Cumulant-based quantum relative Rényi functional

We introduce a new cumulant-based quantum relative R\'enyi functional as a candidate quantum R\'enyi divergence, derived from the cumulant-generating function (CGF) of the quantum relative surprisal operator and extending the classical connection between R\'enyi divergence and statistical cumulants to the quantum setting. Unlike the Petz and sandwiched quantum R\'enyi divergences, the proposed construction is motivated by statistical structure rather than operator-algebraic or operational principles. The functional naturally admits a path-integral-like representation through the Lie-Trotter product expansion, providing a trajectory-based interpretation of quantum divergence in Hilbert space. On its natural non-regularized domain for $\alpha>1$ under the support condition $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$, we establish several fundamental properties, including positivity, reduction to the classical case, additivity, unitary invariance, continuity, and monotonicity with respect to the Renyi parameter $\alpha$. Whether the functional satisfies the quantum data-processing inequality (QDPI) under arbitrary CPTP maps remains open. To extend the analysis beyond the studied regime, we introduce a regularized version of the functional and study its behavior at $\alpha=0$. We show that the resulting relative quantumness quantity vanishes if and only if the underlying states commute, yielding a necessary and sufficient characterization of non-commutativity. For commutativity-preserving (CoP) channels, we further conjecture a QDPI-type monotonicity relation for this quantity. Extensive numerical simulations provide strong evidence in support of this conjecture, with no violations observed for the CoP channels considered in this work.