Implementation of a two-level non-Hermitian qubit via postselection of a three-level system has been demonstrated. The postselection procedure, which discards quantum jumps to the ground-state manifold $|g\rangle$ while retaining excitations in the $|e\rangle$ and $|f\rangle$ manifolds, effectively generates a non-Hermitian qubit exhibiting $\mathcal{PT}$ symmetry. In this work, we perform continuous homodyne measurement of this non-Hermitian qubit and analyze the interplay between decay introduced by postselection and measurement backaction. We compare the ensemble-averaged dynamics obtained from measurement trajectories with the Liouvillian average, highlighting the deviations arising from measurement backaction and postselection. We formulate the no-jump stochastic differential equation describing the postselected non-Hermitian qubit and show that its ensemble-averaged dynamics agree with those of the jump-updated postselected evolution at drive strengths far from the Liouvillian exceptional point (EP). The degree of deviation near the EP depends sensitively on the nature of the drive. This discrepancy is attributed to the interplay between measurement backaction and the non-Hermitian decay introduced by postselection. Furthermore, we determine the optimal path of the non-Hermitian qubit by extremizing the action within the path-integral formulation of the quantum trajectory framework. Our results provide insights into how measurement backaction and non-Hermitian dynamics together shape the transient behavior of open quantum systems and enable controlled manipulation of qubits near exceptional points.

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• Open quantum systems & decoherence
• PT-symmetric quantum mechanics
• Quantum coherence & coherence measures
• Quantum information theory
• Quantum measurements
• Weak values & weak measurements