A Preliminary Study on Some Properties of Cumulant-Based Quantum Rényi Functional

We propose a new kind of quantum relative Rényi functional and propose an information-theoretic interpretation of the Rényi parameter. Unlike several previously proposed forms that often lack clear foundational motivation, our expression naturally arises from the concepts of a cumulant-generating function and quantum surprisal. In addition, we show that it can be written in a path-integral-like formulation. We further numerically investigate several key properties of our functional in the case of qubits. These properties include positivity, monotonicity in the Rényi parameter (α), and monotonicity under several standard completely positive trace-preserving (CPTP) maps.