We analyze and further develop a method to represent the quantum state of a system of
n qubits in a phase-space grid of $N\ifmmode\times\else\texttimes\fi{}N$ points (where
N=2n). 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
GF(2n) 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.
We analyze some features of the class of discrete Wigner functions that was recently introduced by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to represent quantum states of systems with power-of-prime dimensional Hilbert spaces. We consider ``cat'' states obtained as coherent superpositions of states with positive Wigner function; for such states we show that the oscillations of the discrete Wigner function typically spread over the entire discrete phase space (including the regions where the two interfering states are localized). This is a generic property, which is in sharp contrast with the usual attributes of Wigner functions that make them useful candidates to display the existence of quantum coherence through oscillations. However, it is possible to find subsets of cat states with a natural phase-space representation, in which the oscillatory regions remain localized. We show that this can be done for interesting families of stabilizer states used in quantum error-correcting codes, and illustrate this by analyzing the phase-space representation of the five-qubit error-correcting code.
We develop a holomorphic representation of quantum error correction codes (QECCs) within the Segal--Bargmann space. By encoding qubits into Schwinger boson modes
(zaj,zbj) subject to a degree-one homogeneity constraint, we derive closed-form differential operator representations for stabilizers, syndrome extraction, and recovery for fundamental codes (three-, five-, seven-, and nine-qubit codes). Quantum errors are characterized as holomorphic perturbations violating this constraint, while syndrome measurement projects onto eigenspaces of commuting differential operators. Restricting to unit-magnitude variables (
∣z∣=1) reveals a toroidal space $\T^{2n}$ where error syndromes manifest as discrete translations in winding number space
Z2n, and recovery acts as Hamiltonian flows restoring the winding configuration. In the full Segal--Bargmann space, the code space is a holomorphic submanifold of $\CP^{2^n-1}$, with correctable errors as transverse normal directions. Consequently, the Knill--Laflamme condition becomes a Fubini--Study orthogonality condition between the code submanifold and its error-translated images. Topological protection emerges from the
U(1)n fiber bundle structure: global phase noise along fibers is unobservable, while base-space errors require active correction. Finally, we establish a path-integral formulation for semiclassical error correction dynamics and show that geometric entanglement via the Segre embedding naturally quantifies code distance. This framework unifies algebraic, geometric, and topological perspectives on fault-tolerant quantum protocols.