Independent stabilizer Rényi entropy and entanglement fluctuations in random unitary circuits

TL;DR

This paper analyzes the relationship between magic (Stabilizer Rényi entropy M2) and entanglement in Haar-random N-qubit states. Through numerical studies of marginals and the joint distribution, it shows that both magic and entanglement are sharply concentrated around N−2 and N/2, respectively, with variances decaying exponentially in N. Crucially, the covariance between magic and entanglement fluctuations decays as 2^{-3N}, implying that these resources become effectively uncorrelated in the thermodynamic limit. The results suggest that typical quantum states are highly complex, possessing large both resources, while atypical combinations are exponentially rare, informing the understanding of quantum complexity and resource theories in large systems.

Abstract

We investigate numerically the joint distribution of magic ($M$) and entanglement ($S$) in $N$-qubit Haar-random quantum states. The distribution $P_N(M,S)$ as well as the marginals become exponentially localized, and centered around the values $\tilde{M_2} \to N-2$ and $\tilde{S} \to N/2$ as $N\to\infty$. Magic and entanglement fluctuations are, however, found to become exponentially uncorrelated. Although exponentially many states with magic $M_2=0$ and entropy $S\approx S_\text{Haar}$ exist, they represent an exponentially small fraction compared to typical quantum states, which are characterized by large magic and entanglement entropy, and uncorrelated magic and entanglement fluctuations.

Figures (7)

• Figure 1: Convergence with circuit depth. Distribution of magic $P_N(M_2)$ generated by random unitary operators $U \in \mathcal{U}_N$ for $N=5$. In a brickwall construction with only 2-qubit gates, the distribution converges to the one generated by a single unitary operator at circuit depth $D \approx 2N$.
• Figure 2: Distribution of magic $P_N(M_2)$ generated by random unitary operators $U \in \mathcal{U}_N$. Increasing $N$, the distribution becomes concentrated around the value $\tilde{M}_2 \approx N-2$. Inset: distribution of magic density $m_2=M_2/N$.
• Figure 3: Distribution of magic $P_N(M_2)$ centered around the first cumulant $\kappa_1=\langle M_2 \rangle$ on a semilog scale. Black dashed lines correspond to a normal distribution with mean $\kappa_1$ and variance $\kappa_2=\langle (M_2 - \langle M_2\rangle)^2 \rangle$ extracted from $P_N(M_2)$.
• Figure 4: Distribution of entanglement entropy $P_N(S)$ generated by random unitary operators $U \in \mathcal{U}_N$. Increasing $N$, the distribution becomes concentrated around the value $\tilde{S}_2 \approx N/2$. Insets: distribution of entropy density $s=S/N$.
• Figure 5: Distribution of magic $P_N(S)$ centered around the first cumulant $\kappa_1=\langle S \rangle$ on a semilog scale. Black dashed lines correspond to a normal distribution with mean $\kappa_1$ and variance $\kappa_2=\langle (S - \langle S\rangle)^2 \rangle$ extracted from $P_N(S)$.