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arXiv:2607.18400

Magic-protected entanglement and Clifford-irreducible structure in magic state space

Alejandro Borda Kuhlmann, Julian Rincon

quant-phcond-mat.stat-mechSubmitted Jul 20, 2026• Announced Jul 22, 2026• 0 citations
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Abstract

We develop a Clifford-orbit framework for studying magic-protected entanglement, which we refer to as magical entanglement: the part of bipartite entanglement that remains after optimal stabilizer simplification. This construction leverages residual entanglement under Clifford reduction as a state-level organizing principle for magic state space. It defines canonical representatives, spectra, and ranks that characterize the Clifford-irreducible structure of a state. We identify two regimes of the magic-entanglement interplay. In the $T$-magic regime, local nonstabilizer resources can coexist with entanglement, but the protected component remains weak and state-dependent. In the $W$-magic regime, by contrast, entanglement is Clifford-irreducibly tied to nonstabilizerness, producing typical, strongly self-averaging behavior. Analytical examples and random-circuit numerics support a crossover from broad $T$-magic fluctuations to concentrated, Haar-like $W$-magic behavior. These results identify magical entanglement as an orbit-level diagnostic of how nonstabilizerness protects quantum correlations against Clifford reduction.

Comments
18 pages, 7 figures, 1 table

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We develop a Clifford-orbit framework for studying magic-protected entanglement, which we refer to as magical entanglement: the part of bipartite entanglement that remains after optimal stabilizer simplification. This construction leverages residual entanglement under Clifford reduction as a state-level organizing principle for magic state space. It defines canonical representatives, spectra, and ranks that characterize the Clifford-irreducible structure of a state. We identify two regimes of the magic-entanglement interplay. In the TTT-magic regime, local nonstabilizer resources can coexist with entanglement, but the protected component remains weak and state-dependent. In the WWW-magic regime, by contrast, entanglement is Clifford-irreducibly tied to nonstabilizerness, producing typical, strongly self-averaging behavior. Analytical examples and random-circuit numerics support a crossover from broad TTT-magic fluctuations to concentrated, Haar-like WWW-magic behavior. These results identify magical entanglement as an orbit-level diagnostic of how nonstabilizerness protects quantum correlations against Clifford reduction.
Abstract Magic, also known as nonstabilizerness, quantifies the distance of a quantum state to the set of stabilizer states, and it serves as a necessary resource for potential quantum advantage over classical computing. In this work, we study magic in a measurement-only quantum circuit with competing types of Clifford and non-Clifford measurements, where magic is injected through the non-Clifford measurements. This circuit can be mapped to a classical model that can be simulated efficiently, and the magic can be characterized using any magic measure that is additive to the tensor product of single-qubit states. Leveraging this observation, we study the magic transition in this circuit in both one- and two-dimensional lattices using large-scale numerical simulations. Our results demonstrate the presence of a magic transition between two different phases with extensive magic scaling, separated by a critical point in which the mutual magic exhibits scaling behavior analogous to entanglement. We further show that these two distinct phases can be distinguished by topological magic. In a different regime, with a vanishing rate of non-Clifford measurements, we find that the magic saturates in both phases. Our work sheds light on the behavior of magic and its linear combinations in quantum circuits, employing genuine magic measures.
Nonstabilizerness, or quantum magic resource, presents a valuable resource in quantum error correction and computation. We study the dynamics of locally injected nonstabilizerness in unitary Clifford circuits, where the total nonstabilizerness is conserved. However, the absence of physical observables quantifying nonstabilizerness precludes a direct microscopic or hydrodynamic description of its local distribution and dynamics. Using insights from stabilizer quantum error correcting codes, we rigorously show that the spatial distribution of nonstabilizerness can be inferred from a canonical representation of low-magic states, dubbed the bipartite magic gauge. Moreover, we propose two operationally relevant magic length scales. We numerically establish that, at early times, both length scales grow ballistically at distinct velocities set by the entanglement velocity, after which nonstabilizerness delocalizes. Our work sheds light on the spatiotemporal structure of quantum resources and complexity in many-body dynamics, opening up avenues for investigating their transport properties and further connections with quantum error correction.
The interplay between non-stabilizerness and entanglement in random states is a very rich arena of study for the understanding of quantum advantage and complexity. In this work, we tackle the problem of such interplay in random pure quantum states. We show that while there is a strong dependence between entanglement and magic, they are, surprisingly, perfectly uncorrelated. We compute the expectation value of non-stabilizerness given the Schmidt spectrum (and thus entanglement). At a first approximation, entanglement determines the average magic on the Schmidt orbit. However, there is a finer structure in the average magic distinguishing different orbits where the flatness of entanglement spectrum is involved.
Magic, or nonstabilizerness, is the resource that lifts Clifford circuits to universal quantum computation and has become a standard diagnostic of many-body states. For a state shared between two parties, however, a basic question has remained open: how much of the magic resides in the correlations between the parties rather than in their local bases? Isolating this nonlocal magic requires minimizing over all local bases, an optimization that has so far resisted exact solution. Here we solve it for the stabilizer fidelity: the nonlocal magic of every pure multiqubit state is the distance of its entanglement spectrum from the closest spectrum of Bell pairs. The same quantity governs an apparently unrelated task: a family of states universally embezzles entanglement under local operations and classical communication if and only if its nonlocal magic diverges. The deciding property is not the amount of entanglement but the way the entanglement spectrum spreads its weight across factor-of-two windows of rank, so that critical chains and random-singlet states, with identical logarithmic entanglement scaling, carry unbounded and vanishing nonlocal magic, respectively. Nonlocal magic thereby becomes an operationally meaningful property of quantum correlations, directly accessible to tensor-network simulations and, through entanglement spectroscopy, to experiments.
The interplay between non-stabilizerness and entanglement in random states is a very rich arena of study for the understanding of quantum advantage and complexity. In this work, we tackle the problem of such interplay in random pure quantum states. We show that while there is a strong dependence between entanglement and magic, they are, surprisingly, perfectly uncorrelated. We compute the expectation value of non-stabilizerness given the Schmidt spectrum (and thus entanglement). At a first approximation, entanglement determines the average magic on the Schmidt orbit. However, there is a finer structure in the average magic distinguishing different orbits where the flatness of entanglement spectrum is involved.
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 TTT-doped random Clifford brickwork circuits subject to local dephasing noise of strength ppp. 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 ppp-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.