Asymptotically independent fluctuations of stabilizer Rényi entropy and entanglement in random unitary circuits

We investigate numerically the joint distribution of magic (as described by the stabilizer 2-Rényi entropy M2) and entanglement (as described by the von Neumann entropy S) in N-qubit Haar-random quantum states. The distribution PN(M2,S) and the marginals become exponentially localized, and centered around the typical values, M2̃→N−2 and S̃→N/2 as N→∞. Magic and entanglement fluctuations are, however, found to become exponentially uncorrelated. While the set of stabilizer states grows exponentially with N, they nevertheless form a measure-zero subset of Haar-random states and thus occur with vanishing probability. Typical quantum states are characterized by large magic and entanglement entropy, and uncorrelated magic and entanglement fluctuations. The mutual information of the joint distribution vanishes exponentially as N→∞, implying that fluctuations of these two resources are not only uncorrelated but asymptotically independent.