Quantum-Inspired Fidelity Divergence

• Quantum-Inspired Fidelity-based Divergence is a measure derived from quantum fidelity that quantifies differences between probability distributions and quantum states with key properties like symmetry and continuity.
• It offers robust, bounded divergence values that overcome limitations of traditional metrics such as KL divergence, ensuring stability even when distributions have near-disjoint supports.
• Its extensions unify quantum Rényi divergences and incorporate Riemannian-geometric methods, enabling improved regularization in machine learning models and enhanced performance on benchmarks like CIFAR-10 and GLUE.

Quantum-Inspired Fidelity-based Divergence (QIF) is a class of dissimilarity measures rooted in quantum information theory but designed for robust, efficient application in both classical and quantum statistical learning. QIF captures the distance or divergence between probability distributions or quantum states by generalizing the concept of quantum fidelity into a divergence functional. QIF and its parameterized extensions exhibit properties that address key pathologies of traditional measures such as Kullback–Leibler (KL) divergence, including improved stability when distributional supports are near-disjoint, boundedness, and continuity. Recent research has connected QIF to operational tasks in statistical inference and generalized it to encompass the full Rényi divergence hierarchy and optimal transport–inspired metrics.

1. Foundational Definitions

The construction of Quantum-Inspired Fidelity-based Divergence begins by formalizing fidelity-based similarity measures between distributions.

For probability distributions $P = (p_1, \ldots, p_d)$ and $Q = (q_1, \ldots, q_d)$, classical fidelity is

$$F(P, Q) = \left( \sum_{i=1}^{d} \sqrt{p_i q_i} \right)^2$$

For general quantum states, given density matrices $\rho, \sigma \in \mathbb{C}^{d \times d}$, quantum fidelity is defined as

$$F(\rho, \sigma) = \left[\mathrm{Tr} \left(\sqrt{\sqrt{\rho}\, \sigma\, \sqrt{\rho}}\right)\right]^2$$

The Quantum-Inspired Fidelity-based Divergence is then formulated as

$\mathrm{QIF}(P \parallel Q) = -F(P, Q) \ln F(P, Q)$

with $0 \leq F \leq 1$, leading to $0 \leq \mathrm{QIF}(P \parallel Q) \leq 1/e$ and $\mathrm{QIF}(P \parallel Q)\,{=}\,0$ iff $P = Q$ (Peng et al., 31 Jan 2025). For the quantum case, analogous constructions apply by substituting the appropriate quantum fidelity.

2. Mathematical Properties and Robustness

QIF possesses several rigorous properties especially relevant in high-dimensional and irregular statistical settings:

• Symmetry: $F(P,Q)=F(Q,P)$, hence $$\mathrm{QIF}(P\parallel Q) = \mathrm{QIF}(Q\parallel P)$$.
• Nonnegativity and Boundedness: $\mathrm{QIF}$ is nonnegative and upper bounded by $1/e$, eliminating the unbounded divergence that afflicts $D_{KL}(P\parallel Q)$ under support mismatch.
• Continuity: Both $F$ and $-x\ln x$ are continuous on the simplex interior, rendering $\mathrm{QIF}$ continuous in $(P,Q)$.
• Robustness to Support Mismatch: $\mathrm{QIF}$ converges to zero smoothly as the overlap between $P$ and $Q$ decreases, crucially avoiding divergence to infinity when $P$ and $Q$ have near-disjoint or partially overlapping support.
• Joint Convexity: Quantum generalizations satisfy joint convexity and are Lipschitz with respect to the trace norm (Matsumoto, 2014).

In quantum settings, QIF generalizes naturally by treating the fidelity as a functional on density matrices: $$\mathrm{QIF}(\rho \parallel \sigma) = 1 - F(\rho, \sigma)$$ or alternatively $D_{\!QIF}(\rho\|\sigma)=-\ln F(\rho,\sigma)$, with basic properties inherited from monotonicity and joint concavity of $F$ under completely positive trace-preserving maps (Matsumoto, 2014).

3. Parameterized and Riemannian-Geometric Extensions

Advancements in Riemannian-geometric approaches have produced a rich parameterized family of fidelities and divergence measures: $$F_R^{(\alpha)}(\rho, \sigma) = \operatorname{Tr}\bigl[ (R^{1/2} \rho R^{1/2})^{\alpha} R^{-1} (R^{1/2} \sigma R^{1/2})^{1-\alpha} \bigr]$$ for base point $R$ (a positive-definite matrix), and divergence

$$\mathrm{QIF}_\alpha(\rho\|\sigma) = \frac{1}{\alpha-1} \log \bigl( \mathrm{Re} F^{(\alpha)}_R(\rho, \sigma) \bigr)$$

This formalism subsumes the Petz–Rényi, sandwiched Rényi, reverse-sandwiched, and geometric α-divergences by suitable specializations of $R$, providing a unified operational interpretation and interpolation between quantum divergences (Afham et al., 7 Oct 2024).

These constructions inherit several invariance properties:

• Unitary Invariance: $$F_R(U\rho U^\dagger, U \sigma U^\dagger) = F_{U R U^\dagger}(\rho, \sigma),$$ for unitary $U$,
• Geodesic Covariance: On the Bures–Wasserstein manifold, the generalized fidelity reduces to Uhlmann fidelity at geodesic points,
• Operational Characterization via Purification: The generalized fidelity equals the maximal transition amplitude among purifications (Afham et al., 7 Oct 2024).

4. Computational Strategies

One notable feature of QIF is its practicality: in the classical case, computing $\mathrm{QIF}(P, Q)$ requires only $O(d)$ arithmetic operations and negligible additional memory, as shown by the pseudocode:

def qif(P, Q, epsilon=1e-13): dot = 0 for i in range(len(P)): dot += np.sqrt(P[i])*np.sqrt(Q[i]) F = dot**2 F_clamped = max(F, epsilon) return -F_clamped * np.log(F_clamped)

For quantum and matrix-valued deployments (moderate $d$), the computational bottleneck is matrix diagonalization and power operations, with total cost $O(d^3)$, as in:

def qif_alpha(rho, sigma, R, alpha): A = sqrtm(R).dot(rho).dot(sqrtm(R)) B = sqrtm(R).dot(sigma).dot(sqrtm(R)) A_alpha = scipy.linalg.fractional_matrix_power(A, alpha) B_1ma = scipy.linalg.fractional_matrix_power(B, 1-alpha) T = np.trace(A_alpha.dot(np.linalg.inv(R)).dot(B_1ma)) return (1.0/(alpha-1)) * np.log(np.real(T))

For large systems, profiling for computational scalability is necessary, although the core procedures for classical QIF remain linear in output dimension (Peng et al., 31 Jan 2025, Afham et al., 7 Oct 2024).

5. Applications in Machine Learning: QR-Drop Regularization

QIF has immediate utility in machine learning, especially as a replacement for KL divergence in regularization for model output consistency. QR-Drop is a regularization technique that employs QIF to enforce consistency between two stochastic (dropout-perturbed) outputs, replacing the standard R-Drop penalty: $$\mathcal{L}_{QIF} = \frac{1}{2}(\mathrm{QIF}(P_{w_1}, P_{w_2}) + \mathrm{QIF}(P_{w_2}, P_{w_1}))$$ with the practical simplification to a single symmetric term.

Empirical evaluations on standard benchmarks show that QR-Drop achieves systematically lower test loss and higher accuracy than unregularized, ordinary Dropout, and R-Drop. For CIFAR-10 (ResNet-18, dropout rate 0.1):

• Un-Reg: ~86.5% test accuracy
• R-Drop: ~88.3%
• QR-Drop: ~89.1%

Similar performance improvements are observed for LLMs on GLUE tasks, e.g., BERT-base: baseline 77.2, R-Drop 78.2, QR-Drop 78.5; RoBERTa-large: baseline 85.9, R-Drop and QR-Drop both 86.6 (Peng et al., 31 Jan 2025).

6. Theoretical and Practical Limitations

Analyses to date have concentrated on classification and sequence-classification; the potential of QIF in generative models (VAEs/GANs), reinforcement learning, and self-supervised learning remains to be explored. For very high-dimensional output spaces (e.g., vocabularies $>50$K), computational profiling is needed but formal complexity remains linear in $d$ for classical QIF. Future research directions include:

• Rigorous analysis of QIF’s Lipschitz continuity and implications for optimization and convergence,
• Extension to mixed quantum-state embeddings (beyond pure-state or diagonal cases),
• Integration with optimal transport and kernel-based divergences for composite metrics (Peng et al., 31 Jan 2025).

7. Connections to Quantum Information Theory and Generalizations

The QIF framework is tightly linked to foundational concepts in quantum statistics. Its parameterized forms encompass the entire family of quantum Rényi divergences, generalize Uhlmann, Holevo, and Matsumoto fidelities, and admit operational interpretations analogous to state discrimination and recoverability (Afham et al., 7 Oct 2024, Matsumoto, 2014).

By leveraging Riemannian geometry, these divergences associate naturally to the Bures–Wasserstein manifold, inheriting convexity, joint continuity, and invariance properties of their functional origins. Additionally, dual or polar fidelities, as established in convex analysis formalisms, provide further flexibility and insights into extremal behaviors and operational regimes (Matsumoto, 2014).

In summary, Quantum-Inspired Fidelity-based Divergences furnish a unified, mathematically robust, and computationally efficient class of measures, bridging quantum and classical paradigms for statistical inference and machine learning while avoiding pathologies of traditional divergences such as KL. The framework’s extensions and operational interpretations position QIF as a central tool in contemporary statistical and quantum information research.