Semantic Uncertainty Quantification of Hallucinations in LLMs: A Quantum Tensor Network Based Method

Conference: ICLR 2026
OpenReview: https://openreview.net/forum?id=11kPIEkj75
Code: https://github.com/pragasv/semantic-entropy-UQ-
Area: LLM Hallucination Detection / Uncertainty Quantification
Keywords: Hallucination Detection, Semantic Entropy, Uncertainty Quantification, Quantum Tensor Network, Entropy Maximization

TL;DR

To address the blind spot where semantic entropy ignores the "stochastic fluctuations of token sequence probabilities themselves," this paper embeds the kernel mean embedding (KME) of sequence probabilities as a wave function of a Quantum Tensor Network (QTN). It employs perturbation theory to calculate the local uncertainty of each probability in a one-shot manner. These probabilities are then calibrated via "entropy maximization + KL penalty weighted by inverse uncertainty" to derive an interpretable Semantic Rényi Entropy that is more sensitive to confabulation. Across 116 experimental setups involving 4 datasets, 8 models, and 3 quantization levels, the method consistently outperforms SOTA in AUROC/AURAC.

Background & Motivation

Background: LLMs generate "confabulation"—fluent but unreliable content where answers drift randomly under the same prompt. The mainstream unsupervised approach for detecting such hallucinations is Semantic Entropy (SE): multiple responses are sampled for the same question, clustered into semantically equivalent groups using bi-directional entailment (DeBERTa), and the Shannon entropy of the cluster distribution is calculated. Higher entropy indicates a higher likelihood of hallucination. Subsequent works like KLE, SNNE, and SD have improved similarity metrics and clustering methods.

Limitations of Prior Work: Existing methods treat token sequence (TS) probabilities \(P(s\mid y)\) as deterministic inputs to calculate entropy, without considering the aleatoric uncertainty of these probabilities. However, when repeating the same prompt, sequence probabilities jitter due to irrelevant factors like random seeds or phrasing; as long as the probabilities are unstable, the derived semantic entropy remains unreliable. Consequently, high entropy does not always indicate a true hallucination, and low entropy can result from overconfidence, making it difficult to reduce false positives and negatives.

Key Challenge: Hallucination risk should ideally depend on "how sensitive sequence probabilities are to model perturbations"—a local sensitivity issue that global entropy thresholds (e.g., "hallucination if entropy > \(\tau\)") naturally fail to capture. Furthermore, classical Bayesian Deep Learning (BDL) tools for UQ are difficult to calibrate, and next-token probabilities do not behave like true categorical probabilities. Additionally, variational inference via stochastic sampling is prohibitively expensive for large-scale LLMs.

Goal: To provide a local, interpretable, one-shot (no multiple samplings required) uncertainty measure for each TS probability and integrate it into entropy calculations, making the final semantic entropy more sensitive to confabulations.

Key Insight: Drawing from physics-inspired UQ (Principe 2010)—treating the kernel mean embedding (KME) of a probability distribution as a wave function of a quantum system. By perturbing this system, the local sensitivity of the distribution to infinitesimal perturbations can be read from the first-order corrections of the eigenstates/eigenenergies. If the eigenstate is unstable under perturbation, the corresponding probability is uncertain.

Core Idea: Treat TS probabilities as wave functions of a Quantum Tensor Network, use perturbation theory to calculate the local uncertainty of each probability in one shot, and then use entropy maximization to push high-uncertainty regions toward high entropy while anchoring low-uncertainty regions to their original values. This yields an "uncertainty-aware" Semantic Rényi Entropy as the hallucination indicator.

Method

Overall Architecture

The method solves the problem of "incorporating the stochastic fluctuations of sequence probabilities into semantic entropy." The entire pipeline is deterministic and one-shot: the input \(y\) is sampled \(R\) times to obtain \(R\) generations; semantic clustering is performed via DeBERTa bi-directional entailment. Unlike previous works using Shannon entropy, Semantic Rényi (quadratic) entropy is used as the mathematical foundation for the physics-inspired UQ. All TS probabilities are embedded into an RKHS via KME as a QTN wave function. Perturbation theory is then used to calculate the local uncertainty \(\mathrm{UQ}(p_s^{(r)})\) for each probability \(p_s^{(r)}\). Finally, entropy maximization calibration is applied, using a KL penalty weighted by inverse uncertainty to adjust probabilities toward "maximum entropy." This results in adjusted cluster probabilities \(p_c^{(j)*}\), from which \(\mathrm{SE}_R^+\) is calculated for hallucination detection and answer selection.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Input y<br/>Sample R times"] --> B["Semantic Clustering<br/>DeBERTa Bi-directional Entailment"]
    B --> C["Semantic Rényi Entropy<br/>Quadratic Entropy as UQ Base"]
    C --> D["QTN Perturbation Uncertainty<br/>KME→Wave Function→1st-order Correction"]
    D --> E["Entropy Maximization Calibration<br/>KL Penalty Weighted by 1/UQ"]
    E -->|Adjusted Cluster Probabilities| F["Semantic Rényi Entropy SE⁺_R<br/>Hallucination Detection / Answer Selection"]

Key Designs

1. Semantic Rényi Entropy: Converting Entropy to a Quadratic Form for the UQ Framework

Clustering follows the classical approach: sample the input \(y\) repeatedly and cluster semantically equivalent generations \(C\) using DeBERTa. The cluster probability is \(p_c^{(j)} = P(c_j\mid y) = \frac{\sum_{s\in c_j}P(s\mid y)}{\sum_{j}\sum_{s\in c_j}P(s\mid y)}\). To integrate the physics-inspired UQ, an entropy form that can be expressed as a KME and corresponds directly to an "expectation" is required; Shannon entropy does not satisfy this. Thus, quadratic Rényi entropy is used: \(S_{E_R}(y) = -\log\sum_j p_c^{(j)2} = -\log(\mathbb{E}[p_c])\). This is a direct measure of distributional uncertainty, and \(\sum_j p_c^{(j)2}\) can be approximated by the empirical KME of a Gaussian kernel \(\hat\psi_y(x)=\frac{1}{R}\sum_r \kappa_\sigma(p_s^{(r)};x)\,p_s^{(r)}\). This step bridges "entropy" and "wave functions," which is the prerequisite for QTN perturbation.

2. QTN Perturbation Uncertainty: Quantifying Local Sensitivity via Wave Function First-order Corrections

This is the core innovation, targeting the blind spot of unquantified sequence probability instability. The empirical KME \(\{\hat\psi_y\}\) is treated as an eigen-mode of a QTN Hamiltonian \(\hat H\). Perturbing \(\hat H\) is equivalent to perturbing the underlying TS probability distribution; the first-order correction of the eigen-mode/eigen-energy reflects the sensitivity of the distribution to infinitesimal changes. Specifically, a first-order uncertainty "feature" vector is constructed:

\[V_m^{(1)}(x) = E_m^{(1)} + \frac{\sigma^2}{2}\frac{\nabla_m^2|\psi_m^{(1)}(x)|}{|\psi_m^{(1)}(x)|},\quad E_m^{(1)} = -\min_{p_x}\frac{\sigma^2}{2}\frac{\nabla_m^2|\psi_m^{(1)}(x)|}{|\psi_m^{(1)}(x)|}\]

Where the Laplacian \(\nabla_m^2|\psi_m^{(1)}(x)|\) measures the local change of the first-order correction relative to the mean. \(V_m^{(1)}(x)\) can be viewed as the "spectrum" of uncertainty across probability amplitudes. Finally, \(p_s^{(r)}\) is mapped to \(x^{(r)}\) in the RKHS, and the average of the adjacent \(M\) modes (the paper uses \(M=8\)) is taken as the uncertainty: \(\mathrm{UQ}(p_s^{(r)}) = \frac{1}{M}\sum_m V_m^{(1)}(x)\big|_{x=x^{(r)}}\). Large first-order corrections imply high instability; small ones imply local stability. Compared to Bayesian/sampling UQ, this is deterministic, one-shot, and physically interpretable.

3. Entropy Maximization Calibration: Pushing Probabilities toward Max-Entropy via Inverse-UQ Weighting

Having \(\mathrm{UQ}\) is not enough; it must be integrated into the probabilities. This is done via the principle of maximum entropy: when information is partial, the maximum entropy distribution is the "least biased" estimate. For each probability, the following is solved:

\[p_s^{(r)*} = \arg\max_{\hat p_s^{(r)}}\Big\{-\log\big(\hat p_s^{(r)2} + (1-\hat p_s^{(r)})^2\big) - \lambda\cdot\frac{1}{\mathrm{UQ}(p_s^{(r)})}\cdot \mathrm{KL}(\hat p_s^{(r)}\,\|\,p_s^{(r)})\Big\}\]

The first term is the Rényi entropy (pushing toward high entropy/non-arbitrariness), and the second term is a KL penalty pulling the adjusted value \(\hat p_s^{(r)}\) back to the empirical \(p_s^{(r)}\). Crucially, the penalty strength is scaled by \(1/\mathrm{UQ}\): when uncertainty is high, the KL penalty relaxes and the entropy term dominates, pushing the probability toward max-entropy; when uncertainty is low, the KL penalty tightens, anchoring the probability to the original value. This prevents blind flattening of all probabilities while correctly handling unreliable regions. The resulting \(\mathrm{SE}_R^+\) corrects the overestimation bias of confabulation, leading to more reliable detection.

An Example: "Which oil-producing country is a close ally of the US?"

Asking this question 10 times yields generations like Russia / Saudi Arabia (×4) / Iran / Kuwait / Qatar / Iraq. Cluster probabilities \(p_c^{(j)}\) are calculated; Saudi Arabia, the correct answer, originally holds \(0.888\). Hallucinated answers like Qatar, Iraq, and Iran have non-trivial total mass, causing standard \(S_{E_S}\) to underestimate confabulation risk. After uncertainty-aware calibration, the probabilities of high-uncertainty hallucinations are suppressed, and the cluster probability of Saudi Arabia rises to \(0.859\). The overall entropy estimate \(S_E^+\) (0.130) is systematically lower than NE (0.846) and \(S_{E_S}\) (0.225)—enabling the selection of high-certainty, semantically consistent answers even when hallucinations are present in the pool.

Key Experimental Results

Experiments cover TriviaQA, NQ-Open, SVAMP, and SQuAD across 8 models (Mistral-7B/instruct, Falcon-rw-1B, LLaMA-3.2-1B, LLaMA-2-7B/13B, and chat versions), totaling 116 scenarios. Metrics include AUROC, AURAC, and RAC.

Main Results

Evaluation Dimension	Ours	Baselines	Conclusion
AUROC (Correct vs. Incorrect)	SRE-UQ (\(\mathrm{SE}_R^+\))	NE / \(S_{E_S}\) / DSE / KLE etc. + supervised ER, p(True)	Competitive across all models; exceeds supervised methods without requiring ground-truth labels.
AURAC (AU Accuracy Curve under Rejection)	SRE-UQ	Same as above	Accuracy remains higher as high-uncertainty outputs are filtered; superior to discrete SE, naive entropy, and supervised p(True).
Robustness (16/8/4-bit quantization)	SRE-UQ	Same as above	Relative ranking of methods remains stable across three quantization levels; outperforms or matches SOTA under compression.

Ablation Study

Configuration	Key Metric	Description
\(\mathrm{SE}_R^+\) (with UQ Calibration)	Lowest entropy estimation, highest AUROC/AURAC	Full method.
\(S_{E_R}\) (Rényi only, no UQ Calibration)	Second to \(\mathrm{SE}_R^+\)	Removing UQ integration leads to overestimation of confabulation risk.
\(S_{E_S}\) / DSE / NE (Shannon/Discrete/Naive)	More likely to underestimate confabulation	Classical semantic entropy baselines.
4-bit Quantization vs. 16-bit	Slight dip in absolute AUROC	Quantization coarsens calibration, but relative ranking remains unchanged.

Key Findings

• Quantization Robustness is a dimension unexamined by previous works: absolute AUROC drops slightly at 4-bit, but the relative advantage of the method holds, suggesting the gains are not artifacts of precision settings—relevant for edge/mobile deployment.
• Entropy change is non-uniform across the input space: low-entropy regions (near 0) are stable, while intermediate ranges (especially 0.25–0.50) show the highest fluctuations—models often toggle between multiple "confident but semantically opposing" answers in this range. A single global entropy threshold is fragile; these high-risk zones require caution.
• Uncertainty integration systematically reduces entropy estimates and corrects overestimation bias, making it possible to select high-certainty answers even during hallucinations—a feat previous works failed to achieve.

Highlights & Insights

• Linking hallucination detection to quantum perturbation theory: Use the first-order correction of wave function eigenstates under perturbation to measure "how unstable a probability is," providing a deterministic, one-shot, and interpretable local UQ that avoids the pitfalls of BDL and sampling.
• Uncertainty is not just a score, but a KL penalty scaler: The \(1/\mathrm{UQ}\) weighting ensures calibration only where necessary. This "inverse weighting of constraint terms via auxiliary signals" is transferable to other scenarios requiring confidence-based output adjustments.
• Switching from Shannon to Quadratic Rényi Entropy allows the entropy to be formulated as a KME—a deliberate mathematical alignment to fit the physics-inspired framework.

Limitations & Future Work

• Restricted by compute, evaluations focused on smaller models (≤13B), which may not fully reflect the absolute detection precision of frontier models like GPT-4 or Claude.
• Semantic clustering relies on external entailment models (DeBERTa-large + MNLI); errors in entailment propagate to entropy estimation.
• The method requires access to token-level probabilities, making it inapplicable to black-box APIs that only provide text without logprobs.
• Details regarding QTN perturbation, the choice of \(M=8\) modes, and \(\lambda\) are moved to the appendix; the main text lacks comprehensive sensitivity analysis.

Related Work & Insights

• vs. Semantic Entropy SE (Farquhar 2024) / Discrete SE (DSE): These treat TS probabilities as deterministic inputs to Shannon entropy, ignoring stochastic fluctuations; Ours uses Rényi entropy + QTN perturbation to explicitly quantify and calibrate this aleatoric uncertainty.
• vs. KLE / SNNE / SD: These improve similarity metrics and aggregation but remain sensitive to clustering quality and TS probability jitters; Ours models uncertainty directly at the probability level.
• vs. Structure-aware/Perturbation-based (Graph Uncertainty, SPUQ, SIP): These rely on external knowledge graphs or repeated paraphrasing, which are expensive; Ours is a one-shot deterministic calculation.
• vs. Supervised ER, p(True): These require labels/few-shot training and generalize poorly to OOD data; Ours is unsupervised yet outperforms them in AUROC/AURAC.

Rating

• Novelty: ⭐⭐⭐⭐⭐ First to introduce aleatoric uncertainty of TS probabilities into hallucination detection via QTN perturbation theory for deterministic, interpretable local UQ.
• Experimental Thoroughness: ⭐⭐⭐⭐ 116 setups across 4 datasets, 8 models, and 3 quantization levels; includes under-explored dimensions like quantization and generation length.
• Writing Quality: ⭐⭐⭐⭐ The motivation and "intuition" sections explain the physical analogy clearly; however, the QTN formulas are heavy and rely on the appendix for derivation.
• Value: ⭐⭐⭐⭐ Unsupervised, one-shot, and robust to quantization; suitable for resource-constrained deployment, though limited by reliance on token probabilities for closed APIs.

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