Contrast in multipath interference and quantum coherence

We develop a rigorous connection between statistical properties of an interference pattern and the coherence properties of the underlying quantum state. With explicit examples, we demonstrate that even for inaccurate reconstructions of interference patterns, properly defined statistical moments permit a reliable characterization of quantum coherence in terms of the number of coherently superposed state vectors.

Figure 1

Upper part: interference patterns corresponding to different types of coherence in mixed states. Lower part: an interference pattern that results from a state distributed over three path alternatives. Reduced coherence between the three paths can sometimes be expressed as the sum of two-path interference patterns. In that case the pattern can result from a two-coherent state.

Figure 2

Three-path extension of the Mach-Zehnder interference setup. An incoming particle is split into three contributions. After passing the individual phase shifters, the three beams are recombined at the second beam splitter, where they interfere.

Figure 3

Dependence of the average detection ratio $〈R〉$ on width $\sigma$ and error ${\sigma }_G$ (indicated by the color scale or gray scale). The depicted data refer to $Q_3, d=k=7$ and the ensemble given by Eq. (14). There is a broad region (${\sigma }_G\lesssim 0.15$ and $\sigma \lesssim 0.7$) in which $Q_3$ performs nearly as well as in the ideal case ${\sigma }_G=0$ and $\sigma \to 0$.

Figure 4

Dependence of the average detection ratio $〈R〉$ on width $\sigma$ for ${\sigma }_G=0.4$. As discussed above, the performance of $Q_1$ is independent of $\sigma$, but there is an optimal, finite $\sigma$ for $Q_2$ and $Q_3$.

Figure 5

The values $Q_2^{\left(k\right)}$ depict the numerically obtained, purity-dependent threshold values of $k$ coherence for $k=2,3,4,5$ in a $d=5$-dimensional system and $\sigma =1$. The solid curve denotes the analytical result obtained for the state (17). The horizontal lines denote the original threshold values obtained by pure states.