We study the capacity of quantum private information retrieval (QPIR) with multiple servers. In the QPIR problem with multiple servers, a user retrieves a classical file by downloading quantum systems from multiple servers each of which containing the whole classical file set, without revealing the identity of the retrieved file to any individual server. The QPIR capacity is defined as the maximum rate of the file size over the whole dimension of the downloaded quantum systems. When the preexisting entanglement among servers are assumed, we prove that the QPIR capacity with multiple servers is 1 regardless of the number of servers and files. We propose a rate-one protocol which can be implemented by using only two servers. This capacityachieving protocol outperforms its classical counterpart in the sense of the capacity, server secrecy, and upload cost. The strong converse bound is derived concisely without using any secrecy condition. We also prove that the capacity of multi-round QPIR with server secrecy is 1.S. Song and M. Hayashi are with Graduate school of Mathematics,
We study the capacity of quantum private information retrieval (QPIR) with multiple servers. In the QPIR problem with multiple servers, a user retrieves a classical file by downloading quantum systems from multiple servers each of which contains the copy of a classical file set while the identity of the downloaded file is not leaked to each server. The QPIR capacity is defined as the maximum rate of the file size over the whole dimension of the downloaded quantum systems. When the servers are assumed to share prior entanglement, we prove that the QPIR capacity with multiple servers is 1 regardless of the number of servers and files. We construct a rate-one protocol only with two servers. This capacity-achieving protocol outperforms its classical counterpart in the sense of the capacity, server secrecy, and upload cost. The strong converse bound is derived concisely without using any secrecy condition. We also prove that the capacity of multi-round QPIR is 1.
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 m1 of the attacked channels over the entire network uses is less than a half of the network transmission rate m0 (i.e., m1 < m0/2), our code implements secret and correctable quantum communication of the rate m0 − 2m1 by 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.
We study the equivalence between non-perfect secret sharing (NSS) and symmetric private information retrieval (SPIR) with arbitrary response and collusion patterns. NSS and SPIR are defined with an access structure, which corresponds to the authorized/forbidden sets for NSS and the response/collusion patterns for SPIR. We prove the equivalence between NSS and SPIR in the following two senses. 1) Given any SPIR protocol with an access structure, an NSS protocol is constructed with the same access structure and the same rate. 2) Given any linear NSS protocol with an access structure, a linear SPIR protocol is constructed with the same access structure and the same rate. We prove the first relation even if the SPIR protocol has imperfect correctness and secrecy. From the first relation, we derive an upper bound of the SPIR capacity for arbitrary response and collusion patterns. For the special case of n-server SPIR with r responsive and t colluding servers, this upper bound proves that the SPIR capacity is (r − t)/n. From the second relation, we prove that a SPIR protocol exists for any response and collusion patterns.
Quantum private information retrieval (QPIR) is the problem to retrieve one of f classical files by downloading quantum systems from non-communicating n servers each of which contains the copy of f files, while the identity of the retrieved file is unknown to each server. As an extension, we consider the (n−1)-private QPIR that the identity of the retrieved file is secret even if any n−1 servers collude, and derive the QPIR capacity for this problem which is defined as the maximum rate of the retrieved file size over the download size. For an even number n of servers, we show that the capacity of the (n−1)-private QPIR is 2/n, when we assume that there are preexisting entanglements among the servers and require that no information of the nonretrieved files is downloaded. We construct an (n − 1)-private QPIR protocol of rate ⌈n/2⌉ −1 and prove that the capacity is upper bounded by 2/n. The (n − 1)-private QPIR capacity is strictly greater than the classical counterpart. Replicated serverswith collusion- * n, f, and t: the number of servers, files, and colluding servers, respectively. † This paper derives the capacity for any even number n and the number t = n − 1.
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