Multi-Choice Minority Game

The generalization of the problem of adaptive competition, known as the minority game, to the case of $K$ possible choices for each player is addressed, and applied to a system of interacting perceptrons with input and output units of the type of $K$-states Potts-spins. An optimal solution of this minority game as well as the dynamic evolution of the adaptive strategies of the players are solved analytically for a general $K$ and compared with numerical simulations.Comment: 5 pages, 2 figures, reorganized and clarifie

Phase variance of squeezed vacuum states

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number, $n$, as dictated by the Heisenberg limit, i.e., as $n^{-2}$, only for $N>4$. For $N\leq 4$ this fundamental scaling breaks down and it becomes $n^{-N/2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling $n^{-2}$ is recovered for large samples.Comment: Minor changes, version to appear in PRA, 8 pages, 2 figure