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arXiv:1805.10428 · Leibniz international proceedings in informatics

Quantum Network Code for Multiple-Unicast Network with Quantum Invertible Linear Operations

Seunghoan Song, Masahito Hayashi

Published Jan 1, 2018• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.4230/lipics.tqc.2018.10

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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.
It is natural in a quantum network system that multiple users intend to send their quantum message to their respective receivers, which is called a multiple unicast quantum network. We propose a canonical method to derive a secure quantum network code over a multiple unicast quantum network from a secure classical network code. Our code correctly transmits quantum states when there is no attack. It also guarantees the secrecy of the transmitted quantum state even with the existence of an attack when the attack satisfies a certain natural condition. In our security proof, the eavesdropper is allowed to modify wiretapped information dependently on the previously wiretapped messages. Our protocol guarantees the secrecy by utilizing one-way classical information transmission (public communication) in the same direction as the quantum network although the verification of quantum information transmission requires two-way classical communication. In the protocol, some nodes may share secret randomness as resources in advance. Our secure network code can be applied to several networks including the butterfly network.
We consider the secure quantum communication over a network with the presence of a malicious adversary who can eavesdrop and contaminate the states. The network consists of noiseless quantum channels with the unit capacity and the nodes which applies noiseless quantum operations. As the main result, when the maximum number m1of the attacked channels over the entire network uses is less than a half of the network transmission rate m0(i.e., m10/2), our code implements secret and correctable quantum communication of the rate m0- 2m1by using the network asymptotic number of times. Our code is universal in the sense that the code is constructed without the knowledge of the specific node operations and the network topology, but instead, every node operation is constrained to the application of an invertible matrix to the basis states. Moreover, our code requires no classical communication. Our code can be thought of as a generalization of the quantum secret sharing.
Classical network coding uses encoding of information over networks to achieve secure and high throughput communication with low latency. Its quantum analog, namely Quantum Network Coding (QNC) is a promising protocol to achieve quantum communication over quantum networks. In this paper, we propose a method for establishing QNC in Measurement-Based Quantum Computing (MBQC), which is a universal quantum computational model. We demonstrate that the problem of multiple unicast or multiple multicast of distinct computational basis states over a quantum network can be solved by simulating existing classical networks, with the CNOT implementation in MBQC as a basis for transmitting and encoding of states. We also show how it can be further extended to simultaneously distribute the Bell pairs and Greenberger–Horne–Zeilinger (GHZ) states required for quantum communication between distant transmitter and corresponding receiver nodes of a network.
Nowadays, network coding has been a very successful topic in information theory including many applications such as wireless networks while one of the simplest benefits is the efficient transmission in networks by allowing us to encode the transmitted information at every intermediate node. Since quantum information is much more expensive for communication than classical information, it was natural that this benefit made quantum researchers motivate the study of “quantum network coding.” This paper reports the current status of quantum network coding. At present, quantum network coding mostly means sending quantum information (rather than classical information) in a quantum network. Since quantum information cannot be cloned (the quantum no-cloning theorem), multiple unicast networks have been well-studied (in several settings). We present some of the known possibilities and limitations, and future works of quantum network coding, focusing on multiple unicast networks.
We consider the secure quantum communication over a network with the presence of the malicious adversary who can eavesdrop and contaminate the states. The network consists of the noiseless quantum channels with unit capacity and the nodes which applies noiseless quantum operations. As the main result, when the maximum number m1of the attacked channels over the entire network uses is less than a half of network transmission rate m0(i.e., m10/2), our protocol implements secret and correctable quantum communication of rate m0- 2m1by using the network asymptotic number of times. Our protocol requires no classical communication and no knowledge of network structure, but instead, a node operation is limited to the application of an invertible matrix to the basis states. Our protocol can be thought of as a generalization of honest-dealer verifiable quantum secret sharing. A full version of this paper is accessible at: https://arxiv.org/pdf/1801.03306.pdf.
Abstract Quantum and private communications are affected by a fundamental limitation which severely restricts the optimal rates that are achievable by two distant parties. To overcome this problem, one needs to introduce quantum repeaters and, more generally, quantum communication networks. Within a quantum network, other problems and features may appear when we move from the basic unicast setting of single-sender/single-receiver to more complex multiend scenarios, where multiple senders and multiple receivers simultaneously use the network to communicate. Assuming various configurations, including multiple-unicast, multicast, and multiple-multicast communication, we bound the optimal rates for transmitting quantum information, distributing entanglement, or generating secret keys in quantum networks connected by arbitrary quantum channels. These bounds cannot be surpassed by the most general adaptive protocols of quantum network communication.