Interference and Distinguishability in Quantum Mechanics

Quantitative measures are introduced for the indistinguishability U of two quantum states in a given measurement and the amount of interference I observable in this measurement. It is shown that these measures obey an inequality U ≥ I which can be seen as an exact formulation of Bohr’s claim that one cannot distinguish between two possible paths of a particle while maintaining an interference phenomenon. This formulation is applied to a neutron interferometer experiment of Badurek e.a. It is shown that the formulation is stronger than an argument based on an uncertainty relation for phase and photon number considered by these authors. A recent experiment in neutron interferometry [1] can be seen as a realisation of the double-slit thought experiment discussed by Einstein and Bohr. In this famous discussion, Bohr argued that one cannot distinguish between two possible paths of a particle while preserving an interference phenomenon. To reach this conclusion Bohr applied the uncertainty relation for position and momentum in a somewhat informal way. However, it has been shown that the Heisenberg uncertainty relation by itself is not strong enough to justify Bohr’s claim [2]. The discussion has been revived in the light of the new neutron experiments. In ref. 1 an argument is presented supporting Bohr’s claim, based on an uncertainty relation for the phase and photon number of the electromagnetic field. The validity of this explanation was subsequently disputed [3], because of the dubious theoretical status of the phase-number uncertainty relations. This raises the question whether it is possible to give a direct quantitative formulation of Bohr’s claim, without recourse to the uncertainty relations. Work in this direction has been done by Wootters and Zurek[4]. Here we propose an alternative formulation that seems particularly apt for the interferometer experiments. The main obstacle for a direct formulation of Bohr’s claim is the problem of choosing a quantitative measure for the extent to which the two paths are distinguishable in a given experiment. Let ψ1 and ψ2 denote orthogonal quantum states that represent possible paths of a particle. A measurement performed on this particle may be described by a complete set of orthogonal projection operators {Dk}, where k denotes a possible outcome of the measurement. The hypotheses that the particle traveled either one of two paths then provide two probability distributions, viz. pk = 〈ψ1|Dk|ψ1〉, (1) qk = 〈ψ2|Dk|ψ2〉. The two paths may be said to be discriminated if, from the observed outcome of the measurement one can decide between these two hypotheses. Thus, the problem of distinguishing between two paths can be seen as a special case of the general classical problem of discriminating between two statistical hypotheses.