Operational signatures of exceptional quantum channels

Exceptional points (EPs) are usually introduced as defective degeneracies of non-Hermitian Hamiltonians and, in fully open quantum dynamics, of Liouvillian generators. Here we formulate an operational theory of exceptional points that does not invoke a Hamiltonian or a master equation. The primitive dynamical object is a completely positive trace-preserving (CPTP) quantum channel $\mathrm{\Phi }$, applied an integer number $n$ of times, while physical data are probabilities $p_n^{\mathbf{A}}(x\mathrm{\mid }\rho )$ associated with states $\rho$ and observables $\mathbf{A}$ having measurements results $x$. We define exceptional channels through nontrivial Jordan chains of $\mathrm{\Phi }$ and show that a Jordan block of order $m$ generates discrete polynomial-exponential terms with maximal degree $m-1$. This places the physics of exceptional points in contact with what can be directly measured in experiments.