Controlled bidirectional remote implementations of partially unknown quantum operations

Abstract By blending the ideas of controlled teleportation, bidirectional teleportation and remote implementation of quantum operations, we investigate the bidirectional transmission of quantum operations under the control of one supervisor. Firstly, we provide a method for generating a class of multi-particle entangled states, including five-particle and seven-particle entangled states, using a nine-particle entangled state as an example. Then using our constructed five-particle entangled state as a quantum channel, we propose a new controlled bidirectional protocol for remotely implementing partially known quantum operations (PUQOs for short) of one qubit within the two restricted sets satisfying specific conditions. Subsequently, exploiting a nine-particle entangled state as the quantum channel, we extend it to the case of two-qubit quantum operations which come from eight restricted sets. Replacing two PUQO of two qubits by a PUQO of one qubit and a PUQO of two qubits, we put forward another new asymmetric controlled bidirectional protocol for remotely implementing PKQOs via a seven-particle entangled state as the quantum channel. In these protocols, by using universal recovered operations performed by the receivers, the senders can simultaneously exchange their PUQOs as long as the controller collaborates, where each PUQO acts on the corresponding unknown state owned by the remote recipient.