Eternal life of entropy in non-Hermitian quantum systems

We find a different effect for the behavior of von Neumann entropy. For this we derive the framework for describing von Neumann entropy in non-Hermitian quantum systems and then apply it to a simple interacting $PT$-symmetric bosonic system. We show that our model is well defined even in the $PT$-broken regime with the introduction of a time-dependent metric and that it displays three distinct behaviors relating to the $PT$ symmetry of the original time-independent Hamiltonian. When the symmetry is unbroken, the entropy undergoes rapid decay to zero (so-called “sudden death”) with a subsequent revival. At the exceptional point it decays asymptotically to zero and when the symmetry is spontaneously broken it decays asymptotically to a finite constant value (“eternal life”).

Figure 1

von Neumann entropy as a function of time and varied bath size, with $c_1=1, g=0.7, \kappa =0.3$.

Figure 2

von Neumann entropy as a function of time and varied bath size, with $c_1=1, g=\kappa$.

Figure 3

von Neumann entropy as a function of time and varied bath size, with $c_1=1, g=0.3, \kappa =0.7$. The asymptote is at $S_{t\to \infty }\approx 0.3521$.