Role of inefficient measurement in realizing post-selection-based non-Hermitian qubits

Abstract

Post-selecting against quantum jumps into the ground state confines the evolution of the three-level system to the excited states manifold, effectively realizing a PT-symmetric non-Hermitian qubit. In this work, by introducing post-selection efficiencies for both decay channels, the second-excited to first-excited and the first-excited to ground-state transitions, we formulate a hybrid-Liouvillian framework that captures the unmonitored dynamics of the non-Hermitian qubit. We find that the decoherence effects arising from quantum jumps within the second-excited and first-excited manifold also manifest under inefficient post-selection of the second-excited to first-excited transitions, thereby modifying the spectral properties of the Liouvillian and leading to a splitting of the exceptional points. A comparative analysis shows that the trajectory-based approach, obtained by ensemble-averaging stochastic measurement trajectories generated via the Bayesian state update rule, and the Lindblad evolution remain consistent. Our results highlight the fundamental role of measurement inefficiency in realizing post-selection-based non-Hermitian qubits and in shaping the structure of Liouvillian exceptional points. These findings provide new insights into how inefficient measurement processes influence non-Hermitian behavior in open quantum systems.