Remote interactions between two d -dimensional distributed quantum systems: nonlocal generalized quantum control-NOT gate and entanglement swapping

We present a systematic simple method to implement a generalized quantum control-NOT (CNOT) gate on two d-dimensional distributed systems. First, we show how the nonlocal generalized quantum CNOT gate can be implemented with unity fidelity and unity probability by using a maximally entangled pair of qudits as a quantum channel. We also put forward a scheme for probabilistically implementing the nonlocal operation with unity fidelity by employing a partially entangled qudit pair as a quantum channel. Analysis of the scheme indicates that the use of partially entangled quantum channel for implementing the nonlocal generalized quantum CNOT gate leads to the problem of `the general optimal information extraction'. We also point out that the nonlocal generalized quantum CNOT gate can be used in the entanglement swapping between particles belonging to distant users in a communication network and distributed quantum computer.