Measurement-enhanced entanglement in monitored superconductors

Monitored quantum dynamics can exhibit entanglement phase transitions driven by the competition between unitary evolution and measurement. A common expectation is that LOCC measurements do not increase steady-state entanglement in such settings. We challenge this view by studying the dynamics of one-dimensional fermions (system size L) governed by a BCS Hamiltonian with gap Δ and subjected to on-site, spin-selective number measurements at rate p. Using exact Gaussian-state simulations and a quasiparticle analysis, we find that for Δ>0 the steady-state entanglement S_s increases with p over a finite interval 0<p<p_th. The mechanism is a multi-party competition among entanglement generation, measurement, and fermion pairing: pairing suppresses entanglement generation, while measurement suppresses pairing, and together these effects yield a net growth of S_s as p rises. A complementary nonlinear sigma-model analysis further suggests that the scaling of S_s is upper bounded as S_s(L) = O(ln L) for Δ > 0 and small finite p, implying that the measurement-enhanced entanglement vanishes (i.e., p_th→0) in the thermodynamic limit.