Single-quanta interferometry: which-way versus which-phase information stored in an ancillary quantum system

In interferometers, the more information about the quanta's path available in an ancillary quantum system (AQS), the less visibility the interference has. By use of Shannon entropy, we try to compare the amount of which-phase information with the amount of which-way information stored in the AQS of two-path interferometers with symmetric beam merging. We show that the former is lower than or equal the latter if the bipartite system of the single-quanta and the AQS is initially prepared in a pure state and the interaction between the two parts is unitary. Especially when there exists symmetry, the equality holds. No which-way information is obtained by the measurement that we use for extracting the which-phase information and vice versa. In order to verify the results experimentally, we propose assembling a new single-photon interferometer.
Comment: revised slightly