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Non-Hermitian Hamiltonians are a compulsory aspect of the linear dynamical systems that model many physical phenomena, such as those in electrical circuits, open quantum systems, and optics. Additionally, a representation of the quantum theory of closed systems with non-Hermitian observables possessing unbroken PT-symmetry is well-defined.   In this talk, I will second-quantize non-Hermitian quantum theories with paraFermionic statistics. To do this, I will introduce an efficient method to find conserved quantities when the Hamiltonian is free or translationally invariant. Using a specific non-Hermitian perturbation of the Su-Schrieffer-Heeger (SSH ) model, a prototypical topological insulator, I examine how PT-symmetry breaking occurs at the topological phase transition. Finally, I show that although finite-dimensional PT-symmetric quantum theories generalize the tensor product model of locality, they never permit Bell inequality violations beyond what is possible in the Hermitian quantum tensor product model.

Higher-order transformations that act on a certain number of input quantum channels with an indefinite causal order, such as the quantum switch, cannot be described by standard quantum circuits that use the same number of calls of the input quantum channels. But could they be simulated, i.e., could their action on their input channels be deterministically reproduced, for all arbitrary inputs, by a quantum circuit that uses on a larger number of calls of the input channels? In this work, we prove that, when only one extra call of each input channel is available, the quantum switch cannot be simulated. We demonstrate the robustness of this result by showing that even when probabilistic and approximate simulations are considered, higher-order transformations that are close to the quantum switch can be at best simulated with a probability strictly less than one. This result stands in stark contrast with the known fact that, when the quantum switch acts exclusively on unitary channels, its action can be simulated. We also show other particular cases where a restricted simulation of the quantum switch is possible. Finally, we discuss the implications of our findings to the analysis of experiments based on the quantum switch.