Complementarity and path distinguishability: Some recent results concerning photon pairs

Two results concerning photon pairs, one previously reported and one new, are summarized. It was previously shown that if the two photons are prepared in a quantum state formed from bar-A and bar-A' for photon 1 and bar-B and bar-B' for photon 2, then both one- and two-particle interferometry can be studied. If upsilon(sub i) is the visibility of one-photon interference fringes (i = 1,2) and upsilon(sub 12) is the visibility of two-photon fringes (a concept which we explicitly define), then upsilon(sub i) squared + upsilon(sub 12) squared is less than or equal to 1. The second result concerns the distinguishability of the paths of photon 2, using the known 2-photon state. A proposed measure E for path distinguishability is based upon finding an optimum strategy for betting on the outcome of a path measurement. Mandel has also proposed a measure of distinguishability P(sub D), defined in terms of the density operator rho of photon 2. We show that E is greater than or equal to P(sub D) and that upsilon(sub 2) = (1 - E(exp 2))exp 1/2.