  Quantum private information retrieval (QPIR) for quantum messages is the
protocol in which a user retrieves one of the multiple quantum states from one
or multiple servers without revealing which state is retrieved. We consider
QPIR in two different settings: the blind setting, in which the servers contain
one copy of the message states, and the visible setting, in which the servers
contain the description of the message states. One trivial solution in both
settings is downloading all states from the servers and the main goal of this
paper is to find more efficient QPIR protocols. First, we prove that the
trivial solution is optimal for one-server QPIR in the blind setting. In
one-round protocols, the same optimality holds even in the visible setting. On
the other hand, when the user and the server share entanglement, we prove that
there exists an efficient one-server QPIR protocol in the blind setting.
Furthermore, in the visible setting, we prove that it is possible to construct
symmetric QPIR protocols in which the user obtains no information of the
non-targeted messages. We construct three two-server symmetric QPIR protocols
for pure states. Note that symmetric classical PIR is impossible without shared
randomness unknown to the user.
