Two interferometric complementarities

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 ${\mathit{v}}_1$ when amplitudes from two paths are combined. It is shown that [maxD(scrP)${]}^2$+${\mathit{v}}_1^2$=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 ${\mathit{v}}_{12}$ of two-particle fringes: ${\mathit{v}}_{12}^2$+${\mathit{v}}_1^2$≤1. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.