Probabilistic nonlocal gate operation via imperfect entanglement

Nonlocal gate operation is based on sharing an ancillary pair of qubits in perfect entanglement. When the ancillary pair is partially entangled, the efficiency of gate operation drops. Using general transformations, we devise probabilistic nonlocal gates, which perform the nonlocal operation conclusively when the ancillary pair is only partially entangled. We show that a controlled purification protocol can be implemented by the probabilistic nonlocal operation.