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**Published in:** LIPIcs, Volume 111, 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)

This paper considers the communication over a quantum multiple-unicast network where r sender-receiver pairs communicate independent quantum states. We concretely construct a quantum network code for the quantum multiple-unicast network as a generalization of the code [Song and Hayashi, arxiv:1801.03306, 2018] for the quantum unicast network. When the given node operations are restricted to invertible linear operations between bit basis states and the rates of transmissions and interferences are restricted, our code certainly transmits a quantum state for each sender-receiver pair by n-use of the network asymptotically, which guarantees no information leakage to the other users. Our code is implemented only by the coding operation in the senders and receivers and employs no classical communication and no manipulation of the node operations. Several networks that our code can be applied are also given.

Seunghoan Song and Masahito Hayashi. Quantum Network Code for Multiple-Unicast Network with Quantum Invertible Linear Operations. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 10:1-10:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{song_et_al:LIPIcs.TQC.2018.10, author = {Song, Seunghoan and Hayashi, Masahito}, title = {{Quantum Network Code for Multiple-Unicast Network with Quantum Invertible Linear Operations}}, booktitle = {13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)}, pages = {10:1--10:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-080-4}, ISSN = {1868-8969}, year = {2018}, volume = {111}, editor = {Jeffery, Stacey}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2018.10}, URN = {urn:nbn:de:0030-drops-92572}, doi = {10.4230/LIPIcs.TQC.2018.10}, annote = {Keywords: Quantum network code, Multiple-unicast quantum network, Quantum invertible linear operation} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

Since quantum information is continuous, its handling is sometimes
surprisingly harder than the classical counterpart. A typical
example is cloning; making a copy of digital information is
straightforward but it is not possible exactly for quantum
information. The question in this paper is whether or not {em
quantum} network coding is possible. Its classical counterpart is
another good example to show that digital information flow can be done
much more efficiently than conventional (say, liquid) flow.
Our answer to the question is similar to the case of cloning, namely,
it is shown that quantum network coding is possible if approximation
is allowed, by using a simple network model called Butterfly. In this
network, there are two flow paths, $s_1$ to $t_1$ and $s_2$ to $t_2$,
which shares a single bottleneck channel of capacity one. In the
classical case, we can send two bits simultaneously, one for each
path, in spite of the bottleneck. Our results for quantum network
coding include: (i) We can send any quantum state $|psi_1
angle$
from $s_1$ to $t_1$ and $|psi_2
angle$ from $s_2$ to $t_2$
simultaneously with a fidelity strictly greater than $1/2$. (ii) If
one of $|psi_1
angle$ and $|psi_2
angle$ is classical, then the
fidelity can be improved to $2/3$. (iii) Similar improvement is also
possible if $|psi_1
angle$ and $|psi_2
angle$ are restricted to
only a finite number of (previously known) states. (iv) Several
impossibility results including the general upper bound of the fidelity
are also given.

Masahito Hayashi, Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Shigeru Yamashita. Quantum Network Coding. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2006)

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@InProceedings{hayashi_et_al:DagSemProc.06111.14, author = {Hayashi, Masahito and Iwama, Kazuo and Nishimura, Harumichi and Raymond, Rudy and Yamashita, Shigeru}, title = {{Quantum Network Coding}}, booktitle = {Complexity of Boolean Functions}, pages = {1--17}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.14}, URN = {urn:nbn:de:0030-drops-6080}, doi = {10.4230/DagSemProc.06111.14}, annote = {Keywords: Network coding, quantum computation, quantum information} }

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