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

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 R\'enyi 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 R\'enyi entropy.