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arXiv:2607.13134

Magic without a phase: phase-independent stabilizer Rényi entropy in gluon scattering

Jinwei Chu, Savan Kharel

hep-thhep-phquant-phSubmitted Jul 14, 2026• Announced Jul 16, 2026
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Abstract

Magic, also known as non-stabilizerness, measures the usefulness of a quantum state for quantum computation. While magic is defined relative to a choice of computational basis, in some physical settings the available data determine this basis only up to local phase conventions. In this paper, we generalize the notion of magic and formulate it in a phase-independent manner, and hence define a generalized stabilizer Rényi entropy. As a case study, we consider higher-multiplicity tree-level gluon scattering, interpreting the outgoing helicities as qubits. In this setting, the helicity data naturally determine a local basis for each qubit but leave a phase ambiguity. For $3\to 2$ scattering, we find that the final-state phase-independent magic is generically larger than the maximum attainable in $2\to 2$ scattering. For $2 \to 3$ scattering, we find a nonzero minimal value approached in the soft limit. Moreover, when the three outgoing momenta become symmetric, the magic approaches a local minimum only a few percent above the soft-limit value. In all cases considered, the color dependence cancels from the phase-independent stabilizer Rényi entropy.

Comments
28 pages, 8 figures

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Magic, also known as non-stabilizerness, measures the usefulness of a quantum state for quantum computation. While magic is defined relative to a choice of computational basis, in some physical settings the available data determine this basis only up to local phase conventions. In this paper, we generalize the notion of magic and formulate it in a phase-independent manner, and hence define a generalized stabilizer Rényi entropy. As a case study, we consider higher-multiplicity tree-level gluon scattering, interpreting the outgoing helicities as qubits. In this setting, the helicity data naturally determine a local basis for each qubit but leave a phase ambiguity. For 3→23\to 23→2 scattering, we find that the final-state phase-independent magic is generically larger than the maximum attainable in 2→22\to 22→2 scattering. For 2→32 \to 32→3 scattering, we find a nonzero minimal value approached in the soft limit. Moreover, when the three outgoing momenta become symmetric, the magic approaches a local minimum only a few percent above the soft-limit value. In all cases considered, the color dependence cancels from the phase-independent stabilizer Rényi entropy.
Stabilizer Rényi entropy quantifies magic relative to a complete Pauli frame. In settings such as helicity states of massless particles, unsynchronized qubit registers, energy eigenbases, and systems under superselection rules, the computational basis may be fixed while the transverse Pauli axes remain unspecified. The stabilizer Rényi entropy then depends on the choice of those axes. We define a phase-averaged stabilizer Rényi entropy by averaging the Pauli moments before taking the logarithm. It is exactly invariant under independent local rephasings. For pure states, it vanishes precisely when the state is a stabilizer state in every phase convention. For a broad class of sparse supports, it reduces to a closed-form function of the basis-state probabilities alone. Within this class, the ceiling for nnn channels is ln⁡[n3/(6n−5)]\ln[n^{3}/(6n-5)]ln[n3/(6n−5)] for n≥3n\geq3n≥3 and ln⁡75\ln\frac{7}{5}ln57​ for two channels. We apply the invariant to tree-level gluon scattering at four and five points, where the helicity-basis phases are unspecified. It saturates the two- and three-channel ceilings: ln⁡75\ln\frac{7}{5}ln57​ at four points and ln⁡2713\ln\frac{27}{13}ln1327​ in 3→23\to23→2 scattering. In same-helicity 2→32\to32→3 scattering, it lies between a floor of ln⁡87\ln\frac{8}{7}ln78​ and a ceiling of ln⁡184125\ln\frac{184}{125}ln125184​, reached in the resolved soft limit and particular collinear limits, respectively. In every configuration studied, the invariant is independent of the fixed external colors.
The property of non-stabiliserness, or ``magic'', is of interest in quantum computing due to its role in developing fault-tolerant quantum algorithms with genuine computational advantage over classical counterparts. There has been much interest in quantifying magic in various physical systems, in order to probe how to produce and enhance it. The production of magic has previously been quantified in gluon and graviton scattering, in the so-called helicity basis relating particle spins with momentum directions. For a basis-independent statement, one should instead use the recently developed concept of non-local non-stabiliserness, and our aim in this paper is to derive how this varies for gluon and graviton scattering processes. Our results show that, for many initial states, including those produced with polarised beams, the helicity basis coincides with a basis in which the non-local magic is manifest, providing a physical motivation for using the helicity basis to study quantum information quantities. However, this property breaks upon adding additional operators to the Yang-Mills Lagrangian, as would be the case in new physics scenarios.
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.
In quantum computing, non-stabilizerness -- the magic -- refers to the computational advantage of certain quantum states over classical computers and is an essential ingredient for universal quantum computation. Employing the second order stabilizer R\'enyi entropy to quantify magic, we study the production of magic states in Quantum Electrodynamics (QED) via 2-to-2 scattering processes involving electrons and muons. Considering all 60 stabilizer initial states, which have zero magic, the angular dependence of magic produced in the final states is governed by only a few patterns, both in the non-relativistic and the ultra-relativistic limits. Some processes, such as the low-energy e−μ−→e−μ−e^-\mu^-\to e^-\mu^-e−μ−→e−μ− and Bhabha scattering e−e+→e−e+e^-e^+\to e^-e^+e−e+→e−e+, do not generate magic at all. In most cases the largest magic generated is significantly less than the maximal possible value of log⁡(16/7)≈0.827\log (16/7) \approx 0.827log(16/7)≈0.827. The only instance where QED is able to generate maximal magic is the low-energy μ−μ+→e−e+\mu^-\mu^+\to e^-e^+μ−μ+→e−e+, in the limit me/mμ→0m_e/m_\mu \to 0me​/mμ​→0, which is well approximated in nature. Our results suggest QED, although capable of producing maximally entangled states easily, may not be an efficient mechanism for generating quantum advantages.
Stabilizer states constitute an important class of quantum states that can be generated from computational-basis states using Pauli operators and Clifford gates. Although they may exhibit substantial multipartite entanglement, quantum circuits restricted to stabilizer operations can be efficiently simulated classically and therefore cannot, by themselves, provide a quantum computational advantage. Such an advantage requires non-stabilizer resources, commonly referred to as quantum magic. In this paper, we investigate the non-stabilizerness of quantum states arising in a class of spin Hamiltonians by computing their stabilizer Rényi entropies. Using real-space renormalization-group techniques, we obtain a closed-form expression valid in the low-energy, large-distance regime. We analyze how quantum magic evolves under coarse-graining and explore its behavior across different parameter regimes and quantum phases.
Entanglement and non-stabilizerness (magic) quantify two distinct departures of quantum systems from classical description: the former measures non-local correlations, while the latter measures the deviation from stabilizer states that can be efficiently simulated classically. Understanding magic in physically relevant quantum field theories is essential for identifying where quantum advantage may be realized in the early fault-tolerant quantum computing era. We calculate the gauge-invariant entanglement entropy and stabilizer Rényi entropy of the ground state of the (1+1)-dimensional SU(2) lattice gauge theory formulated in a dressed-site basis that enforces Gauss's law exactly. Using tensor networks, we obtain results for system sizes up to L=100L=100L=100 (300 qubits). We find a crossover denoted by g⋆g_{\star}g⋆​ where the ground state passes from a more magic-rich regime into a regime with less magic; this is also tracked by the sharpest change of both the entanglement entropy and lattice particle density. Our large-scale study of non-stabilizerness and entanglement entropy in a non-Abelian lattice gauge theory with matter provides new insight into the interplay of magic and entanglement in gauge theories, both of which are relevant for classical and early fault-tolerant quantum simulations.