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arXiv:2307.16268 · Bolyai Society mathematical studies

Quantum Optimal Transport: Quantum Channels and Qubits

Giacomo De Palma, Dario Trevisan

Published Jan 1, 2024• 8 citations
arXivHTMLPDF
DOI
10.1007/978-3-031-50466-2_4

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Controlling the dynamics of quantum states is a common demand in quantum technology. We theoretically study quantum transport and switching in long-range coupled quantum networks possessing a single excitation. The qubit networks of our study are affected by Markovian environments while one qubit is irreversibly connected to an additional site. Different goals are studied towards controlling quantum transport such as efficient single excitation transfer, switching quantum transport,and manipulating the quantum states of qubit networks. These goals have been previously investigated using different mechanisms, however in this thesis we approach them by proposing appropriate geometrical arrangements of sites. Regarding efficient single excitation transfer we follow two approaches: Checking if there exist a spatial dimension preference for qubit arrangements towards optimal quantum transport, and whether the efficient network designs are robust against geometrical variations. For dimension analysis a random walk optimization method is used to compare the transport efficiency of networks expanded in one and two spatial dimensions. We find that for some choices of network parameters the two-dimensional networks are slightly more efficient than the one dimensional equivalent networks. This assures that designing one-dimensional qubit channels are adequate for efficient transport and the two-dimensional networks should only be considered where the two-dimensional spatial expansion is more compatible with the surrounding architecture. To design a network, after deciding about the network dimensionality, one may attempt to optimize the channel geometry which might result in a non-robust optimal configuration against the geometrical errors. To achieve robust optimal configurations we follow two approaches: We investigate the efficient configurations of sites in both one and two dimensional arrangements and present some patterns of robust geometric arrangements of sites in geometrical parameter space. Another approach to achieve robust efficient transport is defining a quantity as the geometrical robustness (georobustness) of qubit networks. By optimising geo-robustness against the network-environment parameters one can efficiently transfer a single excitation with many arbitrary arrangements of the network sites. Switching of quantum states is another goal towards controlled quantum dynamics. We study the transport characteristics of highly symmetric three-dimensional networks and analytically prove that a fraction of the initially injected excitation can trap for long durations via the creation of dark states. Using this characteristic we suggest switching devices which are robust against environmental noises.
A method to optimize the cost of a quantum channel is developed. The goal is to determine the cheapest channel that produces prescribed output states for a given set of input states. This is essentially a quantum version of optimal transport. To attach a clear conceptual meaning to the cost, channels are viewed in terms of what we call elementary transitions, which are analogous to point-to-point transitions between classical systems. The role of entanglement in optimization of cost is emphasized. We also show how our approach can be applied to theoretically search for channels performing a prescribed set of tasks on the states of a system, while otherwise disturbing the state as little as possible.
We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.