Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem

We analyze and further develop a method to represent the quantum state of a system of $n$ qubits in a phase-space grid of $N\times N$ points (where $N=2^n$). The method, which was recently proposed by Wootters and co-workers (Gibbons et al., Phys. Rev. A 70, 062101 (2004).), is based on the use of the elements of the finite field $\mathrm{GF}\left(2^n\right)$ to label the phase-space axes. We present a self-contained overview of the method, we give insights into some of its features, and we apply it to investigate problems which are of interest for quantum-information theory: We analyze the phase-space representation of stabilizer states and quantum error-correction codes and present a phase-space solution to the so-called mean king problem.