Extreme violation of the Leggett-Garg inequality in nonunitary dynamics with complex energies

We study the Leggett-Garg inequality (LGI) of a two-level system (TLS) undergoing coherent dynamics described by a non-Hermitian Hamiltonian and Lindblad equation with no quantum jumps. The nonlinear Bloch equation for the TLS density matrix predicts violations of LGI above the TLS Lüders bound of 3/2, approaching the extremal case of LGI parameter K3=3 in the PT-symmetric region and the PT-broken region. We show that these findings are reproduced by using postselection to remove instantaneous quantum jumps from a three-level system described by the Lindblad equation with a single spontaneous emission dissipator. We trace the K3 excesses beyond the standard quantum limit of 3/2 to a nonuniform speed of evolution on the Bloch sphere. Finally, we consider the effects of competing Lindblad dissipators on the postselected non-Hermitian dynamics and the viability of observing K3 exceeding the Lüder bound and approaching its algebraic maximum of three in current experimental setups.