Nonlocal nonstabilizerness for slightly entangled quantum many-body states

Abstract

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.

Bipartite nonlocal stabilizer Rényi entropy (SRE) is defined by minimizing a pure state's SRE over arbitrary unitaries within each subsystem. This paper asks whether that optimization can be replaced by evaluating the SRE of a canonical state constructed from the descending Schmidt spectrum, $|\tilde\psi\rangle=\sum_i\sqrt{\lambda_i}|i\rangle|i\rangle$. Equality is conjectured; without it, the reference-state value is an upper bound. The reference state is shown to be stationary under infinitesimal local unitaries. Riemannian manifold optimization agrees with its SRE within numerical precision for sampled rank-four spectra and accessible many-body benchmarks. For rank four, the reference-state 2-SRE is expressed through entanglement Rényi entropies $S_2,S_3,S_4$, with maximum $\log_2(9/5)\approx0.848$. A truncation bound controls the reference-state SRE error through discarded Schmidt weight $\eta$, scaling as $O(\sqrt{\eta}/\zeta_\alpha)$ when the stabilizer-purity change is small. The sorted-reference value is also bounded above by Gaussian-unitary-restricted fermionic nonlocal magic. Applications, conditional on the conjecture, give an $O(1)$ Haar-random limit determined by a Marchenko–Pastur quantile integral. Tensor-network calculations find approximately logarithmic critical scaling in Ising and XXZ chains; fitted XXZ coefficients vary despite fixed central charge $c=1$. PXP simulations at $N=30$ show slower entanglement growth but larger, longer-lived reference-state SRE for Néel-state scar dynamics than empty-state dynamics. Global optimality remains unproved. Efficient evaluation requires manageable or truncatable Schmidt spectra, and accessible critical-system data do not establish asymptotic scaling or exclude eventual saturation.