Sam Cole, Michał Eckstein, Shmuel Friedland, Karol Życzkowski
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引用次数: 1
Abstract
We analyze a quantum version of the Monge–Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix C is minimized over the set of all bipartite coupling states \(\rho ^{AB}\) with fixed reduced density matrices \(\rho ^A\) and \(\rho ^B\) of size m and n. The minimum quantum optimal transport cost \(\textrm{T}^Q_{C}(\rho ^A,\rho ^B)\) can be efficiently computed using semidefinite programming. In the case \(m=n\) the cost \(\textrm{T}^Q_{C}\) gives a semidistance if and only if C is positive semidefinite and vanishes exactly on the subspace of symmetric matrices. Furthermore, if C satisfies the above conditions, then \(\sqrt{\textrm{T}^Q_{C}}\) induces a quantum analogue of the Wasserstein-2 distance. Taking the quantum cost matrix \(C^Q\) to be the projector on the antisymmetric subspace, we provide a semi-analytic expression for \(\textrm{T}^Q_{C^Q}\) for any pair of single-qubit states and show that its square root yields a transport distance on the Bloch ball. Numerical simulations suggest that this property holds also in higher dimensions. Assuming that the cost matrix suffers decoherence and that the density matrices become diagonal, we study the quantum-to-classical transition of the Monge–Kantorovich distance, propose a continuous family of interpolating distances, and demonstrate that the quantum transport is cheaper than the classical one. Furthermore, we introduce a related quantity—the SWAP-fidelity—and compare its properties with the standard Uhlmann–Jozsa fidelity. We also discuss the quantum optimal transport for general d-partite systems.
期刊介绍:
MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas.
The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process.
The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed.
The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.