Connecting Continuous and Discrete Wigner Functions Via GKP Encoding

Wigner function is an intuitive and powerful tool for understanding quantum systems in terms of functions on phase space. However, it is uniquely defined only for continuous systems, and there is no universally accepted Wigner function for arbitrary discrete systems. Several different versions of discrete Wigner functions have been developed, each with its own usage. The basic reason for this phenomenon lies in that for discrete systems, the number-theoretic aspects of the system dimension come into play, and many properties of the system depend crucially on the dimension. The purpose of this work is twofold: First, we present a concise review of continuous and discrete Wigner functions. Second, we establish some connections between continuous and discrete Wigner functions via the Gottesman-Kitaev-Preskill (GKP) encoding, which encodes a qudit into a continuous system and allows us to use the tools and techniques developed for continuous quantum systems to study discrete quantum systems. We formalize GKP encoding from the perspective of linear mapping going beyond the Hilbert space framework, and employ it to define GKP-induced discrete Wigner function. We further reveal its basis properties, and offer a new perspective for understanding discrete Wigner functions in a unified fashion.