Violations of the Leggett-Garg inequality in Hybrid Liouvillian Dynamics: The Nonlinear Role of Detector Efficiency

TL;DR

This paper shows that maximal violations of the Leggett-Garg inequality in hybrid Liouvillian systems are highly sensitive to detector efficiency, implying such violations are fragile and require near-perfect measurement conditions.

Contribution

It introduces a nonlinear analysis of LGI violations in hybrid Liouvillian dynamics, highlighting the fragility of maximal violations under realistic measurement efficiencies.

Findings

Maximal LGI violations approach the algebraic bound at zero detector efficiency.

Even minimal detector efficiency significantly suppresses LGI violations.

Achieving algebraic violations demands near-perfect detector performance.

Abstract

Violations of the Leggett-Garg inequality (LGI) up to its algebraic bound under non-Hermitian dynamics are well established theoretically. Here, we demonstrate that such extreme violations are intrinsically fragile when realistic measurement processes are taken into account. We consider an open two-level system described by a time-local hybrid Liouvillian, with a continuous parameter $q \in [0,1]$, representing detector efficiency, i.e., the fraction of quantum jump trajectories that are retained in the ensemble. This parameter interpolates between trace-preserving Lindblad dynamics ($q=1$) and non-Hermitian ``no-jump" evolution ($q=0$). While $K_3$ approaches its algebraic maximum of 3 in the null-efficiency limit, even an infinitesimal increase in detector efficiency induces a rapid, highly nonlinear suppression toward the classical bound. This logarithmic sensitivity reveals that maximal LGI violations are not robust physical features but rather singular limits of idealized measurement conditions. Our results have direct experimental implications: achieving algebraic LGI violations in systems undergoing continuous time evolution requires near-perfect suppression of detected quantum jumps (i.e., effective post-selection), placing stringent constraints on detector performance. In contrast to discrete protocols based on time-non-divisible dynamics, our framework shows that extreme violations arising within continuous, divisible quantum trajectory evolution constitute a fundamentally fragile regime.