Quantum advantage in transmitting a permutation

We describe a quantum scheme to "color code" a set of objects in order to record which one is which. In the classical case, N distinct colors are required to color code N objects. We show that, in the quantum case, only N/e distinct "colors" are required, where eapproximate to2.718 28 is Euler's constant. If the number of colors is less than optimal, the objects may still be correctly distinguished with some success probability less than 1. We show that the success probability of the quantum scheme is better than the corresponding classical one and is information-theoretically optimal.