We investigate numerically the joint distribution of magic (as described by the stabilizer 2-Rényi entropy M 2 ) and entanglement (as described by the von Neumann entropy S ) in N -qubit Haar-random quantum states. The distribution P N ( M 2 , S ) and the marginals become exponentially localized, and centered around the typical values, M 2 ̃ → N − 2 and S ̃ → 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.
We investigate numerically the joint distribution of magic (as described by the stabilizer 2-Rényi entropy M 2 ) and entanglement (as described by the von Neumann entropy S ) in N -qubit Haar-random quantum states. The distribution P N ( M 2 , S ) and the marginals become exponentially localized, and centered around the typical values, M 2 ̃ → N − 2 and S ̃ → 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.
Entanglement is central to quantum information processing, while stabilizer operations underpin fault-tolerant quantum computation. We ask how much entanglement remains visible or distillable under stabilizer restrictions. We quantify stabilizer-visible entanglement by restricting the measured relative entropy of entanglement to stabilizer measurements, thereby obtaining converse bounds on entanglement distillation under stabilizer operations. We demonstrate magic-free asymptotic entanglement hiding: we construct explicit convex mixtures of pure stabilizer states on N qutrits per party whose unrestricted visible entanglement and LOCC-distillable entanglement both grow as Ω(N/logN), while their stabilizer-visible and stabilizer-distillable entanglement vanish as N→∞. Thus, an unbounded amount of LOCC-distillable entanglement carried by stabilizer states can become asymptotically invisible and undistillable under stabilizer restrictions. We further prove that stabilizer-visible entanglement is O(1) with high probability for Haar-random pure states despite extensive unrestricted visibility, and vanishes uniformly over entangled Werner states as the local dimension grows through odd primes. These results reveal a fundamental separation between entanglement and magic as resources, exposing intrinsic limits on entanglement extraction using stabilizer operations.
The quantum-trajectory description of a noisy quantum circuit is not unique: every Kraus representation of the noise channel defines a different unraveling, i.e., a different ensemble of pure states that averages to the same density matrix. While all unravelings share the same expectation values of observables, they generally differ in the amount of quantum resources carried by the individual trajectories. Here we study non-stabilizerness, or magic, quantified by the stabilizer Rényi entropy, in quantum-trajectory unravelings of Haar-random and T-doped random Clifford brickwork circuits subject to local dephasing noise of strength p. The canonical unraveling in terms of Pauli jumps is Clifford and leaves magic untouched, so that trajectories retain the Haar-random value of non-stabilizerness. Exploring the full two-parameter family of minimal unravelings, we find that the trajectory-averaged magic can be substantially reduced by a state-independent choice of Kraus operators, and that the optimal choice is not in general the projective-measurement unraveling, but a two-outcome generalized measurement in which the Kraus operators correspond to a weak measurement of the noisy qubit. The optimal unraveling reduces the extensive magic density of the trajectories by a p-dependent factor without altering its linear scaling with system size. At the same time, it produces a very broad distribution of magic over trajectories, with a sizable fraction of trajectories retaining Haar-random magic. Since the cost of stabilizer-based classical simulation grows exponentially with the magic of the state, these high-magic trajectories, rather than the average, control the simulation cost. Our results delineate the potential and the limitations of unraveling optimization as a route to the classical simulation of noisy quantum dynamics with stabilizer methods.
We investigate magic and its connection to entanglement in 1+1 dimensional random free fermion circuits, with a focus on hybrid free fermion dynamics that can exhibit an entanglement phase transition. To quantify magic, we use the Stabilizer Rényi Entropy (SRE), which we compute numerically via a perfect sampling algorithm. We show that although the SRE remains extensive as the system transitions from a critical phase to an area-law (disentangled) phase, the structure of magic itself undergoes a delocalization phase transition. This transition is characterized using the bipartite stabilizer mutual information, which exhibits the same scaling behavior as entanglement entropy: logarithmic scaling in the critical phase and a finite constant in the area-law phase. Additionally, we explore the dynamics of SRE. While the total SRE becomes extensive in O(1) time, we find that in the critical phase, the relaxation time to the steady-state value is parameterically longer than that in generic random circuits. The relaxation follows a universal form, with a relaxation time that grows linearly with the system size, providing further evidence for the critical nature of the phase.
The non-stabilizerness, or "magic", of a quantum state is a common metric for the hardness of simulation on a classical computer, and is often quantified by the stabilizer Rényi entropy (SRE). We show how random superpositions of classically simulable states saturate the SRE to the symmetry-restricted Haar-random maximum. For any large-N random pure-state ensemble satisfying the conditions of Schur's lemma, the average SRE of the m-component superposition ⟨M2(m)⟩=N−log2[⟨W(m)⟩] is governed by an average witness ⟨W(m)⟩ that universally decomposes into a Haar floor and an ensemble-specific excess. As the number of components m increases, the excess is suppressed, resulting in a logarithmic scaling of the SRE before plateauing at the Haar-valued maximum. We derive a general closed form for ⟨W(m)⟩ in the thermodynamic limit, and numerically verify our result for random stabilizer states and random Gaussian states.
Non-stabilizerness is a key resource for fault-tolerant quantum computation, yet its interplay with entanglement in dynamical settings remains underexplored. We address this by analyzing a well-controlled, analytically tractable setup, where we show that entanglement acts as a conduit that teleports magic across the system, thereby enhancing magic injection. Using exact calculations, we prove that when a Haar-random unitary U A is applied to a subsystem A of an entangled stabilizer state, the total injected magic increases with the amount of entanglement between A and its complement. More generally, for any unitary U A , we show that this enhancement is maximized when A is maximally entangled with its complement, in which case the total injected magic is exactly given by the unitary stabilizer Rényi entropy we introduce. This quantity provides both a directly computable measure of unitary magic and a lower bound on the minimum number of T gates required to synthesize U A . In particular, we show that the linear unitary stabilizer entropy gives a better estimate for the nonstabilizer content produced by U A than the previously proposed notion of nonstabilizing power. We further extend our analysis to tripartite stabilizer entanglement, non-stabilizer entanglement, and magic injection via shallow-depth brickwork circuits, finding that the qualitative picture remains unchanged.
Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple pattern of entanglement to a universal, complex pattern can be driven by doping a random Clifford circuit with T gates. This work shows that quantum complexity and complex entanglement stem from the conjunction of entanglement and non-stabilizer resources, also known as magic.