Independent stabilizer Rényi entropy and entanglement fluctuations in random unitary circuits

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 $\widetilde{M_2}\to N-2$ 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.