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arXiv:2608.02745

Schmidt-Gauge Non-Local Magic: Representation, Optimality, and Mathematical Properties

Fabio Franchini, Salvatore Marco Giampaolo

quant-phSubmitted Aug 3, 2026• Announced Aug 5, 2026• 0 citations• Revised Aug 6, 2026 (v2)
v1 · Aug 3, 2026v2 · Aug 6, 2026
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Abstract

While the full non-stabilizerness (magic) of a quantum state contains local, basis-dependent contributions, a non-local formulation based on a minimization over local unitaries isolates the component associated with genuinely non-local correlations. Within such a framework, the Schmidt-gauge formulation of non-local magic provides a direct connection between genuinely non-local non-stabilizer correlations and the entanglement spectrum of quantum many-body states. Building on the exact Walsh--Hadamard representation introduced in our accompanying Letter, we develop the mathematical theory associated with this formulation. We extend the formalism to arbitrary bipartitions, prove the exactness of the Schmidt gauge for arbitrary $1\times N$ bipartitions, and derive general analytical properties of Schmidt-gauge non-local magic, including entanglement bounds, selection rules, and exact relations with the moments of the normalized Walsh spectrum. This representation also provides an interpretation of Schmidt-gauge non-local magic as a logarithmic inverse participation ratio normalized by a universal harmonic baseline, thereby relating it to excess delocalization in Walsh space. These results demonstrate that the Walsh--Hadamard representation reveals an underlying discrete harmonic structure that is hidden in the original spectral formulation and provides considerably more than an equivalent expression for Schmidt-gauge non-local magic. Rather, it furnishes the natural mathematical framework for its analytical investigation, placing the theory within the broader context of discrete harmonic analysis.

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While the full non-stabilizerness (magic) of a quantum state contains local, basis-dependent contributions, a non-local formulation based on a minimization over local unitaries isolates the component associated with genuinely non-local correlations. Within such a framework, the Schmidt-gauge formulation of non-local magic provides a direct connection between genuinely non-local non-stabilizer correlations and the entanglement spectrum of quantum many-body states. Building on the exact Walsh--Hadamard representation introduced in our accompanying Letter, we develop the mathematical theory associated with this formulation. We extend the formalism to arbitrary bipartitions, prove the exactness of the Schmidt gauge for arbitrary 1×N1\times N1×N bipartitions, and derive general analytical properties of Schmidt-gauge non-local magic, including entanglement bounds, selection rules, and exact relations with the moments of the normalized Walsh spectrum. This representation also provides an interpretation of Schmidt-gauge non-local magic as a logarithmic inverse participation ratio normalized by a universal harmonic baseline, thereby relating it to excess delocalization in Walsh space. These results demonstrate that the Walsh--Hadamard representation reveals an underlying discrete harmonic structure that is hidden in the original spectral formulation and provides considerably more than an equivalent expression for Schmidt-gauge non-local magic. Rather, it furnishes the natural mathematical framework for its analytical investigation, placing the theory within the broader context of discrete harmonic analysis.
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.
Non-local magic quantifies the non-stabilizerness of a bipartite quantum state that cannot be removed by local unitary transformations. Despite its natural definition, its evaluation generally requires a difficult optimization over local unitaries. Here, we show that for the log-stabilizer fidelity this optimization admits an exact closed-form analytic solution depending only on the Schmidt spectrum. We then introduce the magic of purification, defined as the minimum pure-state magic over all purifications of a mixed state, and show that it naturally induces a resource theory whose free states are normalized stabilizer-code projectors. For the log-stabilizer fidelity, the magic of purification admits a distance-based formulation in terms of the Uhlmann fidelity. Remarkably, we prove that non-local magic coincides with the minimum magic of purification along the unitary orbit of the reduced density operator. Our results provide both an efficient analytical characterization and a mixed-state resource-theoretic interpretation of non-local magic.
Nonergodic features of chaotic quantum many-body systems are commonly characterized through local observables, fidelity, and entanglement entropy. Here, using global and nonlocal measures of nonstabilizerness (magic), we study quantum many-body scars near infinite temperature in (1+1)(1+1)(1+1)-dimensional Abelian Z2\mathbb{Z}_2Z2​ and U(1)(1)(1) lattice gauge theories, both analytically and numerically. We derive an exact closed-form expression for the nonlocal trace distance magic solely from the Schmidt spectrum, reducing the optimization over local unitaries to a finite maximization over stabilizer-compatible Schmidt ranks. We find that scar eigenstates can exhibit extensive global magic despite their anomalously low entanglement, while retaining anomalously large nonlocal magic compared with ergodic states. We further show that constrained Hilbert spaces can generate irreducible nonlocal magic through stabilizer-incompatible Schmidt ranks, even for a flat entanglement spectrum.
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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 444, 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.
Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by establishing a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which we prove at Rényi index α=2α=2α=2 for arbitrary subsystem size, and which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography.