Thermodynamics of an Open $\mathcal{P}\mathcal{T}-$Symmetric Quantum System

For a subclass of a general $\mathcal{P}\mathcal{T}-$symmetric Hamiltonian obeying anti-commutation relation with its conjugate, a Hermitian basis is found that spans the bi-orthonormal energy eigenvectors. Using the modified projectors constructed from these eigenvectors, the generalized density matrix of the $\mathcal{P}\mathcal{T}-$symmetric evolution is calculated, and subsequently, ergotropy for a closed system is obtained. The $\mathcal{P}\mathcal{T}-$symmetric system, in an open system scenario, is studied to understand ergotropy under different regimes of non-Hermiticity of the Hamiltonian. The consistency of the three laws of thermodynamics for the $\mathcal{P}\mathcal{T}-$symmetric system in an open system scenario is also analyzed.