One-particle and two-particle visibilities in bipartite entangled Gaussian states

Complementarity between one-particle visibility and two-particle visibility in discrete systems can be extended to bipartite quantum-entangled Gaussian states implemented with continuous-variable quantum optics. The meaning of the two-particle visibility originally defined by Jaeger, Horne, Shimony, and Vaidman with the use of an indirect method that first corrects the two-particle probability distribution by adding and subtracting other distributions with varying degree of entanglement, however, deserves further analysis. Furthermore, the origin of complementarity between one-particle visibility and two-particle visibility is somewhat elusive and it is not entirely clear what is the best way to associate particular two-particle quantum observables with the two-particle visibility. Here, we develop a direct method for quantifying the two-particle visibility based on measurement of a pair of two-particle observables that are compatible with the measured pair of single-particle observables. For each of the two-particle observables from the pair is computed corresponding visibility, after which the absolute difference of the latter pair of visibilities is considered as a redefinition of the two-particle visibility. Our approach reveals an underlying mathematical symmetry as it treats the two pairs of one-particle or two-particle observables on equal footing by formally identifying all four observable distributions as rotated marginal distributions of the original two-particle probability distribution. The complementarity relation between one-particle visibility and two-particle visibility obtained with the direct method is exact in the limit of infinite Gaussian precision where the entangled Gaussian state approaches an ideal Einstein-Podolsky-Rosen state. The presented results demonstrate the theoretical utility of rotated marginal distributions for elucidating the nature of two-particle visibility and provide tools for the development of quantum applications employing continuous variables.