Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts

We provide a brief introduction to the mathematical aspects of Hermitian and non- Hermitian quantum theory. Subsequently, we derive the general form of a two- dimensional Hamiltonian possessing PT symmetry, where T is the complex conjugation operator. We then investigate the diagonalizability of such Hamiltonians at the transition point. Furthermore, we analyze the time-independent solutions of the Dyson map for a non-Hermitian, time-independent Hamiltonian. During this exploration, we uncover certain features of these solutions at the transition point. As a central finding of this thesis, we consider a quantum oscillator coupled to a bath of N other oscillators. The total system evolves with a quasi-Hermitian Hamiltonian. Associated to it is a family of Hermitian systems, parameterized by a unitary map W. Our main goal is to find the influence of W on the entropy and the entanglement in the Hermitian systems. We calculate explicitly the reduced density matrix of the single oscillator for all Hermitian systems and show that, regardless of W, their von Neumann entropy oscillates with a common period which is twice that of the non-Hermitian system. We show that generically, the oscillator and the bath are entangled for almost all times. While the amount of entanglement depends on the choice of W, it is independent of W when averaged over a period. These results describe some universality in the physical properties of all Hermitian systems associated to a given non-Hermitian one.