Cumulant-based quantum relative Rényi functional

Abstract: We introduce a new cumulant-based quantum relative Rényi functional as a candidate quantum Rényi divergence, derived from the cumulant-generating function (CGF) of the quantum relative surprisal operator and extending the classical connection between Rényi divergence and statistical cumulants to the quantum setting. Unlike the Petz and sandwiched quantum Rényi 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 $α>1$ under the support condition $\operatorname{supp}(ρ)\subseteq\operatorname{supp}(σ)$, we establish several fundamental properties, including positivity, reduction to the classical case, additivity, unitary invariance, continuity, and monotonicity with respect to the Renyi parameter $α$. 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 $α=0$. 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.

• The paper introduces a novel cumulant-based quantum relative Rényi functional that extends traditional quantum divergences by incorporating the cumulant-generating function of the quantum relative surprisal.
• It employs a path-integral-like expansion and extensive numerical simulations to demonstrate data-processing behavior under commutativity-preserving channels.
• The analysis establishes key analytic properties, offers a quantification of relative quantumness, and paves the way for future research in quantum resource theories.

Cumulant-Based Quantum Relative Rényi Functional: A Technical Examination

Overview and Motivation

The paper introduces a novel quantum divergence measure—the cumulant-based quantum relative Rényi functional (Cu-Q relative Rényi functional, $S_\alpha^Q$)—extending classical and quantum information theory by grounding the generalization of Rényi divergence on the cumulant-generating function of the quantum relative surprisal operator. In contrast to the conventional Petz and sandwiched quantum Rényi divergences, which are rooted in operator-algebraic and operational justifications, this cumulant-centric approach is statistically motivated and admits a formal path-integral-like expansion.

The central analytic construct is $S_\alpha^Q(\rho\|\sigma)$, defined for $\alpha > 1$ under the support constraint $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$, with a regularized extension to arbitrary $\alpha\neq 1$, facilitating rigorous treatment in physically relevant, potentially non-full-rank cases. The paper establishes the functional's core analytic properties and explores its behavior under data processing, especially in the context of commutativity-preserving (CoP) quantum channels, supported by extensive numerical simulations.

Construction and Cumulant-Generating Structure

The cumulant-based quantum relative Rényi functional is defined as

$$S_\alpha^Q(\rho\|\sigma) := \frac{1}{\alpha-1} \ln \operatorname{Tr} \left[ \rho\, e^{(\alpha-1)(\ln \rho - \ln \sigma)} \right]$$

for $\alpha > 1$ and appropriately supported $\rho$, $\sigma$.

Unlike Petz’s or sandwiched definitions, this approach is directly built from the cumulant-generating functional of the quantum relative surprisal operator $\Delta \Xi = \ln \rho - \ln \sigma$. The cumulant expansion reveals the first term as the Umegaki quantum relative entropy, $S_\alpha^Q(\rho\|\sigma)$0, with higher terms corresponding to quantum cumulants such as variance and skewness:

$S_\alpha^Q(\rho\|\sigma)$1

Key structural attributes:

• Reduces to the Umegaki relative entropy as $S_\alpha^Q(\rho\|\sigma)$2.
• Recovers the classical Rényi divergence in the commutative limit.
• Encodes all cumulants of the quantum relative surprisal, providing a bridge from classical statistical mechanics to quantum statistical inference.

Path-Integral Representation

A defining feature of $S_\alpha^Q(\rho\|\sigma)$3 is its path-integral-like representation in Hilbert space, derived via a Lie–Trotter product expansion of the operator exponential. This expansion delivers a sum over discrete trajectories in the eigenbases of $S_\alpha^Q(\rho\|\sigma)$4 and $S_\alpha^Q(\rho\|\sigma)$5, with weights informed by overlaps, transition probabilities, and accumulated phases:

Figure 1: Scatter plot of $S_\alpha^Q(\rho\|\sigma)$6 versus $S_\alpha^Q(\rho\|\sigma)$7 for the qubit bit-flip channel with $S_\alpha^Q(\rho\|\sigma)$8, demonstrating strict adherence to the conjectured data-processing bound; the inset magnifies the near-equality region.

The representation reveals a geometric phase-like term and quantifies non-commutativity explicitly; in the commutative case, only classical trajectories survive, collapsing the representation to a direct sum.

Analytical and Information-Theoretic Properties

A comprehensive analytical investigation confirms that $S_\alpha^Q(\rho\|\sigma)$9 possesses the following properties:

• Positivity: $\alpha > 1$0 for $\alpha > 1$1.
• Normalization: $\alpha > 1$2.
• Unitary Invariance: The functional is invariant under joint unitary conjugations of $\alpha > 1$3 and $\alpha > 1$4.
• Additivity: For tensor product states, $\alpha > 1$5.
• Continuity: The functional is continuous over full-rank pairs and under regularization.
• Monotonicity in $\alpha > 1$6: $\alpha > 1$7 is monotonically non-decreasing for $\alpha > 1$8.

A regularization procedure establishes well-posedness for all $\alpha > 1$9 and all density matrices, allowing analysis of boundary cases (notably $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$0).

Relative Quantumness: Order-Zero Behavior and Commutativity

Specializing the functional to $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$1, the paper defines a relative quantumness measure:

$$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$2

This measure satisfies:

• $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$3 iff $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$4.
• $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$5 iff $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$6.

Thus, $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$7 is exactly sensitive to pairwise non-commutativity between quantum states, establishing a necessary and sufficient indicator for quantum structure lacking in purely classical distributions.

Figure 2: Scatter plot for the qutrit generalized bit-flip channel at $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$8, demonstrating violations above the conjectured QDPI threshold due to absence of the CoP property in this channel class.

Data-Processing Inequality and CoP Channels: Theoretical and Numerical Analysis

The quantum data-processing inequality (QDPI) for the functional under general CPTP maps is unresolved; however, strong numerical evidence is presented for monotonicity under commutativity-preserving (CoP) channels:

• For any CoP map $$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$$9 and all $\alpha\neq 1$0, $\alpha\neq 1$1.
• The conjecture is supported by extensive Monte Carlo simulations over qubit and qutrit state pairs for a variety of physically relevant (CoP) channels including depolarizing, semi-classical, and isotropic channels.

  Figure 3: Scatter plot for the qubit depolarizing channel at $\alpha\neq 1$2: all points fall below the equality line, in complete agreement with the CoP-QDPI conjecture.

  

  Figure 4: Qutrit dephasing channel, $\alpha\neq 1$3: while most points are monotone, a small fraction of violations are observed (inset), consistent with the absence of CoP structure in higher dimensions for this channel.

Channels not satisfying the CoP property (such as generalized bit-flip and dephasing channels in dimension $\alpha\neq 1$4) do exhibit QDPI violations, which are explicitly captured in the numerical analysis.

Implications and Future Directions

This cumulant-based framework extends the landscape of quantum divergences in several ways:

• Offers a path-integral and cumulant-based interpretation beyond algebraic operator approaches.
• Derives a quantifier of relative quantumness operationally tied to commutativity.
• Numerically isolates the structural role of CoP channels for the data-processing principle, suggesting a structural foundation for monotonicity in the quantum regime.

Potential future research directions include:

• Analytic resolution of QDPI for $\alpha\neq 1$5 under arbitrary CPTP maps.
• Extension of the cumulant-based formalism to other quantum resource theories (coherence, thermodynamics, etc.).
• Development of operational tasks or protocols (e.g., discrimination, hypothesis testing) where higher-order cumulants captured by $\alpha\neq 1$6 provide a quantifiable advantage.

Conclusion

The cumulant-based quantum relative Rényi functional broadens the class of quantum divergences with a construction fundamentally rooted in the statistical structure of the quantum relative surprisal. It is equipped with a robust set of analytic properties and a deep interrelation with commutativity and quantum data processing. The operational content relating higher cumulants to quantum distinguishability, the explicit path-integral structure, and the emergent theory of relative quantumness all signal avenues for further exploration in quantum information science.