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arXiv:2504.09360

An Exact Link between Nonlocal Nonstabilizerness and Operator Entanglement

Faidon Andreadakis, Paolo Zanardi A

Published Apr 12, 2025• 0 citations
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Abstract

Nonstabilizerness is a quantum property of states associated with the non-Clifford resources required for their preparation. As a resource, nonstabilizerness complements entanglement, and the interplay between these two concepts has garnered significant attention in recent years. In this work, we establish an exact correspondence between the generation of nonlocal nonstabilizerness and operator entanglement under unitary evolutions. Nonlocal nonstabilizerness refers to nonstabilizerness that cannot be erased via local operations, while operator entanglement generalizes entanglement to operator space, characterizing the complexity of operators across a bipartition. Specifically, we prove that a unitary map generates nonlocal nonstabilizerness if and only if it generates operator entanglement on Pauli strings. Guided by this result, we introduce an average measure of a unitary's Pauli-entangling power, serving as a proxy for nonlocal nonstabilizerness generation. We derive analytical formulas for this measure and examine its properties, including its typical value and upper bounds in terms of the nonstabilizerness properties of the evolution.

DOI
10.48550/arxiv.2504.09360

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Nonstabilizerness is a quantum property of states associated with the non-Clifford resources required for their preparation. As a resource, nonstabilizerness complements entanglement, and the interplay between these two concepts has garnered significant attention in recent years. In this work, we establish an exact correspondence between the generation of nonlocal nonstabilizerness and operator entanglement under unitary evolutions. Nonlocal nonstabilizerness refers to nonstabilizerness that cannot be erased via local operations, while operator entanglement generalizes entanglement to operator space, characterizing the complexity of operators across a bipartition. Specifically, we prove that a unitary map generates nonlocal nonstabilizerness if and only if it generates operator entanglement on Pauli strings. Guided by this result, we introduce an average measure of a unitary's Pauli-entangling power, serving as a proxy for nonlocal nonstabilizerness generation. We derive analytical formulas for this measure and examine its properties, including its typical value and upper bounds in terms of the nonstabilizerness properties of the evolution.
Operator entanglement is a well-established measure of operator complexity across a system bipartition. In this work, we introduce a measure of the ability of a unitary channel to generate operator entanglement, representing an operator-level generalization of the state-space entangling power. This operator space entangling power is demonstrated to be linked to the scrambling properties of the unitary channel via the recently introduced concept of mutual averaged noncommutativity of quantum operator algebras. An upper bound for the operator space entangling power is identified that corresponds to unitary channels with scrambling properties akin to those of typical unitaries. Additionally, for Hamiltonian dynamics, we find that the short-time growth rate of the operator space entangling power matches the Gaussian scrambling rate of the bipartite out-of-time-order correlator, establishing a direct link between information scrambling and operator entanglement generation for short timescales. Finally, we examine the average growth of local operator entanglement across a symmetric bipartition of a spin chain. For dual-unitary circuits, a combination of analytical and numerical investigations demonstrates that the average growth of local operator entanglement exhibits two distinct regimes in relation to the operator space entangling power of the building-block gate.
Entanglement and nonstabilizerness capture distinct aspects of quantum complexity, yet their relation through the entanglement spectrum remains poorly understood. Here we develop a spectral framework for bipartite nonlocal nonstabilizerness. We introduce a generalized anti-flatness and derive upper and lower bounds on the nonlocal stabilizer Rényi entropy (SRE) in terms of Rényi entanglement entropy and spectral non-uniformity. We further reduce the local-unitary optimization to a single-unitary problem and prove that the ordered Schmidt reference state is a local minimum of the SRE for integer α≥2α\geq2α≥2. Applying these results to exponentially and algebraically decaying spectra reveals parametrically distinct relations between entanglement and nonlocal nonstabilizerness. For broad spectra, we introduce a dyadic-shell sandwich construction that determines the asymptotic SRE scaling. At one-dimensional conformal critical points, it yields a universal hierarchy of double-logarithmic scaling laws for integer α≥2α\geq2α≥2. Our spectral bounds and dyadic-shell sandwich construction provide general tools for analyzing nonlocal SRE, offering a flexible framework that can be applied to a wide range of entanglement spectra in quantum many-body systems.
Quantum entanglement and quantum nonstabilizerness are fundamental resources that characterize distinct aspects of a quantum state: Entanglement reflects nonlocal correlations, while nonstabilizerness quantifies the deviation from stabilizer states. A quantum state becomes a valuable resource for applications like universal quantum computation only when both quantities are present. Here, we propose that quantum nonlocal nonstabilizerness (NN) serves as an effective measure of this combined resource, incorporating both entanglement and nonstabilizerness. We demonstrate that NN can be precisely computed for two-qubit pure states, establishing a direct relationship with the entanglement spectrum. We then extend the definition of NN to mixed states and explore its presence in many-body quantum systems. We reveal that the two-point NN decays according to a power law in critical states, indicating that it can serve as an effective quantum information probe for phase transitions. Furthermore, we explore measurement-induced NN and uncover an intriguing phenomenon termed ``nonstabilizerness swapping,'' wherein post-measurement NN decays more slowly than any pre-measurement correlations. The occurrence of this phenomenon provides a sufficient condition for both entanglement swapping and the presence of a nontrivial sign structure. Our results thus represent a pivotal step toward accurately quantifying the ``quantumness'' of a state and bring insights from multiple perspectives to the property of NN.
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.
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.