Sorelli et al. Figure 3. Marginals ofthe Wigner function ofa two-mode squeezed vacuum state with mean photon numberNs ¼ 3. The top row shows the distribution ofthe q and p quadratures ofeach ofthe two modes. The distributions in the bottom row demonstrates the correlation between the two modes. DISCRIMINATING QUANTUM STATES AND PROBABILITY DISTRIBUTIONS Let us now consider the problem of deciding between two hypotheses H0 and H1 based on some prior knowledge of the conditional probability densities p0ðRÞ and p1ðRÞ for the random variable R when one of the two hypotheses is true. The problem of deciding between one of the two hypotheses is, therefore, equivalent to sample from two different probability distributions and discriminate them from the sampling results. In the following, we will discuss the case when p0ðRÞ and p1ðRÞ are classical probability densities, as well as the one when they originate from the quantum states r0 and r1 corresponding to the two hypotheses. Let us assume that R spans a real space Z, which we call the decision space. To decide between the two hypotheses, let us divide the decision space Z into two regions Z0 and Z1. When R 2 Z0, we decide that H0 is true, and when R 2 Z1, we decide otherwise. How one constructs the regions Z0 and Z1 in order to minimize the error in choosing which hypothesis is true defines a decision