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Two interferometric complementarities are formulated and demonstrated, and relations between them are analyzed. The first relates the distinguishability D (scrP) of the path of propagation of a particle (where scrP is the preparation of an ensemble) to the fringe visibility v₁ when amplitudes from two paths are combined. It is shown that maxD (scrP) ^2+v₁^2=1, where the maximum is taken over all preparations compatible with a fixed density operator. The second complementarity relates the visibility of one-particle interference fringes to the visibility v₁₂ of two-particle fringes: v₁₂^2+v₁^21. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.
Jaeger et al. (Sun,) studied this question.
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