For a general entangling probe attacking the Bennett-Brassard 1984 protocol in quantum key distribution, I calculate three classes of optimized unitary transformations, all yielding the same maximum information to the probe. The simplest one corresponds to a probe having a two-dimensional Hilbert space of states, and is uniquely determined by the error rate induced by the probe in the legitimate receiver. The second class corresponds to a probe having a four-dimensional Hilbert space of states, and is determined by the error rate and two continuous angle parameters which are mutually constrained by the error rate. The third class corresponds to a probe having a four-dimensional Hilbert space, and is determined by the error rate and two continuous angle parameters, one of which is constrained by the error rate. Furthermore, I show that the simplest quantum circuit representing the optimal entangling probe consists of a single controlled-NOT gate in which the control qubit consists of two polarization-basis states of the signal, the target qubit consists of two probe-basis states, and the initial state of the probe is set by the error rate. A method is determined for measuring the appropriate correlated state of the probe. Finally, a possible implementation of the entangling probe is described.

• Received 20 September 2004

DOI:https://doi.org/10.1103/PhysRevA.71.042312