Links between dissipation and Rényi divergences in $\mathcal{PT}$-symmetric quantum mechanics

Thermodynamics and information theory have been intimately related since the times of Maxwell and Boltzmann. Recently it was shown that the dissipated work in an arbitrary nonequilibrium process is related to the Rényi divergences between two states along the forward and reversed dynamics. Here we show that the relation between dissipated work and Renyi divergences generalizes to $\mathcal{PT}$-symmetric quantum mechanics with unbroken $\mathcal{PT}$ symmetry. In the regime of broken $\mathcal{PT}$ symmetry, the relation between dissipated work and Renyi divergences does not hold as the norm is not preserved during the dynamics. This finding is illustrated for an experimentally relevant system of two-coupled cavities.

Figure 1

Numerical verification of the relations between dissipated work and the Renyi divergences in the regime of unbroken $\mathcal{PT}$ symmetry. The red dashed line shows $\langle {\left(e^{-\beta (W-\mathrm{\Delta }F)}\right)}^z\rangle$ as a function of $z$ for a sudden quench process $\lambda \left(0\right)=0\to \lambda \left(\tau \right)=J/2$, in the regime of unbroken $\mathcal{PT}$ symmetry. The blue short-dashed line shows $e^{(z-1)S_z({\rho }_f\left|\right|\rho \left(\tau \right))}$ as a function of $z$ for the same sudden quench process $\lambda \left(0\right)=0\to \lambda \left(\tau \right)=J/2$, in the regime of unbroken $\mathcal{PT}$ symmetry.

Figure 2

Numerical disproval of the relations between dissipated work and the Renyi divergences in the regime of broken $\mathcal{PT}$ symmetry. The red solid line and the blue solid line show, respectively, the real and imaginary part of $\langle {\left(e^{-\beta (W-\mathrm{\Delta }F)}\right)}^z\rangle$ as a function of $z$ for a linear quench process $\lambda \left(t\right)=2Jt/\tau$. The linear quench process brings the control parameter from $\lambda \left(0\right)=0$ to $\lambda \left(\tau \right)=2J$, which is in the regime of broken $\mathcal{PT}$ symmetry. The red short-dashed line and the blue short-dashed line display the real and imaginary part of $e^{(z-1)S_z({\rho }_f\left|\right|\rho \left(\tau \right))}$ as a function of $z$ for the same linear quench process.