Extrema of discrete Wigner functions and applications

We study the class of discrete Wigner functions proposed by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to describe quantum states using a discrete phase space based on finite fields. We find the extrema of such functions for small Hilbert-space dimensions and present a quantum-information application: a construction of quantum random-access codes. These are constructed using the complete set of phase-space point operators to find encoding states and to obtain the codes’ average success rates for Hilbert-space dimensions 2, 3, 4, 5, 7, and 8.

Figure 1

Encoding states for the $3\to 1$ QRAC using a single qubit. Alice prepares one out of eight states on the vertices of a cube inscribed within the Bloch sphere, depending on her three-bit string. The angle $\theta$ is such that ${\cos }^2(\theta ∕2)=1∕2+\sqrt{(}3)∕6\simeq 0.79$, which is the probability of Bob correctly decoding a single bit out of the three.