Blind Remote Implementation of Arbitrary Two-Qubit Unitary Operation via Pre-Shared Entanglement

We propose a protocol for the remote implementation of an arbitrary two-qubit unitary operation using pre-shared entanglement. In our protocol, Bob possesses an unknown two-qubit state, while Alice implements the required operations using calibrated black-box devices. Alice does not need to know the parameters of these operations, and Bob’s unknown state never leaves his laboratory. The protocol is based on the Euler decomposition of single-qubit unitary operations and the Cartan decomposition of two-qubit unitary operations. First, we construct protocols for the remote implementation of the single-qubit operation $U=R_z(\gamma )R_x(\beta )R_z(\alpha )$ using pre-shared Bell states. We then construct a protocol for the two-qubit operation $R_{zz}(\theta )$ using a single pre-shared Bell state and extend it to $R_{xx}(\theta )$ and $R_{yy}(\theta )$ through local unitary transformations. These elementary protocols are finally combined to implement an arbitrary two-qubit unitary operation. A third party, Charlie, determines the required decomposition and provides Alice with the corresponding black-box devices before the protocol begins. Thus, neither Alice nor Bob requires the classical description of the unitary operation.