# Transition Probabilities in the Two-Level Quantum System with PT-Symmetric Non-Hermitian Hamiltonians

## TL;DR

This paper develops a consistent probability framework for two-level quantum systems with PT-symmetric non-Hermitian Hamiltonians, demonstrating probability conservation and applying it to neutrino oscillations, aligning with experimental data.

## Contribution

It introduces a formalism for defining transition probabilities in PT-symmetric non-Hermitian quantum systems, ensuring probability conservation and applying it to neutrino oscillation phenomena.

## Key findings

- Probability conservation is achieved with a proper final state definition.
- Exact PT symmetry implies a maximal vacuum mixing angle.
- The formalism aligns with current neutrino oscillation experiments.

## Abstract

We investigate how to define in a consistent way the probabilities of the transitions between the "flavor" states of the two-level quantum system, which is described by a non-Hermitian but parity and time-reversal (PT) symmetric Hamiltonian. Explicit calculations are carried out to demonstrate the conservation of probability if a proper definition of the final state is adopted. Finally, this formalism is applied to two-flavor neutrino oscillations $\nu^{}_\mu \to \nu^{}_\mu$ and $\nu^{}_\mu \to \nu^{}_\tau$ in vacuum, where the exact PT symmetry requires the vacuum mixing angle to be maximal, which is compatible with current neutrino oscillation experiments. A possible generalization to the three-flavor case is briefly discussed.