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

In quantum information theory, the quantum relative Rényi functional is a generalisation of quantum relative entropy that quantifies the difference between two quantum states. We study a new type of quantum relative Rényi functional, motivated by the concepts of the cumulant-generating function and quantum surprisal difference, and propose an information-theoretic interpretation of the Rényi parameter. In addition, we demonstrate that this functional can be naturally expressed in a path-integral-like formulation, providing a distinct perspective compared to existing approaches. To support the viability of our proposal, we conduct a preliminary numerical study of its key properties for qubits and qutrits. These properties include the monotonicity in the Rényi parameter (α) and the quantum data-processing inequality (QDPI).