein distances obtained from optimizations over general versus separable bipartite states and shows relations to Uhlmann-Jozsa fidelity and superfidelity, including equality for qubits.
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Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits
Establishes Kantorovich duality for linearized non-quadratic quantum optimal transport realized by channels, determines optimal primal-dual solutions for qubits under state restrictions, and proves the triangle inequality for the square of the induced quantum Wasserstein divergences.
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Relations between different definitions of the quantum Wasserstein distance for qubits
Two quantum Wasserstein distance definitions coincide for qubits with single-operator cost functions, implying the self-distance equals the Wigner-Yanase skew information.
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Quantum Wasserstein distance and its relation to several types of fidelities
Proves equalities among quantum Wasserstein distances obtained from optimizations over general versus separable bipartite states and shows relations to Uhlmann-Jozsa fidelity and superfidelity, including equality for qubits.