Bell’s inequality for the Mach-Zehnder interferometer

If a single photon impinges on a Mach-Zehnder interferometer, the probability of detecting the photon in a certain output channel depends on the difference between the phase delays imposed by the two interfering channels. The photon behaves as if it goes through both interfering channels. On the other hand, if detectors are inserted into each interfering channel, the photon is only detected in one of the channels. In the delayed-choice experiment, the detectors may even be inserted in the last instant. Then the experimenter’s choice whether to insert these detectors or not determines whether the photon shall behave “as if” it goes through one or both interfering channels. It might seem that the detectors impart some sort of “nonlocal collapse” to the photon state. Otherwise, one may imagine that the photon itself goes through one channel only accompanied by an “empty wave” in the other channel. Such models have been developed by de Broglie [1] and later by Bohm [2]. They are known to be nonlocal. A common feature of both the “collapse” picture and the “empty wave” picture is that they refer to systems which appear to be “delocalized” (yielding interference) in a certain experiment and “localized” (yielding anticorrelation) in another. It may seem that such behavior is in some way nonlocal. Although classical optical fields may display interference effects in the same way as single photons, they cannot simultaneously be anticorrelated. One may also imagine stochastic classical fields which yield anticorrelation between two channels, but such two-channel systems may not give rise to interference effects when superposed. One might suspect that two-channel quantum states which yield both interference and anticorrelation in a Mach-Zehnder interferometer violate local realism. In this paper we shall see that this is indeed the case, and