Quasi-Hermitian proxy for the conditional no-jump dynamics in a three-site model

In this work, we show that in the regime of real and non-degenerate spectrum, the quasi-Hermitian framework provides an effective description of the conditional no-jump evolution of certain Markovian open systems within invariant subspaces while the full Lindblad dynamics governs the unconditional dynamics which includes quantum jumps. We present a three-site fermionic model with asymmetric hopping amplitudes whose single-particle generator is intrinsically non-Hermitian yet supports an all-real spectrum in a finite region of the parameter space. We construct an explicit Lindbladian-based embedding in which the same Hamiltonian arises as the no-jump generator of a completely positive and trace-preserving Lindblad equation, up to a particle-decay term. Due to particle-number conservation, this shift reduces to a scalar within each fixed-particle-number sector. We further analyze the single-particle spectrum and identify a third-order exceptional point that lies on the boundary marking the breakdown of quasi-hermiticity in a given fixed-particle-number sector.