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

Abstract

We introduce a quantifier of phase-space complexity for discrete-variable (DV) quantum systems. Motivated by a recent framework developed for continuous-variable systems, we construct a complexity measure of quantum states based on the Husimi Q-function defined over spin coherent states. 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 derive fundamental properties of this measure, including its invariance under SU(2) displacements. The complexity is normalized such that coherent states have unit complexity, while the completely mixed state has zero complexity, a feature distinct from the continuous-variable case. 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. Numerical results reveal a monotonic, but not deterministic, relationship between complexity and purity, leading us to 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. We analyze the performance of common unitary gates and the amplitude damping channel, showing that while low-dimensional systems can achieve maximal complexity via spin squeezing or NOON states, this becomes impossible in higher dimensions. These 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.

Figures (6)

• Figure 1: Complexity $\mathcal{C} (\rho_{\rm qubit})$ of a generic qubit state as a function of its purity ${\rm tr} (\rho_{\rm qubit}^2)$.
• Figure 2: Complexity $\mathcal{C} (\rho_{\beta})$ of a Gibbs equilibrium state as a function of its purity ${\rm tr} (\rho_\beta^2)$ for $j=\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2}$ (from bottom to top). Note that the blue solid line for $j=\frac{1}{2}$ coincides with Fig. \ref{['fig:qubit']} because a generic qubit state can be obtained from a Gibbs state via a rotation, which does not change complexity.
• Figure 3: (a)-(e) Histograms of complexity. Each subfigure shows the complexity of $10^4$ randomly generated states for the given spin value $j$. (f) Smooth kernel histograms of (a)-(e), where the vertical axis shows the value of the probability density. When $j\geq1$, the most probable (typical) value of complexity is around 0.3 to 0.4, regardless of the spin value.
• Figure 4: Complexity against purity. The plot reports the complexity of randomly generated states as a function of their purity. We have generated $10^4$ samples for each value $j$ of the spin. The blue solid line is for qubit ($j=\frac{1}{2}$) and coincides with the blue solid line of Fig. \ref{['fig:thermal']}. The other points are for $j=1$ (orange), $j =\frac{3}{2}$ (green), $j=2$ (red), $j=\frac{5}{2}$ (purple). The complexity is fully determined by the purity only for qubits, whereas for higher dimensions purity is no longer the sole factor. Nevertheless, it is apparent from the plot the complexity is increasing with purity, suggesting the maximal complexity is achieved by pure states.
• Figure 5: Complexity $\mathcal{C} (s_1(\eta))$ of one-axis spin-squeezed states (left) and $\mathcal{C} (s_2(\eta))$ of two-axis spin-squeezed states (right) as a function of the squeezing parameter $\eta$, for $j=1,\frac{3}{2},2,\frac{5}{2}$ (from bottom to top).