Phase-space complexity of discrete-variable quantum states and operations

We introduce a quantifier of phase-space complexity for discrete-variable (DV) quantum systems. The quantifier combines into a single scalar quantity two complementary information-theoretic quantities, the Wehrl entropy, which captures phase-space spread, and the Fisher information, which captures localization. We provide analytic expressions for several relevant families of states, including Gibbs and Dicke states, and perform a numerical analysis of spin-squeezed states, NOON states, and randomly generated states. We conjecture that maximal complexity is attained by pure states, thereby connecting the problem to the optimization of Wehrl entropy via Majorana constellations. Finally, we extend the framework to quantum channels, defining measures for both the generation and breaking of complexity. Our results highlight dimension-dependent limitations in the generation of phase-space complexity and establish a unified phase-space approach to complexity across both continuous and discrete variable regimes.