Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication

Cornelia Spee1,2 and Tristan Kraft1,3

1Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria
2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
3Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany

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Recent advances have led towards first prototypes of quantum networks in which entanglement is distributed by sources producing bipartite entangled states. This raises the question of which states can be generated in quantum networks based on bipartite sources using local operations and classical communication. In this work, we study state transformations under finite rounds of local operations and classical communication (LOCC) in networks based on maximally entangled two-qubit states. We first derive the symmetries for arbitrary network structures, as these determine which transformations are possible. Then, we show that contrary to tree graphs, for which it has already been shown that any state within the same entanglement class can be reached, there exist states which can be reached probabilistically but not deterministically if the network contains a cycle. Furthermore, we provide a systematic way to determine states which are not reachable in networks consisting of a cycle. Moreover, we provide a complete characterization of the states which can be reached in a cycle network with a protocol where each party measures only once, and each step of the protocol results in a deterministic transformation. Finally, we present an example which cannot be reached with such a simple protocol, and constitutes, up to our knowledge, the first example of a LOCC transformation among fully entangled states requiring three rounds of classical communication.

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Cited by

[1] Kiara Hansenne, Zhen-Peng Xu, Tristan Kraft, and Otfried Gühne, "Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques", Nature Communications 13, 496 (2022).

[2] Patricia Contreras-Tejada, Carlos Palazuelos, and Julio I. de Vicente, "Asymptotic Survival of Genuine Multipartite Entanglement in Noisy Quantum Networks Depends on the Topology", Physical Review Letters 128 22, 220501 (2022).

[3] Nicky Kai Hong Li, Cornelia Spee, Martin Hebenstreit, Julio I. de Vicente, and Barbara Kraus, "Identifying families of multipartite states with non-trivial local entanglement transformations", Quantum 8, 1270 (2024).

[4] Owidiusz Makuta, Laurens T. Ligthart, and Remigiusz Augusiak, "No graph state is preparable in quantum networks with bipartite sources and no classical communication", npj Quantum Information 9, 117 (2023).

[5] Simon Morelli, David Sauerwein, Michalis Skotiniotis, and Nicolai Friis, "Metrology-assisted entanglement distribution in noisy quantum networks", Quantum 6, 722 (2022).

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