Experimental demonstration of enhanced violations of Leggett-Garg inequalities in a $\mathcal{PT}$-symmetric trapped-ion qubit

The Leggett-Garg inequality (LGI) places a bound for the distinction between quantum systems and classical systems. Despite that the tests of temporal quantum correlations on LGIs have been studied in Hermitian realm, there are still unknowns for LGIs in non-Hermitian conditions due to the interplay between dissipation and coherence. For example, a theoretical hypothesis to be experimentally validated, suggests that within non-Hermitian systems, the non-unitary evolution of the system dynamics allows the boundaries of the LGIs to surpass the constraints imposed by traditional quantum mechanics. Here, we demonstrate the experimental violation of LGIs in a parity-time ($\mathcal{PT}$)-symmetric trapped-ion qubit system by measuring the temporal correlation of the evolving states at different times. We find that the upper bounds of the three-time parameter $K_3$ and the four-time parameter $K_4$ show enhanced violations with the increasing dissipation, and can reach the upper limit by infinitely approaching exceptional point. We also observe the distinct behavior of the lower bounds for $K_3$ and $K_4$. While the lower bound for $K_3$ remains constant, the case for $K_4$ shows an upward trend with increasing dissipation. These results reveal a pronounced dependence of the system's temporal quantum correlations on its dissipation to the environment. This opens up a potential pathway for harnessing dissipation to modulate quantum correlations and entanglement.