Non-hermitian quantum systems have attracted enormous interest recently due to the plethora of unexpected phenomena associated to them, including exceptional points, biorthogonal topology, spontaneous PT-symmetry breaking, non-unitary dynamics, unidirectional invisibility, complex Bloch oscillations etc. While the eigenvalues of a non-hermitian Hamiltonian can still be interpreted in terms of energy bands, already the meaning of its eigenvectors cannot be treated conventionally as they are not orthogonal, and therefore possess finite overlap already in the absence of any additional perturbation.

The goal of this thesis is to focus in simple yet experimentally relevant non-hermitian quantum systems such as the transverse field Ising chain in imaginary magnetic field or PT-symmetric quantum critical models. Then, we would evaluate the full counting statistics of physical observables, such as quantum work done or magnetization density.