Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer Rényi entropy (SRE), the minimum is conjectured to be attained by the computational-basis (CB) representative, the state obtained by assigning the Schmidt coefficients in decreasing order to matching computational-basis labels. We prove this conjecture for two families of states at every system size and bipartition: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. For arbitrary bipartite pure states, we show that the CB value exceeds the nonlocal SRE by at most
4, fixing the leading term of any divergent scaling of nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy. Consequently, one-dimensional area-law states have bounded nonlocal SRE, while critical states with logarithmic entanglement entropy permit at most doubly logarithmic growth with system size. We apply these results to the transverse-field Ising chain, where we tightly bound the nonlocal SRE in the gapped phases and identify its double-logarithmic growth at criticality.
Nonlocal nonstabilizerness quantifies the irreducible magic resource encoded in bipartite entanglement, but its evaluation is generally hindered by a highly nonconvex optimization over local unitary transformations. Here we propose a Schmidt-reference-state framework that replaces this optimization by a direct construction from the sorted Schmidt spectrum. We conjecture that the nonlocal stabilizer Rényi entropy (SRE) is given by the SRE of the corresponding reference state, and support this conjecture through analytical and numerical evidences. Our framework makes nonlocal nonstabilizerness efficiently accessible for weakly entangled many-body states whenever the entanglement spectrum is available. Applying it to Haar-random states, critical spin chains, and PXP dynamics, we show that nonlocal SRE captures nonstabilizer structures in the entanglement spectrum that are invisible to conventional entanglement measures. Our results establish entanglement spectra as a powerful window into irreducible nonstabilizer correlations, opening a broadly applicable route to studying nonlocal magic resources of quantum many-body systems in and out of equilibrium.
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.
Non-local magic has recently emerged as a fundamental resource for characterizing genuinely non-local non-stabilizer correlations. However, its direct calculation is an intractable numerical problem, except for small systems, and its understanding remains limited. We derive a representation of non-local magic in terms of the Walsh--Hadamard autocorrelations of the entanglement spectrum. Our representation makes the underlying harmonic structure explicit and enables a systematic analysis of its behaviors for various scenarios. We prove that non-local magic can be upper-bounded by an entanglement entropy and we derive exact analytical results for broad classes of quantum states, characterizing the scaling of non-local magic for volume-law states, as well as ground states of one-dimensional gapped and critical systems. Our results identify the spectral organization of the entanglement spectrum as the key ingredient governing non-local magic and provide a framework for further systematic analytical investigation.
We study non-stabilizerness (magic) in the ground state of
(2+1)-dimensional
SU(2) Hamiltonian lattice gauge theory with matter, formulated in the dressed-site basis in the hardcore-gluon truncation and restricted to the zero baryon-number sector. Using matrix product states, we compute three facets of magic: the second-order stabilizer Rényi entropy (SRE)
M2, its non-local component
M2NL, estimated by the SRE
M2(ψ~) of the computational-basis Schmidt representative
ψ~, and a lower bound in terms of the anti-flatness
F of the entanglement spectrum. We also prove a stronger form of the sandwich relation:
−log2(1−4F)≤M2NL≤M2; the lower bound rests on a stronger inequality that we obtain for arbitrary Schmidt bases and rank, with an optimal constant. Combined with recent results, this yields the unconditional bracket
max{−log2(1−4F),M2(ψ~)−4}≤M2NL≤M2(ψ~). We emphasize a structural distinction that makes the non-local quantities the physically preferred diagnostics: whereas the full SRE depends on the (non-unique) encoding of the gauge-invariant local Hilbert space into qubits, the non-local magic, its estimate
M2(ψ~), and the anti-flatness are invariant under site-local re-encodings and are therefore intrinsic to the state and bipartition. By varying the gauge coupling for two values of the fermion mass on lattices up to
8×8 (
∼288 qubits) with bond dimension up to
128, we find that the non-local magic, as estimated by
M2(ψ~), furnishes a sharper and more bond-dimension-friendly probe of the gauge-matter delocalization crossover compared to the full SRE or the gauge-invariant entanglement entropy, retaining a clear signal at bond dimensions well below those needed to converge the ground state itself.
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.