Van Hove singularities in stabilizer entropy densities

TL;DR

This work analyzes Haar-induced probability densities of measures of non-stabilizerness, focusing on Stabilizer Rényi entropies (SREs) for random pure states. By mapping SREs to a density-of-states problem on the Bloch sphere, it uncovers Van Hove-type logarithmic singularities at saddle points, notably a divergence at m_c = log(4/3) for the order-2 SRE and an exact PDF derived for α = 2, with the divergence disappearing for Hilbert-space dimension d ≥ 3. The study further links the one-qubit linear stabilizer entropy to partial incompatibility of measurements, tying magic to a fundamental quantum-mechanical property. Collectively, the results reveal a geometric structure in the statistics of magic and its dimension-dependent behavior, with implications for quantum resource theories and the role of measurement incompatibility.

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 Rényi 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 $α= 2$, 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.

Figures (6)

• Figure 1: Probability density function of SRE of order two according to the Haar measure for one qubit. The divergence, of logarithmic type, takes place at $m_c=\log(4/3)=0.287\ldots$, corresponding to the 12 magic states in the Clifford orbit of $\ket{H}=(\ket{0}+e^{i \pi /4}\ket{1})/\sqrt{2}$.
• Figure 2: Intersection between $\ell^{2\alpha}$ and $\ell^{2}$spheres, here for $\alpha=4$. Left panel $n=0.12<n_{c}$, right panel $n=0.13>n_{c}$.
• Figure 3: Integration regions to compute the stabilizer purity PDF for $\alpha=4$. The grey arrow is the critical point $\boldsymbol{n}_{c}=\left(1/\sqrt{2},1/\sqrt{2},0\right)$ and the black curves are the hyperbolae where integration takes place for $n$ close to $n_{c}=1/8$. Left panel $n=0.124<n_{c}$, right panel $n=0.126>n_{c}$.
• Figure 4: In blue the numerical probability density function of $P_{N_2}(\textit{n})$ using $N_{sample}=10^{7}$ random pure states extracted according to the Haar measure. In red the theoretical analytical distribution.
• Figure 5: Probability density function of $P_{N_2}(\textit{n})$ for 2 and 6 qubits extracted numerically for $N_{sample}=2 \times 10^5$ pure Haar random states.

Theorems & Definitions (1)

• Definition 1: Stabilizer entropies leone2022StabilizerRenyiEntropy