On the Capacity of Quantum Private Information Retrieval From MDS-Coded and Colluding Servers

Abstract

In quantum private information retrieval (QPIR), a user retrieves a classical file from multiple servers by downloading quantum systems without revealing the identity of the file. The QPIR capacity is the maximal achievable ratio of the retrieved file size to the total download size. In this paper, the capacity of QPIR from MDS-coded and colluding servers is studied for the first time. Two general classes of QPIR, called stabilizer QPIR and dimension-squared QPIR induced from classical strongly linear PIR are defined, and the related QPIR capacities are derived. For the non-colluding case, the general QPIR capacity is derived when the number of files goes to infinity. A general statement on the converse bound for QPIR with coded and colluding servers is derived showing that the capacities of stabilizer QPIR and dimension-squared QPIR induced from any class of PIR are upper bounded by twice the classical capacity of the respective PIR class. The proposed capacity-achieving scheme combines the star-product scheme by Freij-Hollanti et al. and the stabilizer QPIR scheme by Song et al. by employing (weakly) self-dual Reed-Solomon codes. Partial results have been published at ISIT 2021 [1]. C. Hollanti and M.

Unified Approach to Secret Sharing and Symmetric Private Information Retrieval with Colluding Servers in Quantum Systems

“…Since 𝑃 {1,2} (𝐺 (1) , 𝐹) = 1 0 2 0 1 0 1 0 0 1 1 0 0 1 0 2 , 𝑃 {2,3} (𝐺 (1) , 𝐹) = 1 0 1 0 2 2 1 2 0 1 0 2 0 2 1 2 (36) are invertible, the MMSP (𝐺 (1) , 𝐹) accepts 𝔄. Since the matrices 𝑃 {1} 𝐺 (1) = 𝑃 {2} 𝐺 (1) = 1 0 0 1 , 𝑃 {3} 𝐺 (1) = 2 2 0 2 , (37) are invertible, the MMSP (𝐺 (1) , 𝐹) rejects 𝔅 1 . Hence, (𝐺 (1) , 𝐹) is an (𝔄, 𝔅)-MMSP.…”

“…As quantum versions of SPIR, many existing papers [32][33][34][35][36][37][38][39][40] studied classical-quantum SPIR (CQSPIR), in which the file to be sent is given as a classical information. However, no existing paper studied the relation between CQSPIR and quantum versions of SS while such relation in the classical version was studied in [10].…”

“…That is, under this problem setting, we can discuss the case when only a part of servers answers the query sent by the user. In fact, no existing study addressed general access structure for CQSPIR because all existing studies [32][33][34][35][36][37][38][39][40] considered only the case when the file information is recovered with answers from all severs. This paper is the first paper to address the case when a set of servers is qualified to recover the file information.…”

Two‐Server Oblivious Transfer for Quantum Messages

“…It proved that a protocol can be constructed without any communication loss when the message is classical information, a noiseless quantum channel is available, and prior entanglement among servers is allowed. The papers [40,41] and Allaix et al [42,43] also considered this problem with colluding servers in which secrecy of the protocol is preserved even if some servers may communicate and collude. Kon and Lim [44] constructed a two-server oblivious transfer protocol with quantum-key distribution and Wang et al [45,46] implemented two-server oblivious transfer protocols experimentally.…”

Two-Server Oblivious Transfer for Quantum Messages

“…It proved that a protocol can be constructed without any communication loss when the message is classical information, a noiseless quantum channel is available, and prior-entanglement among servers is allowed. The papers [35,36] and Allaix et al [37,38] also considered this problem with colluding servers in which secrecy of the protocol is preserved even if some servers may communicate and collude. Kon and Lim [39] constructed a two-server oblivious transfer protocol with quantum-key distribution and Wang et al [40,41] implemented two-server oblivious transfer protocols experimentally.…”