Replica Keldysh field theory of quantum-jump processes: General formalism and application to imbalanced and inefficient fermion counting

Measurement-induced phase transitions have largely been explored for projective or continuous measurements of Hermitian observables, assuming perfect detection without information loss. Yet such transitions also arise in more general settings, including quantum-jump processes with non-Hermitian jump operators, and under inefficient detection. A theoretical framework for treating these broader scenarios has been missing. Here we develop a comprehensive replica Keldysh field theory for general quantum-jump processes in both bosonic and fermionic systems. Our formalism provides a unified description of pure-state quantum trajectories under efficient detection and mixed-state dynamics emerging from inefficient monitoring, with deterministic Lindbladian evolution appearing as a limiting case. It thus establishes a direct connection between phase transitions in nonequilibrium steady states of driven open quantum matter and in measurement-induced dynamics. As an application, we study imbalanced and inefficient fermion counting in a one-dimensional lattice system: monitored gain and loss of fermions occurring at different rates, with a fraction of gain and loss jumps undetected. For imbalanced but efficient counting, we recover the qualitative picture of the balanced case: entanglement obeys an area law for any nonzero jump rate, with an extended quantum-critical regime emerging between two parametrically separated length scales. Inefficient detection introduces a finite correlation length beyond which entanglement, as quantified by the fermionic logarithmic negativity, obeys an area law, while the subsystem entropy shows volume-law scaling. Numerical simulations support our analytical findings. Our results offer a general and versatile theoretical foundation for studying measurement-induced phenomena across a wide class of monitored and open quantum systems.