Capacity of Quantum Private Information Retrieval with Colluding Servers

Quantum private information retrieval (QPIR) is a protocol in which a user retrieves one of multiple files from $\mathsf{n}$ non-communicating servers by downloading quantum systems without revealing which file is retrieved. As variants of QPIR with stronger security requirements, symmetric QPIR is a protocol in which no other files than the target file are leaked to the user, and $\mathsf{t}$-private QPIR is a protocol in which the identity of the target file is kept secret even if at most $\mathsf{t}$ servers may collude to reveal the identity. The QPIR capacity is the maximum ratio of the file size to the size of downloaded quantum systems, and we prove that the symmetric $\mathsf{t}$-private QPIR capacity is $\min\{1,2(\mathsf{n}-\mathsf{t})/\mathsf{n}\}$ for any $1\leq \mathsf{t}< \mathsf{n}$. We construct a capacity-achieving QPIR protocol by the stabilizer formalism and prove the optimality of our protocol. The proposed capacity is greater than the classical counterpart.