Van Hove singularities in stabilizer entropy densities

Abstract: The probability distribution of a measure of non-stabilizerness, also known as magic, is investigated for Haar-random pure quantum states. Focusing on the stabilizer R\'enyi entropies, the associated probability density functions (PDFs) are found to display distinct non-analytic features analogous to Van Hove singularities in condensed matter systems. For a single qubit, the stabilizer purity exhibits a logarithmic divergence at a critical value corresponding to a saddle point on the Bloch sphere. This divergence occurs at the $|H\rangle$-magic states, which hence can be identified as states for which the density of non-stabilizerness in the Hilbert space is infinite. An exact expression for the PDF is derived for the case $\alpha = 2$, with analytical predictions confirmed by numerical simulations. The logarithmic divergence disappears for dimensions $d \ge 3$, in agreement with the behavior of ordinary Van Hove singularities on flat manifolds. In addition, it is shown that, for one qubit, the linear stabilizer entropy is directly related to the partial incompatibility of quantum measurements, one of the defining properties of quantum mechanics, at the basis of Stern-Gerlach experiments.