Nonstabilizerness of a Boundary Time Crystal

Abstract: Boundary time crystals exhibit measurement-induced phase transitions in their steady-state entanglement, with critical behavior that depends on the particular unraveling of the Lindblad dynamics. In this work, we investigate another key measure of quantum complexity -- nonstabilizerness (or ``magic'') -- and show that it follows a markedly different pattern. Importantly, in contrast to entanglement, for large system sizes, nonstabilizerness remains invariant under different unraveling schemes -- a property we attribute to the inherent permutational symmetry of the model. Although the steady-state stabilizer entropy does not display a genuine phase transition, it exhibits a singular derivative (a cusp) at the mean-field critical point. Furthermore, we demonstrate that finite-size simulations of the average Lindblad evolution fail to capture the asymptotic behavior of nonstabilizerness in the time-crystal phase, while quantum trajectory unravelings correctly reveal its extensive scaling with system size. These findings offer insights into how different quantum resources manifest in open systems.