# Phase Space Insights: Wigner Functions for Qubits and Beyond
## 1 Introduction  
Phase space methods offer a compelling alternative to the traditional Hilbert-space approach in quantum theory, providing useful tools for visualizing and analyzing quantum states. Within this framework, observables are represented as functions instead of operators, mirroring their classical counterparts while adhering to distinct algebraic rules. By interpreting quantum mechanics as a statistical theory on phase space, this perspective facilitates a more seamless and intuitive transition to the classical world [1, 2, 3, 4]. The rudiments of the method were laid in the groundbreaking works of [PERSON] [5] and [PERSON] [6]. Later, [PERSON] [7] and [PERSON] [8] built upon this foundation, which has since evolved into a comprehensive discipline with applications in numerous fields. The cornerstone of this approach lies in a bona fide mapping that assigns operators to functions (commonly known as the associated symbols) defined on a smooth manifold with a well-established mathematical structure [9]. However, this mapping is not uniquely determined; rather, a whole family of symbols can consistently correspond to a given operator. In particular, quasiprobability distributions are the symbols of the density operator [10, 11, 12, 13, 14, 15]. For continuous variables, such as Cartesian position and momentum--the paradigmatic example that originally spurred interest in this field--the most widely used mappings include the time-honored \(P\) ([PERSON]) [16, 17], \(W\) ([PERSON]) [6], and \(Q\) ([PERSON]) [18] functions. The formalism has been generalized to systems exhibiting diverse symmetries. Beyond the harmonic oscillator, an important example is the SU(2) symmetry, where the Bloch sphere serves as the phase space [19; 20; 21]. Likewise, the Euclidean group E(2), associated with the cylinder as its phase space, has produced valuable results [22; 23; 24; 25; 26; 27; 28], and plays a key role in analyzing the orbital angular momentum of twisted photons [29]. More recently, systems with SU(1,1) symmetry have been described using a two-sheeted hyperboloid as the basic manifold [30; 31], which plays a central role in modeling two-photon effects [32; 33; 34; 35]. This topic has gained renewed interest with the advent of nonlinear interferometers, which promise remarkable enhancements in phase measurement sensitivity [36; 37; 38; 39]. Furthermore, generalizations to broader classes of dynamical groups have also been investigated [40; 41; 42; 43; 44; 45]. In the late 1980s, researchers began exploring how to represent qubits--and more generally, qudits--in phase space. In 1987, both [PERSON] [46] and [PERSON] [47] independently proposed versions of the Wigner function for discrete systems. [PERSON]'s work, based on earlier informal ideas about negative probabilities, focused specifically on spin-1/2 particles. [PERSON], on the other hand, developed a more general approach using a discrete toroidal lattice, which works for systems whose dimensions are prime powers. Since then, there has been ongoing effort to extend this framework and make it more widely applicable [48; 49; 50; 51; 52; 53; 54; 55; 56]. As quantum technologies advanced into the twenty-first century, the direct measurement of phase space shifted from a theoretical goal to a practical and powerful technique. The Wigner function, with its distinctive visual and mathematical properties, has become essential in a wide range of research areas. There are two primary experimental approaches to access the Wigner function: indirect and direct. The indirect method involves first reconstructing the full quantum state--typically by determining the density matrix--and then computing the Wigner function through a mathematical transformation. This approach supports both continuous and discrete variables. The review presented in [57] includes a complete and updated list of references to recent experiments. The direct method, on the other hand, enables a point-by-point measurement of the Wigner function without reconstructing the entire quantum state.