Semantic Uncertainty Quantification of Hallucinations in LLMs: A Quantum Tensor Network Based Method
Authors: pragatheeswaran vipulanandan
Kamal Premaratne
Dilip Sarkar
Venue: ICLR 2026 Poster (2026)
Keywords: Semantic uncertainty
Large language models
quantum physics
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We model token probability uncertainty as a quantum wave function to enhance confabulation detection in LLMs, improving reliability across output lengths and quantization levels for more trustworthy AI.

Abstract: Large language models (LLMs) exhibit strong generative capabilities but remain vulnerable to confabulations, fluent yet unreliable outputs that vary arbitrarily even under identical prompts. Leveraging a quantum tensor network–based pipeline, we propose a quantum physics-inspired uncertainty quantification framework that accounts for the aleatoric uncertainty in token sequence probability for semantic equivalence-based clustering of LLM generations. In turn, this offers a principled and interpretable scheme for hallucination detection. We further introduce an entropy-maximization strategy that prioritizes high-certainty, semantically coherent outputs and highlights entropy regions where LLM decisions are likely to be unreliable, offering practical guidelines for when human oversight is warranted. We evaluate the robustness of our scheme under different generation lengths and quantization levels, dimensions overlooked in prior studies, demonstrating that our approach remains reliable even in resource-constrained deployments. A total of 116 experiments on TriviaQA, NQ, SVAMP, and SQuAD across multiple architectures (Mistral-7B, Mistral-7B-instruct, Falcon-rw-1b, LLaMA-3.2-1b, LLaMA-2-13b-chat, LLaMA-2-7b-chat, LLaMA-2-13b and LLaMA-2-7b) show consistent improvements in AUROC and AURAC over state-of-the-art baselines.

Published as a conference paper at ICLR 2026
SEMANTIC UNCERTAINTY QUANTIFICATION OF HAL- LUCINATIONS IN LLMS: A QUANTUM TENSOR NET-
WORK BASED METHOD
Pragatheeswaran Vipulanandan', Kamal Premaratne', Dilip Sarkar2 DDpartment of Electrical and Computer Engineering 2Department of Computer Science University of Miami Coral Gables, FL 33146, USA
pxv24 122450ma 150mmi ami. edu, kamal@miami edu, sarkar@miami, edu
ABSTRACT
Large language models (LLMs) exhibit strong generative capabilities but remain vulnerable to confabulations, fluent yet unreliable outputs that vary arbitrarily even under identical prompts. Leveraging a quantum tensor network-based pipeline, we propose a quantum physics-inspired uncertainty quantification framework that accounts for the aleatoric uncertainty token sequence probability for semantic equivalence-based clustering of LLM generations. In turn, this offers princi- pled and interpretable scheme for hallucination detection. We further introduce strategy that prioritizes high-certainty, semantically co- herent outputs and highlights entropy regions where LLM decisions are likely be unreliable, offering practical guidelines for when human oversight is ranted. We evaluate the robustness of our scheme under different generation lengths and quantization levels, dimensions overlooked in prior studies, demonstrating that our approach remains reliable even in resource-constrained deployments. A total of 116 experiments on TriviaQA, NQ, SVAMP, and SQuAD across multi- ple architectures (Mistral-7BB Mistral-7B-instren Falcon-rw-16 LLaMA-3.2- LLaMA-2- 13b-catt LLaMA-2-7b-Ca LLaMA-2-131 and LLaMA-2-7b) show consistent improvements in AUROC and AURAC over state-of-the-an art baselines.
matropy-maximize
INTRODUCTION
Large language models (LLMs) have demonstrated remarkable capabilities understanding. in- terpreting, and reasoning over human language and natural language generation (NLG) at high level of fluency and coherence. LLMs, trained on massive text corpora, have led to the creation of powerful tools that brace state-of-the-art (SOTA) approaches in tasks such summarization, question answering AQ&A), dialogue generation, and are reshaping how humans interact with machines (e.g., ChatGPT, which provides conversational AI experiences (Hosseini al. (2023)). GitHub Copilot (Chen et al. (2021). DALL-E (Lai et al. (2023)), and others).
As LLMs are being integrated in new applications in nearly all aspects of daily life and in almost every scientific field, the reliability of their outputs has gained increasing attention. Errors and inconsistencies they introduce can have direct and significant impact on downstream performance. This issue is particularly evident in interactive systems such as chatbots, where the responses are expected to be not only fluent but also factually grounded (Lei al, (2023) ), especially when they are being relied upon in more sensitive application scenarios (e.g.. healthcare, defense, law, journalism, autonomous driving) (Kaddour et al. (2023): Zhao et al. (2023): ; Lei et al. (2023)). One particular vuinerability of LLMs hallucinations, which refer to LLM generations that appear coherent and plausible but are in fact incorrect, unverifiable, or entirely fabricated (Ji et al, (0023)). Hallucinated outputs range from minor factual inaccuracies to completely spurious claims, and they pose serious obstacle to the deployment of LLMs in high-stakes domains; use hallucinatory content has significant ethical concerns well.

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Published as a conference paper at ICLR 2026
1.1 PREVIOUS WORK
Most approaches for detecting and mitigating hallucinations lean heavily on supervised learning frameworks (Rateike et al. (2023); Quevedo et al. (2024) or in-context learning techniques grounded in complex syntactic/semantic analyses (Wang et al. (2022); Manakul et al. (2023); /Zhang al.]
(2023). ). While showing promise, their computational demands often introduce latency making them unfit for real-time applications. Numerical features (e.g., statistical patterns and structural output properties) and correlations between them may also serve as effective indicators of hallucinations (Azaria & Mitchell ): Lee (2023); [Su et al. (2024)), yielding lightweight, resource-efficient detection methods that complement or even outperform more computationally intensive techniques. Although Bayesian deep learning (BDL) methods have long been used to quantify uncertainty in standard classification tasks (Schweighofer et al. (2024)), they are far less effective for hallucination detection in LLMs: they are challenging to calibrate, they do not reliably reflect a model's s uncertainty about factual correctness, and next-token probabilities do not behave like true class presscrreiitioon probabilities. As a result, classical BDL tools struggle to capture the semantic and contextual instabilities that drive hallucinations, limiting their usefulness in this setting (Kang et al. (2025).
Other works focus on specific types of hallucinations with particular attention paid to confabulations, fluent but incorrect generations that appear arbitrary (Berrios (1998). They are sensitive irrelevant factors (e.g., random seeds and prompt phrasing) and are distinct from other types of hallucinations, such incorrect answers from models that have been trained widespread misconceptions (Lin itt al. (2022). "lies" strategically generated in pursuit of reward signals in reinforcement learning setups (Evans al, (202T). ). and systematic failures triggered from flawed reasoning or generalization. To detect confabulations, Quevedo et al. (2024) employ token sequence (TS) probabilities and their induced entropy, a natural and interpretable measure of uncertainty (Lindley) (9566); Kadavath et al.) (2022)) ), but naive estimates of entropy may not serve as good indicator of hallucinations (Xiao) ct al. (2220) because lexical/syntactic differences in LLM generations often do not imply a semantic difference. Therefore, TTeruuore,Faarquaar et al. (2024) and Kossen al. (2024) employ semantic entropy (SE), not lexical/syntactic variations, of generated outputs to quantiffy hallucinatory behavior. The output (TSs) generated when an LLM confronts the same imput (context + prompt) repeatedly are clustered based on their bidirectional entailment (as assessed by DeBERTa). Higher semantic entropy indicates higher likelihood of hallucination. In similar vein, Kernelized Likelihood Entropy (KLE) (Nikitin) et al. (2024) employs a graded similarity measure to account for semantic differences; Semantic Nearest Neighbour Entropy (SNNE) (Nguyen et al. (2025) aggregates pairwise similarities with LogSumExp smoothing, avoiding explicit clustering and improving robustness on summarization and translation: Semantic Density (SD) (Qiu & Miikkulainen (2024)) weights semantic distances by generation probabilities. These approaches improve confabulation detection, but are sensitive to clustering quality, similarity functions, and TS probability fluctuations.
Structure-aware and perturbation-base methods have also been explored for detecting hallucinations. Graph Uneertainty (Tiang et al. (2024)) and KEA Keerne-Ennriched AI) explain (Haskins & Adams (2025) model relationships between knowledge graphs (KGs) constructed from LLM outputs and ground truth claims for detection of hallucinations. But they incur heavy computational overhead and rely on an external knowledge base and accurate KG construction. Sampling with Perturbation for Uncertainty Quantification (SPUQ) (Gao et al.) (2024)) and Semantic- Cemannic-Invariant Perturbation Sampling (Cox et al. (2025) ) estimate uncertainty by probing model sensitivity to paraphrased perturbed inputs, exposing prompt sensitivity but at the cost of repeated queries.
In spite of the advances they have made, SOTA methods still struggle with reliably detecting fabricated content across diverse tasks and domains (Li et al. (2023)). ). A possible reason is that they leave unaddressed the aleatoric uncertainty of the TS probabilities. Hallucination risk should depend on how sensitive these TS probabilities, and hence the semantic entropies they induce, are to model perturbations. This highlights the need for local, rather than global, sensitivity metrics to assess TS probability uncertainty. Prior studies have also not examined the robustness of their approaches with different quantization levels despite the latter playing a major role in real-world deployment of L.MMs.
1.2
OUR CONTRIBUTIONS
"White box" inherently interpretable approaches (e.g., random sampling- based variational inference methods (Ghahramani (2015) )) can be computationally prohibitive for UQ of massively scaled L.LMs
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and iinsiired methods have emerged compelling alternative. Our work employs the mmysics-inspiree approach, advocated in Principe| (2010), and adopted in |Singh & Principe (2020)
and Vipulananthan et al. (2024), that leverages perturbation theory of an approprate quantum wave function to offer a deterministic, one-shot, interpretable (Rudin (2019); Linardatos et al. (2021) method for quantifying uncertainty locally and different resolutions.
The main contributions and key advantages of our approach are as follows: (a) We introduce 7S probability uncertainty as a novel indicator of confabulation detection. For this purpose, view the TS probabilities as the wave function of a quantum tensor network (QTN) (Qi & Ranard (019);; Vipulananthan al. (2024); Vipulanandan al. (2026)) and leverage perturbation theory for UQ (b) We offer a method, based on entropy maximization, to calibrate the TS probabilities to account for this uncertainty associated with TS probabilities. (c) This leads us to cluster LLM generations based on not only SE but uncertainty of TS probabilities. So, instead of simply flagging high entropy generations as hallucinations (Farquhar et al. (2024)), we offer a more meaningful scheme that flags outputs associated with higher uncertainty that one may reduce the likelihood of false negatives and false positives. (d) We evaluate the robustness of our framework under different LLM quantization levels well under different generation lengths, ranging from short phrase answers to sentence- outputs. Our results show that the method robust and remains reliable even in chouucc-constrrined or effciinnyy-pttmizzed deployments, thus extending the utility of hallucination detection to real-world settings. The overall flow our approach is illustrated in Fig.
2 PROPOSED METHOD FOR UQ OF LLM GENERATIONS
Semantic
Cluster Analysis
Fetch Token Probabilities
Analysis
Entailment- Based
Semantic Rényi's
Semantlc Chustering
Entropy Calculation
Quantum Network
Spin Chain
Entropy Maximization
LLM Hallacination Detection
UQ
Uncertainty
with UQ Penalisation
Decision
Quantiftcation
Figure I: Overview of our hallucination detection pipeline. Sequences are clustered via directional entailment, and UQ obtained through QTN is used for entropy maximization to enable reliable LLM hallucination detection.
2.1 PRELIMINARIES: SEMANTIC CLUSTERING
Consider the output TS that the LLM generates in response to input y. Let s; denote the i-th token and sei the sequence of previous tokens. Then, the probability 8, conditioned on y, is
P(s v) II P(s, 22
(1)
Clustering is determined by the semantic dissonance between output TSs as determined by bidirec- tional entailment (for which we employ DeBERTa). Let C' denote the set of semantically dissonant classes and |C| its cardinality. Treating cluster label ey as a random variable (on the space of C), the cluster probability that the TS belongs in Cj E C' and the induced semantic (Shannon) entropy
Exec, ((s v)
( P(e y) EA P( y)
SEs(y) Σ p! log p!.) GEC
(2)
When the TS probabilities P(s; Se y) are available, pe are computed from equations] lndd] otherwise, we use p(,)) x 1ic = c,)/ the membership count each cluster. The latter referred to as discrete semantic (Shannon) entropy in Farquhar et al. (2044)
respectively.
2.2 SEMANTIC RÉNYT ENTROPY
The framework for UQ in Principe (2010) requires non- parametrii feature mapping of the TS probabilities in the form of its kernel mean embedding (KME) in the reproducing kernel
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be precise, sampled Published as a conference paper at ICLR 2026
Hilbert space (RKHS)
determined by suitable positive semi-definite kernel (Aronszajn 1950);
Schölkopf et al. ). We employ the Gaussian kernel Ki (Ps; where _ P(s y) and æ E R. The resulting KM and its empirical estimate are, respectively,
(2(x))
f No (Ps, dpss
2)(2))
1
Σ (p 12) r=0
R
where {8(*), p= 0, ..... R 1} denotes the LLM generations when the input y is repeated R times
and ps P(g(r) y). As Principe 2010) establishes, (.)()) serves as kernel-based empirical estimator of the more general notion Rényi (quadratic) entropy
IC
0))² SEn (y) = log Σ pe
= =- log (E(pe)),
j=1
which in turn is a direct measure of uncertainty in a distribution. this work, we exclusively focus on the quadratic form of Rényi entropy and, for brevity, refer to it throughout the paper semantic R' d'enyi entropy. Here, E[] denotes expectation (over the space of C). Indeed, (x)) can be viewed how uncertainty varies with æ. For purposes of UQ, we view this KME
version {2b of (x) - as a 'data' wave function associated with the time-independent Schrödinger
equation of QTN (Qi & Ranard 20199) Vipulananthan et 2024): Vipulanandan et al. (2026)).
Given that it is Rényi entropy which underpins our probabilities, we opt use the same (instead of Shannon entropy as in SOTA methods) for assessing semantic entropy of output clusters associated with LLM generations too.
framework for UQ of TS
2.3 ACCOUNTING FOR UNCERTAINTY IN TS PROBABILITIES
UQ of TS Probabilities For purposes of UQ of TS probabiltiies, we identify a QTN whose Hamiltonian H has {0, as one of its eigen-modes, and apply perturbation theory to compute the
corrections (we only look at the first-order corrections) to all the eigen-modes/energies H. Then, construct the first-order uncertainty 'feature' vectors
22 ( (x)
0²
V2
(1) (x))
m (x) E(1) +
where E(1)
min Ps
(5)
2
(1))) (x)]
2 (x)]
Here, {eb n Em and (w0)) E!l)) }. for m 2 0, are the eigen- pairs of the Hamiltonian A ordered as Eg < Ej S ... and their first-order correction terms, respectively. For each mode m 2 0, the Laplacian V 1 (x) measures the change in the first-order correction from its average in the local neighborhood across the modes and the vector V!!l (x) can be viewed as a 'spectrogram' of uncertainties across TS probability amplitudes. If p!"" is mapped to (r)r in the RKHS, the uncertainty UQ ( associated with the TS probability z!5)) is taken as
UQ (p!))
Σ M
V (x) x=vi()
m
where the summation is carried over M (we M = 8) modes that are adjacent to f Additional
details on QTN based UQ are provided in Appendix A.2
Calibrated Adjustment of TS Probabilities
We now adjust the TS probabilities as
p(r)" arg max p(r)
log (p(r) + + (1-
KL (2! (ndd")
UQ (p
The log(-) term the right-hand side is the Rényi entropy; the second term penalizes the Kull- back-Leibler (KI.) divergence between the adjusted probability p!r) and p(² scaled inversely with
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the uncertainty UQ )): the hyperparameter À > 0 controls the trade-off between entropy max- imization and the adjusting the TS probability. In essence, we are maximizing entropy of the TS probabilities while penalizing the error between p."' and its adjusted value p This scheme which appears in Fig. | enables one to account for uncertainty associated with TS probabilities so that more informed decision regarding existence/absence of hallucinatory behavior could be made. Appendix D contains additional discussion of computational overheads together with the pseudocode implementations of key components.
2.4 INTUITION BEHIND THE PROPOSED APPROACH
Our method integrates perturbation- based uncertainty quantification and maximum entropy inference into a unified framework for correcting TS probabilities in principled manner.
Perturbation-based uncertainty quantification. To estimate uncertainty in the TS probabilities, we map the KME of the TS probability distribution into the eigen structure of a QTN. The empirical KME is embedded as an eigen-mode of an admissible Hamiltonian H associated with the QTN. Therefore, perturbing H corresponds to perturbing the underlying TS probabilities; the resulting first-order corrections to its aggn-modes/energiee quantify how sensitive the modeled distribution to infinitesimal changes in its inputs. This mirrors standard quantum mechanical perturbation theory where the instability of an eigen-state under perturbation reveals its local variability.
Large first-order corrections, therefore. indicate highly unstable TS probabilities; small corrections indicate locally stable regions. This produces an interpretable, physically grounded uncertainty measure, formalized in eq. (6). that captures the local variability of the TS probabilities in the RKHS amplitude domain.
Maximum entropy inference under partial knowledge. When only partial information is avail- able, the maximum entropy principle provides the least biased estimate of an unknown probability distribution. Among all distributions consistent with the known constraints, the one with the highest entropy is maximally non-committal with respect to the missing information (Jaynes (1957 2003)). Equivalently, the maximum entropy distribution minimizes its KI. divergence from the uniform distribution. This idea has deep roots in robust statistics and variational inference, where uncertainty in the likelihood leads naturally to regularization toward higher-entropy, less overconfident solutions.
Our formulation implements precisely this. The Rényi entropy term in eq. specifies how much the probabilities should be lifted to remain maximally non-committal, whereas the KL term ensures that the adjusted probabilities stay close the model's empirical TS estimates. When uncertainty is high, the KL penalty relaxes and the entropy term dominates; when uncertainty is low, the KL term keeps the adjusted probabilities anchored near their empirical values.
Bringing the two components together. The overall procedure thus follows a coherent logic: (i) infer local uncertainty in the TS probabilities by perturbing the Hamiltonian and reading out the first-order eigen-mode mode corrections; (ii) use these corrections to scale the KL penalty in the entropy maximization step; and (iii) adjust the TS probabilities to the maximum entropy distribution allowed by the empirical TS estimates. High-uncertainty regions are pushed toward higher entropy, while low-uncertainty regions remain close to the model's original predictions.
This yields a deterministic, single-shot, principled method for correcting TS probabilities in accor- dance with their underlying uncertainty, unifying perturbation-based UQ with maximum-entropp reasoning in a single framework.
3 EXPERIMENTS
Datasets Our evaluations cover Q&A in trivia knowledge (TriviaQA (Joshi et al. (2017))) general knowledge (SQuAD 1.1 (Stanford NLP Group)), and open-domain natural questions INQ-Open (Lee et al. (2019)) )) derived from actual queries to Google Search (Kwiatkowski et al. (2019). ) and mathematical word problems (SVAMP (Patel et al. (22TT). Only the results summary is reported here; while detailed results are deferred Appendix
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Models We use diverse set of LLMs obtained via Huggingface (huggingfaat com), including
Falcon-RW 1B (Penedo et al. (2023)) LLaMA-3.2 1B (Grattafion et al. (2024)). LLaMA-2-7B-chan
LLaMA-2 7B, LLaMA-2-13BB chatt, L 13B (Touvron et al. (2023)). Mistral- Disstal-7B-instruue -B-instruct-vO.3 and Mistral-v0.1 7B @Jang 2024)). The inclusion of smaller-scale models and quantized versions of larger models serves two purposes. From a practical standpoint, resource limitations necessitate the use of lighter models to enable extensive sampling and repeated querying. From a scientific viewpoint, evaluating smaller and compressed models allows us to explore hallucinations and uncertainty under constrained model capacity and precision, providing insights into how semantic entropy behaves not only in ideal settings but also in more realistic, resource- constree- deployments. This is particularly relevant as smaller and quantized models are increasingly being deployed in edge, mobile, and enterprise environments. All experiments were run on a workstation with 64 GB memory and NVIDIA A6000 GPU.
Model Prompting, Entailment, and Answer Selection Appendix provides the prompt template used for all datasets to generate L.LM responses and the prompt we used to detect entailment. While other classifiers could have been used, we employ a DeBERTa-large model fine-tuned on the MNLI dataset (Williams et al. (2017)) for entailment prediction. This method builds upon previous research in paraphrase detection based on embedding similarity (Socher et al. (20T):: Yu et al. (2014) and BERT-style models (He et al. (2020): [aa et al. (2021)). For simplicity. entailment is checked by concatenating the question with one answer and comparing it to the concatenation of the question with another answer. Instruction-tunet LLMs, such as LLaMA 2, GPT-3.5 Turbo or GPT-4, could also have been used to predict entailment between generated outputs.
Entropy Maximization Following UQ of the LLM generations, entropy maximization (see equa- tion 7) was applied to identify the 'optimum' answer based on the associated uncertainty estimates. The effectiveness of this approach (in the sense of whether incorporating uncertainty information improves hallucination detection) was assessed using the evaluation metrics described below. We also investigated the optimal choice of the hyperparameter X. Implementation details and extended analyses are deferred Appendix
Comparison Methods For evaluation, we use the following methods for comparison against the proposed Rényi semantic entropy pre- and post-TS uncertainty integration (SER and SE respectively): naive entropy (NE), semantic entropy (SEg), and discrete semantic entropy (DSEg), all employed in |Farquhar al. (2024), and two strong baseline methods, embedding regression (ER) and verifier p (True). ER is a supervised method inspired by the P(IK) approach (Kadavath et al. (2022)). Rather than fine-tuning the entire LLM as in the original work, this method trains logistic regression classifier on the final hidden representations to predict answer correctness, improving both simplicity and reproducibility. This baseline performs well with in-distribution data but poorly with out-of- -istittritttion data. In (TTue) (Kadavath et al. (2022), the model samples multiple candidate answers and is then asked, via a multiple-choice prompt, whether the top answer is "True" or "False". Confidence is measured by the probability assigned to the "True" response. A few-shot strategy, using up to 20 labeled examples, further enhances performance, although context window limitations sometimes require reducing the number of few-shot examples.
Evaluation Metrics We three primary metrics each based on automated assessment of factual consistency with reference answers from the datasets employed - for evaluation purposes: (a) AUROC (Area Under the Receiver Operating Characteristics) measures the ability of classifier to distinguish between correct and incorrect answers while accounting for both precision and recall. Intuitively, represents the probability that randomly selected correct answer receives higher confidence score than randomly selected incorrect one. A perfect model achieves an AUROC of (b) RAC (Rejection Accuracy Curve) measures the question-answerte percent of the most confident model predictions. An effective uncertainty estimation approach should ensure that the high-confidence predictions being retained are more accurate, with RAC improving as less confident examples are progressively excluded. (c) AURAC Area Under the Rejection Accuracy Curve) quantifies how accuracy changes as increasingly uncertain predictions are rejected. Higher AURAC values indicate that the uncertainty method more effectively distinguishes accurate from
accuracy that a specified
code available: tttp://gthhccc.mmnggaaaavaamaima

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inaccurate responses. Unlike AUROC, AURAC is directly sensitive the model's overall accuracy, providing a complementary perspective on the quality of uncertainty estimation methodology.
4 RESULTS
4.1
UNCERTAINTY-AWARE SEMANTIC ENTROPY: A WORKED EXAMPLE
This section presents worked example illustrating the computation of Semantic Rényi entropy with UQ maximization (SRE-UQ), denoted as SE together with NE and SE. The query used for this illustration is, "Which oil producer is close ally of the United States?" repeated ten times. The corresponding TS probabilities are in Table
The results demonstrate that incorporating uncertainty quantification into semantic entropy (i.e., SE yields systematically lower entropy estimates relative to both NE and 'standard' Shannon entropy- based SEg. This reduction arises because semantically equivalent generations are clustered together for this question, while their TS probabilities exhibit reduced alcatoric variability. Importantly, accounting for such uncertainty mitigates overestimation biases in entropy, providing a more faithful measure of confabulation in LLM outputs.
LLM Generation
Cluster ID
((r)
p4
(r)
Pe
(1)"
NE
SEg
SE
Russia
1
0.05899
0.01814
0.01814
0.02223
-0.03159
-0.03159
0.00049
Saudi Arabia
2
0.57761
0.17765
0.88824
0.85880
-0.13331
-0.04572
0.73754
Saudi Arabia
2
0.57761
0.17765
-0.13331
Iran
3
0.07227
0.02223
0.02223
0.02697
-0,03674
+0.03674
0.00073
Saudi Arabia
2
0.57761
0.17765
+0.13311
Kuwait
0.08940
0.02749
0.02749
0.03488
-0.04291
-0.04291
0.00122
Qatar
5
0.02185
0.00672
0.00672
0.01214
-0.01460
-0.01460
0.00015
Saudi Arabia
2
0.57761
0.17765
-0.13331
Iraq
6
0.12086
0.03717
0.03717
0.04498
-0.05315
-0.05315
0.00202
Saudi Arabia
2
0.57761
0.17765
-0.13331
Total
3.25144
1,00000
1. 1.00000 1.00000
0.84557
0.22471
0.12951
Table 1: An example calculation of TS probability and uncertainty metrics corresponding to the query above. Columns: p!" are the raw (unnormalized) sequence probabilities; (() are the normalized sequence probabilities (equation p P(c, y) are the cluster probabilities (equation
post-normalized; (5)* are the cluster probabilities post-TS uncertainty integration (7) Methods: NE and SËş denote naive entropy and semantic entropy, respectively (Farquhar et al. (2024); SE denotes semantic Rényi entropy post-TS uncertainty integration. Values for only representative
LLM generation per cluster are shown.
Table illustrates this effect for the example query. It compares the semantic response distributions produced by 'standard' Rényi entropy-based SER and its uncertainty-adjuse variant SE While several hallucinated outputs (e.g., Qatar, Iraq, Iran) receive non-trivial probability mass, SEş under- estimates the associated confabulation risk. In contrast, the proposed uncertainty- penalizes responses with higher uncertainty, redistributing probability mass toward more semantically coherent answers. For instance, Saudi Arabia contextually appropriate response respevesrreeeives higher probability assignment after adjustment. This highlights a promising direction: even when hallucinations are present, localized uncertainty quantification enables prioritization of high-certainty, semantically meaningful answers, a factor not considered in prior work.
adjustment
4.2 DETECTING CONFABULATIONS
Benchmark Datasets. Building on the worked example above, we now present the results obtained from TriviaQA, NQ. SVAMP, and SQuAD datasets, across multiple LLM models such as Mistral 7B, LLaMA-2 7B, LLaMA-3.2 IB, Falcon-rw-1B and LLaMA-2 13B as well as uncertainty estimation
methods; detailed validation results appear in Appendix
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(a) AUROC-hased win rate.
(b) AURAC-based win rate.
Figure 2: Pairwise win rate matrices across SOTA hallucination detection methods on diverse datasets and LLM models. This summarizes 116 experimental scenarios. Each cell indicates the probability that the row model outperforms the column model. Semantic Rényi entropy with UQ maximization consistently outperforms baselines, even surpassing methods reliant on supervised learning.
Fig. 2(a) reports the SRE-UQ's performance against SOTA methods evaluated via AUROC perfor- mance. Higher AUROC scores indicate better separability between factually correct and hallucinated outputs. Across all evaluated models, SRE-UQ is competitive with SOTA baseline methods. Notably, SRE-UQ achieves better AUROC despite not relying ground truth labels for training. This is evidence of its effectiveness in hallucination detection with no reliance on supervised fine-tuning.
While AUROC reflects model's global ranking ability,, practical deployments often involve thresh- olding on confidence scores to reject outputs associated with higher uncertainty. To assess this, we compute the RAC over rejection thresholds ranging from 80% to 100% confidence. Fig. 2(b) presents the SRE-UQ's performance against SOTA methods evaluated via AURAC performance. Consis- tently, SRE-UQ maintains strong AURAC as more uncertain outputs are filtered out. Particularly it offers superior robustness compared to discrete semantic entropy, naive entropy, and supervised p(True) baseline. This is evidence of its effectiveness pruning predictions that are associated with higher uncertainty. Appendix provides in-depth results at the dataset level for Trivia- -AAa SQuAD, NQ-Open, and SVAMP.
Quantized LLMs While most prior work hallucination detection benchmarks full-precision models, real-world deployments almost always rely on quantization (e.g., 16-bit, 8-bit, or 4-bit) to reduce inference latency and memory consumption. However, quantization does merely compress model weights - it also perturbs probability distributions and modifies token-level uncertainty. These shifts can affect semantic entropy measurements and alter the relative competitiveness of different detection methods. Thus, rigorous study of hallucination detection must examine the stability of detection methods across quantization levels, ensuring that performance gains are not an artifact of precision settings but persist in practical deployment regimes. Figure presents pairwise win rate matrices of hallucination detection methods under different quantization settings bii, 8-bit, and 4-bit), evaluated using both AUROC and AURAC metrics. The results show that SRE-UQ maximization remains robust across quantization levels, consistently outperforming or matching SOTA basclines even under aggressive compression.
The quantization-Ve wise analysis reveals that SRE -UQ maximization consistently outperforms alterna- tive detection strategies across 16-bit, 8-bit, and 4-bit models. While quantization slightly reduces absolute AUROC values due to coarser uncertainty calibration, the relative ordering of methods remains largely stable, demonstrating that the proposed approach is resilient reduced precision. This robustness is critical for deployment at scale, where low-bit quantization is increasingly adopted to balance efficiency with reliability in sefety-sensitive applications.
4.3 ENTROPY UNCERTAINTY ANALYSIS
To further assess the effect of uncertainty- adjustments proposed in equation 7 analyze how SRE changes across different levels of generation diversity. Here, we measure the number of distinct semantic clusters produced by the model when answering the same question multiple times
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Figure 3: Evaluation of SOTA hallucination detection methods across different quantization levels. This summarizes 116 experimental scenarios. Each subfigure shows pairwise win rate matrices under different quantization precision (16-biz bit, 8-bit, 4-bit). SRE-UQ maximization consistently outperforms baselines across both AUROC and AURAC, demonstrating robustness hallucination detection against reduced precision.
(10 times). As shown in Fig. the x-axis represents the observed entropy for given question. A value of on the x-axis indicates that all sampled generations were semantically identical hence entropy is zero. This analysis directly probes how model uncertainty and semantic variability interact under the proposed adjustment mechanism.
While much of the prior work in hallucination detection relies on applying a global threshold to entropy-based scores, such approaches ignore how model behavior varies across different ranges of historical entropy. To address this, we evaluate how the change in entropy (normalized difference) distributes across old entropy bins for each LLM. This perspective provides a more fine-grained decision-making criterion: instead of arbitrarily choosing fixed threshold, one can calibrate decisions depending on the region of the entropy spectrum.
The results demonstrate that entropy change is not uniform across the input space: low-entropy regimes (close to zero) remain relatively stable, while intermediate ranges (particularly 0.25-0.75) highly volatile. This implies that decision rules based solely absolute entropy thresholds are brittle and may fail to capture these nuanced behaviors. Instead, practitioners should treat certain entropy regions with caution and, some cases, defer to auxiliary signals before committing to prediction.
CONCLUSION
This work presents a novel framework for hallucination detection in LLMSs which leverages semantic Rényi entropy and quantum TN-based UQ method that explicitly models and accounts for aleatoric uncertainty of sequence probabilities. Through extensive evaluations on several datasets, diverse LLM architectures, and various quantization precision (16-bit, 8-bit, and 4-bit), we demonstrate that semantic entropy measures, especially when combined with principled uncertainty penalization, significantly improve the detection of hallucinated outputs. By adjusting output probabilities to maximize entropy while penalizing deviations weighted by uncertainty, the proposed method yields consistent gains in AUROC, AURAC and RAC across 116 experiments, outperforming strong

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Figure Signed normalized difference (ND) in entropy across old entropy bins for LLaMA-2 variants under different quantization settings (16-bit, 8-bit, 4-bit). Bars represent the mean signed ND with error bars indicating standard error. Notably, all models exhibit the largest variability in the 12500.50 entropy range, suggesting this is a high-risk region where models frequently oscillate between multiple confident yet semantically divergent answers.
baselines, without requiring supervised fine-tuning, Additionally, this provides principled approach to select answers from an L.M even when it is hallucinating.
Incorporating uncertainty not only sharpens the sensitivity of entropy-based metrics but also enables principled answer selection, even under confabulation. By adjusting TS probabilities to balance entropy maximization with uncertainty-aware penalization, our method promotes more coherent and trustworthy outputs. Furthermore, the uncertainty quantification method operates utilizing QTNs and perturbation theory, making it suitable for scalable, explainable real-worlh deployments.
Overall, this study highlights the importance of uncertainty-avee aware semantic evaluation in achiev- ing reliable language generation, and opens up new avenues for integrating ceermation-thhoreee principles with uncertainty quantification to advance the robustness of LLMs.
Limitations While the proposed method combining quadratic semantic Rényi entropy and quantum TN-based UQ shows great promise in detecting hallucinations, few limitations remain. Most notably, our ovaluations are conducted using relatively small language models due to computational resource constraints. While these models enable efficient experimentation, they may not fully capture the behavior larger frontier models such as GPT-4 or Claude. Consequently, both the absolute detection accuracy and the complexity of hallucinations observed could differ larger scales. Furthermore, our semantic clustering relies on an external entailment predictor (DeBERTa-large fine-tuned on MNLI), and any inaccuracies in this entailment model may propagate into the entropy estimates. Finally, the proposed approach requires access to token-level probabilities, which restricts its applicability to geen-weight models; black-box LLMs, where such information is not exposed, remain outside the scope of this method.
The uncertainty of TS probabilities in equation 6 may very well serve to arrive at meaningful definition of an uncertainty associated with each semantic cluster. For instance, for the cluster Cj. one could employ UQ (C, y) = (1/IC,) (esIC,,)e,eee Ee UQ (n!")). This may facilitate one to identify semantic clusters whose membership is more sensitive to changes in TS probabilities which in turn may lead practical guidelines for identifying situations where automated outputs should be supplemented with careful, perhaps human-assisted, review. We hope to undertake this direction of research in the future.
ACKNOWLEDGMENTS
This work is based on research supported by National Science Foundation (NSF) award #2530256.
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A EXPANDED VERSION OF SECTION 2
A.1 SEMANTIC EQUIVALENCE- CLUSTERING
Computation of semantic entropy requires clustering LLM-generated output TSs, such that the TSs in each cluster are semantically equivalent, while those in different clusters are semantically dissonant. To identify such semantically equivalent TSs, Farquhar et al. (002)) employ bidirectional entailment, notion that has attracted considerable attention in natural language inference (NLI). We, use pretrained models such as DeBERTa- LaBBRRT-Large-MNLJJ or GPT-3.5 to assess entailment between sentence pairs. When confronted with new TS, we check for bidirectional entailment with representative TS from each cluster. If no entailment is found with any of these, new cluster created. This process of semantic clustering allows semantic entropy to be computed for the clusters associated with each question. Higher entropy indicates higher likelihood that the LLM hallucinating when responding to that question. The top rung of Fig. summarizes this process in Farquhar et al. (2024).
language models like
However, semantic entropy itself not a reliable measure for identifying hallucinations because TS probabilities can be highly sensitive to model perturbations. This warrants adjusting the confidence with which hallucinatory behavior is deemed to be present or absent. We therefore need local information (as opposed to global metrics) that allows us to assess sensitivity of the TS probability of each LLM generation.
A.2 QUANTUM TN-BASED UQ METHOD FOR DEEP LEARNING MODELS
Physics-inspired methods offer such local uncertainty information and have emerged as a compelling way for UQ of deep learning models (Singh & Principe (2020) 2021): (Sighh al. (2024)). However, these methods are tethered to the Hamiltonian of the quantum harmonic oscillator (QHO) and are hence severely constrained in their application. In contrast, the recent work in Vipulananthan et al. (2024) introduces a deterministic, interpretable, and single-shot UQ method built on a QTN-based Hamiltonian. Moreover, it leverages the estabblitsd method of perturbation theory ferret out multi-resolution, local uncertainty estimates that are ideally suited for our current purpose.
In our work, we expand this QTN-based framework-orrginalll developed for time-series signals- to the domain of semantic datasets, where TS probability distributions can be naturally viewed as signal-like processes. This extension allows us to capture semantic uncertainty in natural language through the same principled, physics-inspired lens. The importance of this contribution is twofold: first, bridges the gap between temporal signal analysis and semantic modeling, demonstrating that techniques rooted in quantum physics generalize beyond time-series complex language domains. Second, by bringing interpretability and multi-resolution uncertainty estimates into the study of semantic entropy, our approach lays the groundwork for more trustworthy and explainable Al systems, especially in high-stakes applications where hallucination detection and careful decision- making are critical. Fig. 5 Jshows the QTNN-bases UQ method that we leverage in our work.
A.3 RATIONALE FOR ADOPTING SEMANTIC RÉNYI ENTROPY
While SE is widely used for characterizing uncertainty in LLM outputs. its inherent properties introduce limitations in the present setting. For any fixed predictive distribution, SE is always greater than or equal quadratic SRE (Harremöes (009)), ), but this pointwise ordering does not in general determine the relative spread of the two measures across corpus of examples. Empirically, however, SE exhibits a larger variation in most cases (between hallucinatory and non-hallucinatory behavior) in our experiments, which correlates with slightly better AUROC performance of SE (see Fig. 2(a) (Hand & Till (200T)) SE also emphasizes rarer outcomes. But the tail probabilities do not play a role in AURAC (see Fig. ((b) which measures how well increasing entropy corresponds to decreasing correctness.
A more decisive consideration arises from the structural compatibility between SRE and the proposed UQ framework. Quadratic Rényi entropy is a second-ordes functional of the underlying probability distribution and is directly estimated by the KME. This correspondence enables the KME to be encoded as an first-order and higher-order spectral corrections that quantify local sensitivity in the amplitude domain.
of a QTN Hamiltonian. Perturbations of this Hamiltonian, in turn, yield
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Figure 5: Overview of the QTN-based UQ pipeline used in work. The procedure begins with computing the quantum correlation matrix (QCM) from TS probabilities and extracting its null- space vectors. These are used to construct the local Hamiltonian H. whose eigen-modes provide the foundation for spectral and perturbation analysis. First-order perturbation corrections yield uncertainty feature vectors, which are organized into a tensor network representation. Finally, sampling across modes produces uncertainty bounds on probability amplitudes, enabling uncertainty quantification of semantic entropy.
Such perturbation- baaeed based formulation of uncertainty cannot be obtained from SE because it lacks a corresponding RKHS-base representation.
Although SE achieves marginally better separability in AUROC due its wider dynamic range, the empirical performance gap relative to SRE is small. In contrast, the methodological advan- tages facilitated by SRE-namely, its direct alignment with RKHS theory, its compatibility with QTN-based spectral analysis, and its ability to support principled perturbation-based uncertainty quantification are substantially more consequential. For these reasons, SRE constitutes the most appropriate entropy measure for the proposed framework.
A.4 RÉNYI ENTROPY AND KME OF DATA PDF
To provide the essentials of the work in Vipulananthan et al. (2024). take the Rényí (quadratic) entropy H (X)
log (x)X) of the continuous random variable X with PDF p(x) Principe
(2010). Here, (X) = p² (x) dr = E [p(z)) where E[-] is the expectation operator. With N
samples {xr) from X, the non non-parmmetric Parzen Gaussian density estimator of the PDF p (z) is
p(x)= ΣE im1 Ko (z; m/,/N, where Kg (x; xr,) (1/ 22/v2 exp (-(x / Parzen 1962)..
This turns out to be the (empirical) KME 76 of the data PDF in the RKHS determined by (.:
(Aronszajn 1950); Schölkopf et al. 2015 ). i.e., p(x)) = ((x). H(X) = log (v(()) being an entropy measure, the scalar (x( is taken viewed as how it varies with x, the data amplitude values.
measure of uncertainty and the (empirical) KME v(x))
To get more interpretable tool for UQ Vipulananthan et al. 2024) seeks finite QTN having local nearest neighbor random couplings whose Hamiltonian has this KME (xx)) as of its eigen- modes. Suffice it to say that (a) a spin chain having L spin particles can represent a sampled version ( where M = 24, of the KME (x)); (b) the Hamiltonian of the TN takes the form of
linear combination of a finite set Ā Dr of pairwise orthonormal (w.r.t. the Hilbert-Schmin inner product) Hermitian operators, where the extent of local nearest neighbor influence determines
T: and (c) T. in turn, determines whether the set H 'rich' enough for Hamiltonian of the type A EI= wr H (i.e., A E span (F)) to have ie as an eigen-mode.
To identify such a Hamiltonian, the quantum correlation matrix (QCM) M3 (7) Qi Ranard (011
because ② is an mogen-mode of H iff belongs in the null space of the QCM. Here, = (wx)/e
the linear combination weight vector that generates H Σi'D wA H. Additional theoretical results and supporting lemmas related to the QCM formulation provided in Appendix A.5
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With this Hamiltonian in hand, the well-established method of perturbation theory is applied to the associated time-indepee -ineenddeenndnt Schrödinger equation. This yields first- and higher order "corrections' to all the eigen-mn meeenmoods///eegeis (including the KMM)) when the underlying Hamiltonian undergoes perturbation -anhnn-ia et al. (1777). The higher order modes provide better discriminative resolution of those data regions for which less information is available (these correspond to the tails of the PDF) (Singh & Principe 2021)); sensitivity analysis and UQ can be carried out via the corrections to the eigen-modes. This the UQ method in Vipulananthan al. (2024) that allows
one to survey aleatoric uncertainties locally and at different levels of resolution using algebraic and spectral properties of QTNs.
A.5 SOME PROPERTIES OF THE QCM
In this section, present series of results related to the QCM that form the theoretical backbone of our work, discussed in Section 2.1
Note that F2 operators. For convenience, we will use (H),. to denote a (real-valued) linear combination of the operators in H. i.e.,
H denotes a finite set (H., = o,..., . 1} of pairwise orthonormal Hermitian
(H) E
Σ i=0
will.. where = (wi)T ERT.
(8)
Lemma The QCM (H) (M.(H)) associated with H. is real and symmetric.
Proof. From Definition, we have
(M (M)(H)) 0.5 (1 Hz)) -(H)) (Hi)
= =0.5 H (H.) (H (M, (M))))
because (H, (H H.) and both H ) and (H are sealars. So, M symmetric.
In addition,
M, (H)); = 0.5 (( H,
H
=.5 (El 111 1 E) (u 1. 2) (2 E)*
0.5 (v H, -0)* +.. H u) (0 H. -0 H, u0)
But we note that
( 11 1, w) (H H11)* (,,r 11,2) (v) H, ui
(2 H1 l)) [ Hey -H. m) = (2 H. w)
(e H H e) and (u H, e)* (e H g). When substituted into the expression above for (MHH)) we see that (M (M,(0)) = (P)))) So, M is real.
In similar manner,
also show that (v
Lemma The variance Var Mg A - ( of the Hermitian operator H. w.r.t. an arbitrary normalized vector sb
4 1p 1, satisfies
Var Hg =
(11) 1 0:
the equality holds iff vb is an eigen-mode of H.
Proof. For arbitrary C and th. we know that
0 < (H - aab, H. -aaaa v))
Furthermore, the equality holds true iff Å tb a 0, i.e., iff di is eigen-mode of Ā.
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So, let us consider the case when 1/ is not eigen-mode of H so that
0 < (H a/ H a H db H + (H v ar))
(ar
H w))
(argb a
= (H H a H vp 40) a² (xb H vt) + |a|? (1p xb)
Since a arbitrary, select
= abb
2b
a = (
Substitute into the above inequality:
< +
16)
((0b
A 1/)
((bb
H 1b + Keb 2
(1 (1
This establishes the claim.
The following result follows immediately:
Corollary Given the Hermitian operator H and the normalized vector vb ( | v
Var H Ø iff is eigen-mode of H.
Corollary Suppose, for some arbitrary real-valued vector w E {w;} E RT, H = (2) Then the following true:
(i) For arbitrary E RT, (H) is p.s.d.
(ii) w .NV (H)))), the null space of the QCM (H) associated with H. w.rt iff g, (e
n) = 1, is an eigen-mode of H.
Proof.
(i) This follows directly from Lemma 2 and Qi & Ranard (2019).
iii First, suppose w E M (M.(H) so that
My(H) w = (M, (H) = w M H w
=.
Then, Var (i) = (H()) = 0 Qi & Ranard 2019
Corollary then implies that 2
is an eigen-mode of H.
Conversely, suppose (M, (H)) = wT (T,,(()) w = 0. We know from Lemma that M, (H) is real and symmetric. Let its SVD be
M UNUT.
where U E RTxT is unitary and E RTXT is diagonal with its diagonal entries being the non-negative singular values of M (H). Let - =-'200" Then,
w ( -= of
= (H) w. M (H) 0, M, (H) (H)
meaning that we must have
M, (H) m - M,1 0,
i.e., w. E (M,(HH)
A.6 ON THE NULL SPACE OF THE QCM: THE ZERO DIMENSION CASE
We now study a special case where the QCM has full rank (zero-dimensional null space) and investigate how small perturbations can introduce one-dimensional null space. Understanding this
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transition is important because the emergence of nontrivial null space corresponds to identifying new uncertainty directions, which are critical for robust decision decision-making in our framework.
Consider the case in the QCM, when the dimension of the null space of the QCM M ya (H.) is zero
and we 'pertur'' M Ma (H) to M of (H') as
M 4/4 (H²) M g/4 (7L) 6M
(9)
s.t. dim (M (H))) =
Henceforth in this section, for convenience, we will denote Mys (H) by M and Mys (H') by M'.
Recall that (e n E 0....... T 1, denote the eigen-pairs of M ordered as 0 < Pe S 1 S
...S 1T-1-
We next investigate the stability of the QCM under small perturbations. Specifically, we examine how the eigen-modes/energies of the QCM change when small perturbation is introduced, particularly in the context where the original QCM has full rank (no null space) and perturbations induce a null direction. We use classical matrix perturbation theory derive first-order approximations and cosine similarity results.
Let (wen Hhi n E T 1, denote the eigen-pairs of M' ordered as 0 0= p4 μ
MT-1-1 Let us now view the eigen-pairs {w,,, A, of M as the 'perturbed' versions of the eigen-pairs {w's, X of M', so that we may apply perturbation theory to M'. For this purpose, as is customary in perturbation theory, let
M M' t 6M,
(10)
where e > is simply a place-holder. The eigen-pairs of M can then be expressed as
{un, )n
(k)
Ee
(11)
(0) where (wh, ) - (wh, Xa), An 2 O. W.L.o.g. we also assume that (wn
(0)
Wn
(k)
0, Ak 2
1, vn 2 0.
Now we can express the eigen-pair relationship
M w,) = An w.n
(12)
as
(M' c 5M)
k (k)) = Σ
(()))
(8)
k
w (k)
(13)
k=0
t=
k=0
or equivalently,
DO
x6
M'
wilk)
(k)
5M
(k)
A/
(k-t)
(
(14)
K=
k=1
k=
8=
Gather similar powers of e: for k > 1,
1-1
(M' 10)) 10
(k))
(k (k=
m
(2)
5.M
1)).
(15)
m
Note that, with k- O, we get (M' (
((0)) which is of course trivially true.
Push the bra (w)) |: for 2 1,
k-1 x
(1 (w)
w (Ki)
(k- (m)
( (w 88M (R-))).
(16)
n
00
Put m in equation 16 get the order-k correction for (
(w. (0)
8M
w
(k-1))
1)}, k 2 1.
(17)
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For n. put & 1 in equation 16 to get
(wm (0)
6M w
(wh)
(
w
(18)
X
where we assume that M0))) is non-degenerate. Since (0) correction for ten
{wh (0)
forms an ONB, we can get the order-1
Wmm!
(mom
10 (0)
(0)
8M
Win
(0)
(19)
An
Am
m==
m= mfffn
Notice that, since (urn (0)
(2)) O, we have
(1)) 112 (0)
(1)) +
(1)
((l)
1
1
(20)
because is already normalized. Therefore, denoting by (0) the normalized order-1 corrected w
we have
(0)
Wn
W
(21)
(1 |w
12)
where
|| = (w!
whll)
-Σ
(wim
6M
W (
8MM
Wn
(0)
(0)
=
Am
X
Esen molin
Σ
8M
(22)
Xm
(0)
myfn
Thus,
(0)
(0)
(0)
(0)
(
(w
Wn
(wn
+
wm (1)
112)1/2
21
1 11,
(23)
(1
+ 1|w
11)
12)112
2
where we used the Taylor series expansion
1
3
(1 + x///2 =1
+
-5/2
5/2 22 x 0.
(24)
21
for some E 20, which in turn implies that
1
(1 (1+x)
2
1
2
x.
(25)
because (1 + c)/2 0, c C x0,
In summary, we have
(w
(0)
W
(0)
2 2
Σ
(wo
(0)
8M
Wnn
(0): 2
(26)
X
We know that (w we have
(0)
(
= (wh.
A = 0) is the 'smallest" eigen-pair of M'. So, putting n = 0.
(0)
0
> 1 Σ
(wi
(0)
8M
Wi
(0)
(27)
X
(0)
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To conclude, this section shows that introducing a small perturbation to full-rank QCM systemati- cally creates a well-defined null space, corresponding to am emergent low-variance direction in the QTN. The perturbation analysis. grounded in classical matrix perturbation theory, quantifies how close the perturbed eigen-mode remains to the original eigen-structure through a first-order approxi- mation. This understanding critical in practice, it ensures that small modeling errors noise do not destabilize the uncertainty quantification pipeline and that the emergence of low-uncertainty directions remains theoretically sound and controllable.
B PROMPT TEMPLATES
Phrase length Model Prompting and Answer Selection to generate LLM responses is
The prompt template used for all datasets
Answer the following question as briefly as possible: [question] Answer:
Sentence length Model Prompting and Answer Selection The prompt template used for all datasets generate LLM responses
Answer the following question in single brief but complete sentence: {question) Answer:
Entailment Estimation/Semantie Clustering To detect entailment utilized the following prompt template:
We are evaluating answers the question: {question]
Here are two possible answers:
Possible Answer 1: [textl}
Possible Answer 2: [text2]
Does Possible Answer 1 semantically entail Possible Answer 2?
Respond with: entailment, contradiction, or neutral.
C EXTENDED EVALUATION OF HALLUCINATION DETECTION ACROSS DIVERSE SETTINGS
Due to page limitations, reported the results summary in the main paper. In this section, present in depth experimental results to validate that our confabulation detection framework works across diverse datasets. Building the example analysis in Section 4 extend the evaluation to TriviaQA, NQ, SVAMP, and SQuAD across multiple quantization levels, thereby encompassing a broader spectrum of reasoning complexities and linguistic variations. In addition, break down results for both phrase-level and sentence-level generations, via prompts discussed in Section B) examine the impact of output length on detection robustness. For each dataset, we consistently apply the eight LLM models Mistral-7B-vO.. Mistral-7B 7iiitaa---B-sttuett-- -instruct-vO.3 Falcon-rw-I1 1b, LLaMA-3.2. 1b. LLaMA-2- --1 13b-chat, LLaMA-2-7bh 7b-chat,, LLaMA-2- --0M-----131 1-b,,, and LLaMA- 2-bb) and maintain the original entailment- This controlled experimental design ensures that any observed performance trends can be attributed to dataset characteristics rather than methodological changes. The results demonstrate that our framework generalizes effectively across heterogeneous tasks, reinforcing its applicability to varied real-world deployment scenarios.
clustering and uncertainty quantification pipelines without modification.
C. AUROC PERFORMANCE ACROSS VARIOUS PRECISION MODELS AND DATASETS
To more rigorously validate the robustness of our confabulation detection framework, we extend the evaluation to sentence-level and phrase-level generations under different quantization precision (16-bit, 8-bit, and 4-bit). This setup allows us to examine how model compression impacts uncertainty estimation and confabulation detection performance, when sentence length answers as well as short answers generated. We consistently apply the set of LLMs and datasets in the main paper, thereby ensuring that observed variations are attributable solely to quantization effects rather than changes in methodology. Importantly, quantization is necessary step for deploying large models
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in conouure--onntrainee environments, yet hallucination behavior under quantized models has been largely overlooked in prior work. By explicitly studying confabulation detection in this setting and validating our approach against the SOTA methods discussed in Section we address critical gap in the literature and highlight the practical importance of uncertainty methods for L.Mss.
C.1.1 SENTENCE LENGTH OUTPUT PERFORMANCE.
Figs. 6 and illustrate the AUROC performance 16-bit, 8-bit, and 4-bit precision, respectively, for six different hallucination quantification methods across various LLMs and four datasets, using sentence-lengle outputs. In total, we conducted 52 experiments spanning all quantization levels, providing a comprehensive basis for sentence length analysis. The experimental results reveal several notable trends and insights:
1. Methods based on semantic information-sude SRE with UQ, SE, and OSSE-consisstette outperform baselines based on raw token sequence probabilities (p( (TTrue)) and embedding regression across all datasets. These results confirm effectiveness of semantic diversity rather than raw output probabilities in detecting confabulations.
2. Across all quantization levels, our framework maintains competitive AUROC performance relative to baseline methods. Importantly, we observe that reductions in model precision (e.g., moving from 16-bit to 4-bit quantization) do not significantly degrade the reliability of our uncertainty-basee detection.
3. When model precision is reduced, SOTA methods such as SE, NE have performance degradation, especially in TriviaQA and SQuAD datasets. This highlights the resilience our approach to model compression, making it suitable for deployment in resource- constrained environments where lightweight LLMs are required.
4. Smaller models such as LLaMA 3.2-1B show slightly reduced AUROC compared to larger models like LLaMa- 13B, revealing that model size and internal representational power significant factors in uncertainty calibration.
5. We also observe a minor performance drop for the NQ dataset when using the I.LaMA 3.1B model, which warrants further investigation in future work.
C.1.2 PHRASE LENGTH OUTPUT PERFORMANCE ON INSTRUCT MODELS.
In this section, we extend the analysis to phrase-level generations, focusing on instruction tuned LLMs, such under multiple quantization precisions (16-bit, 8-bit, and 4-bit). Phrase-level evaluation cap- tures shorter, context-sensitive completions, which are partieularly relevant for instruction-folloviii scenarios where answers often concise. By applying the same uncertainty quantification pipeline across datasets and models, we ensure methodological consistency, allowing observed performance variations be attributed to quantization and model adaptation effects. While quantization has become standard for enabling deployment of instruction- influence hallucination behavior remains unexplored. Our experiments directly addresses this gap by examining confabulation detection under compressed, instraction-followin settings.
LLMs in practical applications, its
Figs. 10 and 11 present the AUROC scores for six hallucination quantification approaches across diverse datasets and instruction-tuned I.Mss, under 16-bit, 8-bit, and 4-bit precision. Across all quantization levels, we performed total of 24 experiments, providing robust foundation for phrase-level evaluation on instruction fine tuned models. The results reveal several consistent patterns and distinctive insights:
1. Semantic-based criteria such as SRE with UQ maximization, SE, and ASE--cnnsstennle outperform probability- (p( (True) and embedding- based baselines, indicating that semantic variability is a reliable uncertainty signal even for shorter responses.
2. Across all quantization levels, our framework maintains competitive AUROC performance relative to baseline methods. Importantly, we observe that reductions in model precision (e.g., moving from 16-bit to 4-bit quantization) do not significantly degrade the reliability of our uncertainty- based detection
3. By contrast, prior SOTA baselines such SE and NE show larger declines under reduced precision, particularly in more challenging datasets (e.g., NQ, SQuAD), underscoring the sensitivity of token-level entropy measures compression.
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UO entropy maxim zation
Semantic Entropy
piTrue)
Semantic Entrepy
Neive Entropy
Embedding Regression
AUROC 16-bit Quantization: Sentence length
trivia
Squad
svamp
6.
0.7
0.-
077
03
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00
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us-
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00
0.0
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05-
64
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DV4
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02
0.2
Figure 6: summarizing 12 experimental scenarios, AUROC scores for confabulation detection across three LLMs (LLaMA 2 13B. LLaMA 2 7B, LLaMA 3.2 1B) 16 bit precision and four datasets (TriviaQA, SQuAD, NQ, SVAMP) for sentence length output. The performance of the proposed SRE with UQ is in par higher than SOTA methods.
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SRE with UQ entropy maxim zation Semantic Entropy
Discrete Semantic Entropy Entropy
pfTrue) Embedding Regression
AUROC 8-bit Quantization: Sentence length
trivie_0t _0
Squad
ng
svamp
0.-
066
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04
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Metral-20-00
84
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6.
0.0
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0.0
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0.6
0.7
n6-
544
04-
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03
01
03
03
Figure 7: summarizing 20 experimental scenarios, AUROC scores for confabulation detection across five LLMs (LLaMA 2 13B, LLaMA 2 7B, LLaMA 3.2 IB, Mistral-7B, (iistral-BB-cht)) at bit precision and four datasets (TriviaQA, SQuAD, NQ. SVAMP) for sentence length output. The performance of the proposed SRE with UQ is in par even higher than SOTA methods.
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SRE with UQ entropy maxim zation Semantic Entropy
Discrete Semantic Entropy Entropy
pfTrue) Embedding Regression
AUROC 4-bit Quantization: Sentence length
trivia_ga 1a
Squad
ng
svamp
0.-
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0.1
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06-
2.78
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031
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Mostral-DDii Dndre--Biinsttuuteee
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Metral-20-00
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0.
0.0
0.0
0.0
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0u6
05
044
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01
03
031
Figure 8: summarizing 20 experimental scenarios, AUROC scores for confabulation detection across five LLMs (LLaMA 2 13B, LLaMA 2 7B, LLaMA 3.2 IB, Mistral-7B, (iistral-BB-cht)) at bit precision and four datasets (TriviaQA, SQuAD, NQ. SVAMP) for sentence length output. The performance of the proposed SRE with UQ is in par even higher than SOTA methods.
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4. Instruction-ture tuned models provide better calibrated AUROC overall compared to their base versions, though a consistent gap remains between the smaller LLaMA-2-7B-- and the larger LLaMA-2-13BB 2-1B--hhatBBh highlighting the role of model scale in uncertainty reliability.
SRE entropy Sementic Entropy
Semantic Entropy
piTrue) Embediing Regression
Naive Entropy
AUROC 16-bit Quantization: Phrase length
Squad
svamp
0.7
1.0chht
066
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0.0
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0.2
0.2
08
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0.0
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04
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00
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00
Figure 9: summarizing 8 experimental scenarios, AUROC scores for confabulation detection across two LLMs (LLaMA 2 7B chat, LLaMA 2 13B chat) 16 bit precision and four datasets (TriviaQA, SQuAD, NQ. SVAMP). The performance of the proposed SRE with UQ is in par or even higher than SOTA methods.
SRE
entropy Semantic Entropy
Semantic Entropy
p(True)
Naive Entropy
Embedding Regression
8-bit Quantization: Phrase length Squad
svamp
0.8
AUROC Mb-citt
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6.6
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us-
Unie
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60
0.0
00
50
Figure 10: summarizing 8 experimental scenarios, AUROC scores for confabulation detection across four LLMs (Mistral-7B. Falcon-IB: LLaMA 3.2B, LLaMA 7B 4-bit) and four datasets (TriviaQA, SQuAD, NQ. SVAMP), The performance of the proposed semantic Rényi entropy is in par or even higher than SOTA methods.
C.1.3 PHRASE LENGTH OUTPUT PERFORMANCE ON NON INSTRUCT MODELS.
In this section, we analyze phrase-level generations from non con-instruction-tumee LLMs under varying quantization levels (16-bit, 8-bit, and 4-bit). Phrase-level completions in base models especially
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SRE with UQ entropy maxim ization
Discrete Semantic Entropy
pfTrue) Embedding Regression
Semantic Entropy
Entropy
AUROC 4-bit Quantization: Phrase length
trivie
squed
Svammp
077
08
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03
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0.4-
Unme
04-
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03
Figure 11: summarizing 8 experimental scenarios, AUROC scores for confabulation detection across four LLMs (Mistral-7B, Falcon-ID LLaMA 3.2B, LLaMA 2 7B 4-bit) and four datasets (TriviaQA, SQuAD, NQ, SVAMP). The performance of the proposed semantic Rényi entropy is in or even higher than SOTA methods.
prone to drifting and incoherent continuations because, unlike instruction -atnnettiontuneddd models, they lack explicit task alignment. This makes them a critical yet underexplored regime for studying hallucina- tion citaavoor-patteullari models in constrained environments. Prior SOTA methods have largely overlooked this intersection, focusing either on sentence-level or instruction-follooinn setups, leaving open questions about un- certainty reliability in compressed base models. Our study directly addresses this gap by evaluating uncertainty-aware hallucination detection in non-instruction-tunees phrase-level settings.
when combined with quantization, which essential for deploying large
Figs. 12 13 and 14 summarize AUROC performance across six uncertainty quantification methods, four datasets, and multiple base LLMs under 16-bit, 8-bit, and 4-bit quantization. In total, we conducted 40 experiments, enabling systematic comparison of quantization and uncertainty estimation in this overlooked regime. The results reveal the following consistent findings:
1. Semantic-based methods dominate: Across datasets and precisions, SRE with UQ, SE, and DSE consistently outperform p(True) and embedding regression, confirming that semantic diversity remains strong uncertainty signal even in base models with less task alignment.
2. Across all quantization levels, our framework maintains competitive AUROC performance relative to baseline methods. Importantly, we observe that reductions in model precision (e.g., moving from 16-bit to 4-bit quantization) do not significantly degrade the reliability of uncertainty- detection
3. Token-level entropy basclines (NE) suffer performance drops most evident in NQ and SQuAD highhihtng their fragility under quantization and emphasizing the robustness of semantic-based signals.
4. Larger base models (e.g., LLaMA-2-13B) consistently achieve higher AUROC than smaller ones (e.g., -LLMM--3.2-BB Falcon-RW-IB), This indicates that representational richness is particularly crucial for phrase-level uncertainty calibration in non-instruct settings.
C.2 RAC PERFORMANCE ACROSS VARIOUS PRECISION MODELS AND DATASETS
To more rigorously evaluate the robustness of our confabulation detection framework, we complement the AUROC analysis (Appendix C.Il with RAC and their area-under-curve summary, AURAC, across multiple quantization precisions (16-bit, 8-bit, and 4-bit). While AUROC measures separability under idealized conditions, RAC and AURAC provide stricter evaluation by quantifying how reliably uncertainty estimates improve accuracy under progressive rejection. Given that quantization is now standard for deploying L.LMs in resource-constraineed environments, understanding its impact
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SRE with UQ entropy maximization
Discrete Semantic Entropy
p[True]
Semantic Entropy
Naive Entropy
Embedding Regression
AUROC 16-bit Quantization: Phrase length
trivia ga
squad
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Svamp
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Figure 12: summarizing 16 experimental scenarios, AUROC scores for confabulation detection
across four LLMs (Falcon-1B IB, LLaMA 3.2B, LLaMA 2 7B, LLaMA 2 13B) and four datasets (TriviaQA, SQuAD, NQ, SVAMP). The performance of the proposed SRE with UQ is in par or even higher than SOTA methods.
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SRE with UQ entropy maximization Semannti Entropy
Discrete Semantic Entropy
p[True]
Naive Entropy
Embedding Regression
AUROC 8-bit Quantization: Phrase length
trivia
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Figure 13: summarizing 12 experimental scenarios, AUROC scores for confabulation detection across three LLMs (LLaMA 3.2B, LLaMA 7B, LLaMA 2 13B) and four datasets (TriviaQA, SQuAD, NQ, SVAMP). The performance of the proposed SRE with UQ is in par even higher than SOTA methods.
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SRE with UQ entropy maximization Semannti Entropy
Discrete Semantic Entropy
p[True]
Naive Entropy
Embedding Regression
AUROC 4-bit Quantization: Phrase length
trivia
squad
svamp
3.8
0.7
0.7
0.6
0.4
0.3
0.2
AUROC
0.7
0.7
0.8
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0.5
0.6
0.1
0.7
0.7
0.6
0.4
0.4
0.3
02
0.2
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01
6.1
0.0
0.0
Figure 14: summarizing 12 experimental scenarios, AUROC scores for confabulation detection across three LLMs (LLaMA 3.2B, LLaMA 7B, LLaMA 2 13B) and four datasets (TriviaQA, SQuAD, NQ, SVAMP). The performance of the proposed SRE with UQ is in par even higher than SOTA methods.
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rejection behavior remains critical but largely overlooked dimension. By systematically analyzing RACs and AURAC under quantization, we disentangle the effects of compression and model scale, ensuring methodological consistency via identical clustering- based uncertainty pipelines. In this setting, steeper RAC slopes indicate stronger capacity to prioritize reliable generations while filtering confabulations.
Expanding on the AURAC summary results reported in Section 4.2 our findings establish that the proposed SRE with UQ only surpasses prior methods in robustness but also enables fine-grained and safe filtering of LLM outputs-demmonstraii both higher accuracy and greater deployability than existing SOTA uncertainty estimators.
C.2.1 RAC PERFORMANCE: SENTENCE LENGTH
Figs. 15 16 and present RAC curves 16-bit, 8-bit, and 4-bit precision, respectively, for six hallucination quantification methods across multiple LLMs and four datasets under sentence-length generation. In total, we conducted 52 experiments spanning all quantization levels, providing comprehensive basis for sentence length analysis, drawing parallel with its AUROC counterpart evaluation (Appendix C.I.I. Several notable patterns emerge:
1. The proposed SRE with UQ consistently yields the steepest accuracy gains as rejection increases, demonstrating superior ability to prioritize reliable outputs and suppress confabu- lations. Other semantic-based criteria (SE and Discrete SE) follow closely but remain less effective than SRE with UQ. while probability-buse basclines lag significantly behind.
(Tru)) and embedding regression
2. The advantage of SRE with UQ is particularly pronounced in challenging datasets such as NQ and SQuAD, where its RAC curves show clear separation from both semantic and non-semantic baselines, highlighting robustness under high variability.
3. Moving from 16-bit to 4-bit precision induces only minor changes in the RAC slopes of SRE with UQ whereas competing methods (e.g.. SE and NE) degrade more noticeably. This underscores the resilience of SRE with UQ to quantization.
4. Model size plays a role, with smaller models (e.g.. LLaMA-3.2-IBB. exhibiting flatter RAC curves than larger counterparts (e.g., LLLMM-2-13BB) Nonetheless, SRE with UQ consistently preserves its relative advantage across all scales, confirming its effectiveness even under reduced representational capacity.
5. We also observe a minor performance drop for the NQ dataset when using the LLaMA 3.IB model, consistent with the results in C.I.I which warrants further investigation in future work.
C.2.2 RAC PERFORMANCE: PHRASE LENGTH ON INSTRUCT MODELS
Figs. 18 19 and 20 present RAC curves at 16-bit, 8-bit, and 4-bit precision, respectively, for six hallucination quantification methods across multiple LLMS and four datasets under phrase-length generation on instruct fine tuned models. Phrase-level completions are particularly relevant for instuuctionffolloiing use cases, where responses tend to be short and context-sensitive. While prior SOTA hallucination detection methods have largely overlooked this setting, understanding iijection-acuraae behavior under compressed instruction- models is crucial for safe deployment in real-world applications.
This analysis reveals how quantization interacts with model adaptation to instruction tuning, and whether semantic-based uncertainty measures such SRE with UQ retain their reliability when generations are concise and more sensitive to uncertainty calibration. In total, we conducted experiments spanning all quantization levels, providing a comprehensive basis for phrase length analysis, drawing parallel with its AUROC counterpart evaluation (Appendix C.I.I) Several notable patterns emerge upon analysis:
The proposed SRE with UQ consistently yields the steepest RAC slopes across datasets, demonstrating stronger rejection behavior than both brabe (pl (TTur))) and embed- ding regression baselines.
2. Instruct models reveal that semantic criteria-especiiall especially SRE with UQ-rretaie substantially higher accuracy at rejection rates above 90%, with the advantage most pronounced on challenging datasets such as TriviaQA and NQ.
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RAC 15-bit Quantization: Sentence length
SRE with UQ entropy maximization Semantic Entropy
.ee Discrete Sementic Entrapy
piThue]
Naive Entropy
Embedding Regression
trivia qa
squad
ng
svamp
0.32
Liame-2-133
0.70
0.50
0.30
0.425
8.6
0.400
Accuracy
Liama.2-7b
9.75
0.34
0.525
0.52
0.500
a.70
0.30
Uiams 3.2-13
0.38
0.24
9.36
0.40
0.34
0.20
0.22
0.35
0.32
6.30
0.50
0.95 1.00
6:50
0550
0.95
1.00
0.80
0.50
0.55
1.00
0.00
0,50
0.55
1.00
Rejection rate
Rejection rate
Rejection rate
Rejaction rate
Figure 15: summarizing 12 experimental scenarios, RAC scores for confabulation detection across three LLMs (LLaMA 3.2 IB, LLaMA 7B, LLaMA 13B) 16 bit precision and four datasets (TriviaQA, SQuAD, NQ. SVAMP), The performance of the proposed SRE with UQ is in par or even higher than SOTA methods.
3. Across quantization levels, SRE with UQ remains stable from 16-bit to 4-bit precision, whereas baselines degrade more noticeably, underscoring its resilience to model compres- sion.
4. Larger instruct model (e.g.. LLaMA-2- 1-1B--hhaa amplify the gains of SRE with UQ while smaller model (e.g., ALLMMA-2-7B-chal show flatter RAC curves; nonetheless, SRE with UQ consistently outperforms alternatives across scales.
C.2.3 RAC PERFORMANCE: PHRASE LENGTH ON NON INSTRUCT MODELS
Phrase-level RAC plots for non-instruct LLMs (Figs. 21 22 and 23 provide further evidence that remantic-based hallucination detection, and in particular our proposed SRE with UQ, maintains strong robustness under quantization and across diverse datasets. Unlike noken-probabln based baselines, which exhibit flatter or unstable RAC slopes, SRE-UQ consistently yields steeper drection-accuraee improvements, highlighting its ability reliably prioritize correct outputs even when compression reduces representational capacity. These findings extend the AUROC trends to phrase-level rejection settings, demonstrating that SRE-UQ not only effective in distinguishing confabulations but also excels in dynamically filtering them under realistic deployment constraints. Importantly, the proposed SRE with UQ demonstrates consistently stronger rejection-aceet accuuracy behavior, highlighting its suitability for lightweight non-instruct models where calibration is more challenging. Several notable patterns emerge upon analysis:
and embedding-
1. SRE-UQ leads across precisions, across 16-bit, 8-bit, and 4-bit quantization, SRE-UQ produces the steepest RAC curves, confirming that it most effectively raises accuracy as rejection increases. This advantage especially evident in TriviaQA and NQ, where probability-base baselines stagnate or even flatten.
2. While baselines such as Discrete Semantic Entropy and Naïve Entropy show visible degra- dation under 4-bit quantization, SRE-UQ's RAC slopes remain sharp and consistent, under- scoring resilience to aggressive model compression.
3. Gains are most pronounced in datasets with higher linguistic variability (c.g.. NQ and SVAMP), where SRE-UQ sharply separates from weaker baselines. By contrast, probability- based p(Truee) often collapses under phrase-level rejection, failing provide meaningful filtering.
4. Smaller models (e.g., LLaMA-3.2-1D exhibit generally flatter curves due limited repre- sentational power, but even in these challenging cases, SRE-UQ maintains relative gains, outperforming all other methods consistently.
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RAC 8-bit Quantization: Sentence length
SRE with UO entropy maximization Semantic Entropy
Discrete Semantic Entropy Naive Entropy
(Tuue Embedding Regression
trivia
squad
svamp
0.70
0.50
Llame-2-130
3.65
0.28
0.400
0.45
0.26
0.275
0.54
Liama-2-75
0.75
0.56
0.55
0.34
0.50
0.32
0.48
0.50
Accuracy
Mistral- Nastral-7B-o.11 Mistral-Te 78------n at-innsrucc-v..3
0.75
0.36
0.70
0.475
a.20
0.450
0.425
9.75
0.38
0.475
0.34
0.00
Liama-3.2-18
0.400
0.22
0.20
0.375
0.32
0.18
0.38
0.50
050
Rejection
Rejection
Rejection
Rejection
Figure 16: summarizing 20 experimental scenarios, RAC scores for confabulation detection across five LLMs (LLaMA 2 13B, LLaMA 2 7B, LLaMA 3.2 IB, Mistral-7B, Mistral-7B-chha) 8 bit precision and four datasets (TriviaQA, SQuAD, NQ. SVAMP), The performance of the proposed SRE with UQ is in par even higher than SOTA methods.
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RAC 4-bit Quantization: Sentence length
entropy
Discrete Semantic Entropy MB. Naive Entropy
.ra Semantic Entropy
van Embedding Regression
trivia qa
squad
ng
svamp
Laaoo---13
0.65
0.28
0.375
0.26
0.40
Midgll-g..ugg Mistral ww..ininrucctttO... Liamo-2-75
0.75
0.30
0.46
0.35
0.70
0.44
0.50
0.78
0.42
0.70
3.8s
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750
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728
300
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0.450
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L.mma-3.-1B
0.18
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Rejection
Rejection
Rejection
Rejection
Figure 17: summarizing 20 experimental scenarios, RAC scores for confabulation detection across five LLMs (LLaMA 2 13B, LLaMA 2 7B, LLaMA 3.2 IB, Mistral-7B, Mistral-7B-chatt at 8 bit precision and four datasets (TriviaQA, SQuAD, NQ. SVAMP). The performance of the proposed SRE with UQ is in par even higher than SOTA methods.
RAC 16-bit Quantization: Phrase length
SRE with UQ entropy maximization
Discrete Semantic Entropy
[TTue)
Semantic Entropy
Naive Entropy
Embedding Regression
Liama 2-1b-chat Liama 1lama-2-13b- 2-1.--2hhhh chat
trivia_ga
squad
nq
svamp
0.51
0.75
0.52
8.70
0.30
0.28
0.70
0.28
0.6s
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6.00
cán 0.95 1.00
6.50
640 0.55
1.0
6.0
6.95
1.00
050
0.50
0.95
Rejection rate
Rejection rate
Rejection rate
Rejection rabe
Figure 18: summarizing 8 experimental scenarios, RAC scores for confabulation detection across two LLMs (LLaMA 2 13B-chat, LLaMA 2 7B-chat) at 16 bit precision and four datasets (TriviaQA, SQuAD, NQ, SVAMP). The performance of the proposed SRE UQ is in par even higher than SOTA methods.
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RAC 8-bit Quantization: Phrase length
SRE with UQ entropy maximization
Discrete Semantic Entrapy
A[Thue)
Semantic Entropy
Nalve Entropy
Emibedding Regression
Llsma-2 7b--bhhht Liams 2-13b-chat
trivia qa
squad
svamp
0.550
0.75
0.52
Accuracy
a.20
0.500
0.50
0.50
0.44
3.05
0.42
0.455
0.26
3.00
Rejection rate
Rejection rate
Rejection rate
Rejection rate
Figure 19: summarizing experimental scenarios, RAC scores for confabulation detection across two LLMs (LLaMA 2 13B-chat, LLaMA 7B-chat) at 8 bit precision and four datasets (TriviaQA, SQuAD, NQ, SVAMP). The performance of the proposed SRE UQ is in par or even higher than SOTA methods.
RAC 4-bit Quantization: Phrase length
SRE with UQ entropy maximization Semantic Entropy
Discrete Sementic Entrapy
p[true) Embedding Regression
Entropy
Llama-2-11 chat Largo 133 ccigt
trivia qa
Squad
1..50
0.54
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725
Accuracy
700
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0/455
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Rejaction rate
Rejection rate
Rejection rate
Rejection rate
Figure 20: summarizing experimental scenarios. RAC scores for confabulation detection across two LLMs (LLaMA 2 13B-chat, LLaMA 7B-chat) at 4 bit precision and four datasets (TriviaQA, SQuAD, NQ, SVAMP). The performance of the proposed SRE UQ is in par or even higher than SOTA methods.
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RAC 16-bit Quantization: Phrase length
entropy
Discrete Sementic Entroyy
pf Embeddinng Regression
.... Semantic Entropy
... Neive Entropy
ng
Svamp
tiama 1ama---1330
0.75
0.34
0.475
0.55
0.70
0.32
0.50
0.55
Liama 2-7b
9.75
0.50
Accuracy
3.10
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felcon-ew-15
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Usmm 3.2-18
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Rejection rate
Rejection rate
Rejection rate
Figure 21: RAC scores for confabulation detection across four LLMs (Mistral-7B, Falcon- ILaMA 3.2B, LLaMA 2 7B 4-bit) and four datasets (TriviaQA, SQuAD, NQ. SVAMP), The performance of the proposed semantic Rényi entropy is in par even higher than SOTA methods.
Quantization: Phrase length
SRE with UQ entropy maximization Semantic Entropy
Discrete Semantic Entrapy
van piTuue
Nalve Entropy
Embedding Regression
trivia_gu qa
squad
ng
svamp
Jagga-8 1-38p
775
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350
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10700
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Accuracy
Laama Lama-2-7b
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Lisma-3:--1B
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022
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035
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035
1000
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035
100
Rejection
Rejection rate
Rejection rate
Rejection rate
Figure 22: RAC scores for confabulation detection across four LLMs (Mistral-7B, Falcon- 1B, LLaMA 3.2B. LLaMA 2 7B 4-bit) and four datasets (TriviaQA. SQuAD, NQ. SVAMP). The performance of the proposed semantic Rényi entropy is in par even higher than SOTA methods.
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4-bit Quantization: Phrase length
SRE with UQ entropy maximization Semantic Entropy
Discrete Sementic Entrapy Naiva Entropy
piTuue
Embedding Regression
trivia qa
ng
svamp
Ligma2-130
750
0.36
725
700
0.425
0.52
0.400
Accuracy
0.75
Liama 2.73
0.30
0.44
0.70
0.42
0.26
0.38
Liams-3.7-5
9.26
0.18
0.27
0.26
0.145
0.36
0.340
6.30
0,50
0.95
6:50
0.50
0.95
1.00
0.80
0.50
0.55
0.50
0.50
0.55
1.00
Rejection rate
Rejection
Rejection rate
Rejaction
Figure 23: RAC scores for confabulation detection across four LLMSs (Mistral-7B, Falcon- LLaMA 3.2B, LLaMA 2 7B 4-bit) and four datasets (TriviaQA. SQuAD, NQ. SVAMP), The performance of the proposed semantic Rényi entropy is in par or even higher than SOTA methods.
C.3 HYPERPARAMETER SELECTION
In this section, we examine the effect of hyperparameter tuning for À in the entropy maximization framework introduced in equation Recall that this formulation adjusts the output TS probabilities by jointly maximizing semantic entropy and penalizing deviations from the original model distribution, weighted by the model's uncertainty estimates. A critical innovation our approach lies in integrating UQ directly into the token adjustment mechanism, thereby enhancing the model's sensitivity to confabulations beyond conventional semantic entropy analysis.
To assess the impact of X, we measure AUROC scores for hallucination detection across different each case, the Rényí's semantic entropy is computed after probability
values the hyperparameter. adjustment, and the results are compared against the baseline Rényi's entropy without uncertainty correction (denoted by the red line). The yielding the highest AUROC green. This comparison isolates the contribution of UQ-aware entropy maximization to the confabulation detection task.
As shown in Fig. 24 incorporating UQ- -Q-based probability adjustments yields substantial improvements over the baseline entropy approach across all models tested.
Several key observations are notable:
Moderate values of systematically boost AUROC scores, confirming that calibrated trade-off between entropy maximization and fidelity to the original model predictions is beneficial.
2. While the precise optimal may vary slightly across models and datasets, the performance gains robust across wide range of À values, underscoring the general effectiveness of the proposed correction.
3. This experiment validates one of the most crucial contributions of our work: by intro- ducing UQ into semantic entropy calculations, significantly sharpen the model's ability to discriminate confabulations, especially in the presence of noisy or low-confidence predictions.
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Bacoline SRE
Mittral TB 2B
Nitooeel 16
Ibrra 3.2 15
LLANA 2 TO (E
X190
3791
AIN
XTAA
3799
15
130
1105
A134
XTA4
A7173
2-100
SCU4D
6:90
7109
2377
1000
6.79
AC00
3370
690
MC2
SUGMP
15
Lambaa
150
15
Landuk
Lanboo
Figure 24: Effect of À hyperparameter selection on AUROC performance. The red line denotes the baseline SRE without UQ adjustment. Integrating uncertainty via equation consistently enhances hallucination detection capability across all LLM models and datasets. The À yielding the highest AUROC is in green.
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D ADDITIONAL DETAILS ON ALGORITHMIC STRUCTURES AND COMPUTATIONAL COST
D.1 COMPUTATIONAL OVERHEAD
The computational overhead introduced by the QTN-based UQ module is modest and remains well within real-time inference constraints. The method requires storing only M = 8 q-bit perturbation features in a 2 X 2 matrix, which is significantly smaller than the embedding caches required by clustering- brased semantic entropy methods. Constructing the Hamiltonian associated with the sampled KME incurs one-time pre-processing cost of approximately 45-60 on CPU, though this can be substantially reduced through GPU-accelerated linear algebra routines. At inference time, the per-query overhead is minimal: evaluating the perturbation-basse uncertainty features adds merely 6-10 ms on an NVIDIA A6000 GPU, which negligible compared to the intrinsic latency LLM decoding. Overall, the QTN perturbation step introduces only a lightweight computational cost while enabling richer uncertainty estimates.
D.2 ALGORITIMIC STRUCTURE AND COMPLEXITY
In this subsection, present pseudocode illustrating key components of our approach, complement- ing the pipeline diagram provided in Fig. |
This includes pseudocode for (i) Hamiltonian construction from the sampled KME, (ii) extraction of first-order perturbation features, and (iii) the antropy-maximizzn procedure from Sec. 2.3.
Algorithm 1 Construction of Empirical KME (x)
1: Input: Samples {x, N from random variable X; kernel bandwidth o; sampling grid {2y)/=0"
2: Output: Sampled KME vector &B ( 3-
M-1
3: for j 0 to M - do
4: Compute Parzen PDF estimate:
N No (z)
p(x,))
= N
where Hig ( is the Gaussian kernel
5: Set KME value bi p(x,)
6: end for
7: return +b
Algorithm 2 QTN Hamiltonian Construction via Quantum Correlation Matrix
1: Input: Sampled KME vector & of length M = 24; operator basis Ā = (H& TT
2: Output: Local Hamiltonian H
3: Construct the QCM matrix using the inner products, where the (k, E) entry is given by
M (3 (( H. Φ6
4: Compute null space of QCM:
W = [wx T-1 E Null(M
5: Form Hamiltonian linear combination:
T H カ=
Σ
wy
H
k=
6: return H
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Algorithm 3 First-Order Perturbation Corrections and TS UQ
1: Input: Base Hamiltonian
Wk H
k=0
where {wa} are the learned weights multiplying the Hermitian operator basis; perturbation vector
Aw specifying perturbation of these weights.
2: Output: First-order corrected eigen-modes {v/m) and the resulting UQ feature vector.
3: Construct the perturbed Hamiltonian:
T-1 AH = Σ Awy H
This corresponds to perturbing the coefficients (weights) associated with the Pauli-like operators forming the QTN Hamiltonian. 4: Compute eigen- decoppsccooooitiionn of the unperturbed Hamiltonian:
H c Em Dmm
.,..., M 1.
5: for m = 0 to M do
6: Compute first-order energy correction:
E(!) = (Wm, AH vmm).
7: Compute first-order cigenvector correction:
(Wa, AH v/m)
(1)
Em
En
8: end for
9: We utilize Eq. and { to compute UQ feature vectors in the KME amplitude domain, yielding the local UQ scores that appear in Eq. 6
return {w() UQ(p!").
Algorithm 4 Calibrated Entropy-Maximizatng Update for TS Probabilities
1: Input:
Original TS probability p!!))
Uncertainty score UQ(p!" !r) computed via Eq.
Regularization parameter À
2: Output: Adjusted probability ps" (r)
3: Define objective:
C(D) log + +1 (1-p)²
KL p p!
UQ(D!")
where KL(p)pp) Slogg & - (1 - pp log 1-0
4: Initialize: (1() + Ps
5: for to Tiaa do
6: Compute gradient V C(
7: Update via projected gradient ascent
8: Stop early if (p(t) f 1)
9: end for
10: Set p!
(r)
< ((T)
11: return (')""
40


Reviews:
[{"review_id": "luivN4S37p", "reviewer": "Reviewer_YDUM", "summary": "The authors introduce a quantum tensor network (QTN)-based uncertainty quantification (UQ) framework that models aleatoric uncertainty of token sequence (TS) probabilities and leverages semantic clustering, producing a principled scheme for hallucination detection. Authors show experiments on multiple datasets and model scales to highlight the efficacy of their approach.", "questions": "- How does the hallucination detection robustness change with different semantic clustering models? Can you quantify error propagation from supplementary entailment models?\n\n- What are the runtime/memory costs versus baseline methods for real-time or production deployment at scale?\n\n- How do you use the local entropy/uncertainty scores in practice?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- The paper is generally well written although a bit heavy on theoretical notations. It could be simplified a bit more though.\n\n- It brings quantum physics-inspired UQ to the LLM hallucination detection problem.\n\n- The paper tackles a very significant issue. Hallucination risk in LLMs is a key challenge for safe AI.", "weaknesses": "- Heavy reliance on advanced mathematical machinery. Makes it a bit harder to adopt.\n\n- The apporach does not come significant performance boost.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vZdrsHJkpA", "reviewer": "Reviewer_tnuf", "summary": "The paper introduces a quantum-inspired framework for estimating uncertainty in Large Language Models (LLMs) to better detect hallucinations.\nBuilding on Semantic Entropy (Farquhar et al., 2024), the authors reinterpret the model’s sequence probabilities as a wave-function in a reproducing-kernel Hilbert space and construct a quantum tensor-network (QTN) representation to analyze perturbations in this space.\nThey derive first-order perturbation features that quantify the local sensitivity of token probabilities and use these to re-weight output probabilities through an entropy-regularized optimization objective.\nIn parallel, they compute semantic-cluster probabilities and a Renyi-entropy-based uncertainty score.\nExperiments across Mistral-7B, Falcon-1B, and LLaMA-2/3 on TriviaQA, NQ, SVAMP, and SQuAD show consistent AUROC improvements over Semantic Entropy, naive token entropy, and other baselines, including under 4-bit quantization.", "questions": "1. How sensitive is the performance to the Rényi entropy order (α)? Did you experiment with α > 2 or other generalized entropies?\n2. What is the additional computational cost of the QTN perturbation step compared to standard Semantic Entropy?\n3. Would the uncertainty-aware reweighting change the model’s decoding behavior if used during generation (not only for scoring)?\n4. Did you test the approach in long-form generation (e.g., summarization) where clustering and token perturbations interact differently?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Clear motivation: addresses the instability of entropy-based uncertainty when probability mass collapses into a few clusters.\n2. Conceptual novelty: combining semantic clustering with quantum-mechanical perturbation analysis is original and theoretically interesting.\n3. Methodological soundness: the derivations are mathematically consistent, and the optimization formulation is well justified.\n4. Robustness: results are shown across several models, datasets, and quantization levels, indicating the approach is not overly brittle.\n5. Empirical clarity: tables separate the effect of the perturbation term, Rényi entropy, and adjusted probabilities; ablations are informative.\n6. Good writing: the exposition is clean, equations are readable, and notation is consistent.", "weaknesses": "1. Complexity and interpretability: the proposed QTN embedding and perturbation analysis are computationally heavier than simpler uncertainty measures; inference-time cost is not reported.\n2. Baselines could be broader: comparisons omit recent energy- or logits-based approaches (e.g., Semantic Energy, LogTokU).\n3. Sensitivity analysis: the method introduces several hyperparameters (kernel bandwidth, λ, Rényi α), but their tuning process and sensitivity are not discussed in depth.\n4. Clustering dependency: since semantic clustering remains the outer layer, the total uncertainty measure still inherits its limitations (e.g., sensitivity to embedding choice).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "j0sMbwnkbI", "reviewer": "Reviewer_Bhsm", "summary": "This paper uses a physics-inspired UQ method to modify cluster probabilities in a semantic entropy-type decomposition, based on prediction uncertainty. It’s a neat idea, and a way of incorporating distributional uncertainty that I haven’t seen much in the literature. There seems to be a small but persistent improvement over standard SE, across a variety of settings. However, I am confused on quite a few aspects, see the comments below.", "questions": "First, I am not sure on whether my interpretations are correct, so I am going to summarize my interpretation (in the hopes that the authors can correct any misunderstanding!). For context, I am familiar with the semantic entropy line of work, Bayesian NNs, and KMEs, but not with the physics-inspired framework of Principe (I skimmed the Singh and Principe work; I suspect that the Vipulananathan et al paper would be helpful background but I couldn’t find it without a paywall):\n\n\n* Semantic Entropy is underspecified, because it does not consider uncertainty in the predictive distributions (loosely, epistemic uncertainty... not exactly, I know, but I’m going to refer to it as “epistemic” uncertainty in my review as a shorthand)\n* The physics-inspired approach of Principe offers a way of getting at this uncertainty, via permutation.\n* Rather than Shannon entropy, we can look at Renyi entropy, which can be expressed in terms of the kernel mean embedding of the sequence probabilities.\n* This is natural for the physics-inspired approach, because you can find a QTN whose Hamiltonian is aligned with the KME, and directly perturb that.\n    * i.e., the authors aren’t trying to claim that Renyi entropy is necessarily a “better” option than Shannon entropy, rather that adding in the “epistemic” uncertainty is important.\n\n\nI’d really appreciate any corrections on my understanding of the above — if I’m lost at this point, who knows how wrong my interpretations of the rest are!\n\nBased on this: I like the idea! I agree with the general idea that we should include uncertainty about the logits, and I am intrigued by the physics-based approach. But, I think the exposition could be improved so that people who aren’t already familiar with the physics-based approach (i.e., me! But also I suspect a lot of the people in the LLM/UQ community) can get better understanding and intuition. Also, the implementational details and details of the estimator are a little vague, and the experiments don’t fully explore the method (in one case, the authors seem to refer to an experiment that isn’t in table 1, but that I would love to see in table 1... perhaps something that got left out in last-minute editing?)\n\nGiven that, I have a number of questions/requests:\n\nIntuition:\n\n1. More background/intuition about the physics-inspired approach, and QTNs. In particular, in line 179, you say you identify a QTN whose Hamiltonian has $\\{\\underline{\\hat{\\psi}}_\\underline{y}\\}$ as an eigen-mode, and then perturb this Hamiltonian. How does one do this? What is the intuition for this giving reasonable estimates of the variability of the probabilities?\n2. What is the connection to other ways of estimating “epistemic” uncertainty? Eg: the perturbation-based approach seems similar to the Laplace approximation to a Bayesian posterior — is there a connection there? More generally, there probably should be some mention of the Bayesian deep learning literature, since they are coming from a similar viewpoint. I don’t think you need to compare with BDL, but it would be good to see some discussion of why you choose the physics-based approach vs alternatives.\n3. In Section 2.2, you say that the KME is an approximation of the semantic entropy, but it is not explicit how you get from eq 3 to eq 4. On the first read, I thought you were claiming that the KME of the entire distribution was equivalent to the Renyi Semantic Entropy, and I was confused how this incorporates the clustering. Reading on, it is clear that you are using deberta clustering... I assume that you calculate the KME for each cluster, then combine them, but I’m not exactly sure how I should combine these KMEs based on eq 4.\n4. I would like a little more intuition on the calibrated adjustment of the probabilities. Why is increasing the entropy in this way the “right” thing to do? I think I get it in a loose way — your KL term says “the raw probability is roughly right, where ”roughly“ is base on how uncertain I am” and the RE term say “but, we are underestimating the entropy so pull it up” — but are there theoretical underpinnings behind this specific formulation? Is the assumption that lambda is learned via cross-validation?\n5. Once you have the calibration-adjusted probabilities, are you then combining them using eq 4, or are you using the KMEs?\n\nExperiments:\nWhat I would hope to see from the experiments are evidence of the following: 1) using Renyi semantic entropy (without additional UQ) is as good, or better, than using Shannon semantic entropy; 2) adding in the entropy maximization correction improves on this; 3) ablations; 4) qualitative analysis of a specific case where you get differing results. Most of these are at least fully answered, but not completely.\n\n\n1. Using Renyi semantic entropy vs using Shannon. This is included in the heat maps, but could be discussed more. As far as I can see, shannon typically does better than renyi in terms of AUROC, but comparably in terms of AURAC. Is there any intuition behind this?\n2. Overall performance. Looking at win rate feels misleading, when (looking at the figures in the appendix) the actual differences between the various SE methods is pretty small (mostly invisible to the eye... I suspect the confidence intervals are greater than the difference in most cases). I know that the differences in AUROC in these sorts of LLM UQ paper tend to be small, so I’m not surprised to see that the improvement is slight.... however I've would like to see in the main paper, something that more gets at a specific effect size (e.g., notable increase for a given TPR/FPR tradeoff).\n3. Overall performance #2: I would like to see baseline SRE in figures 12-14. Based on Figure 24, at least in Falcon, UQ-based SRE can dramatically improve on baseline SRE — implying that in this case, baseline SRE is much worse than baseline shannon SE. Why is this?\n4. Ablations: Would it make sense to apply the UQ adjustment to Shannon Semantic Entropy? It seems you could take the adjusted probabilities from eq 7 and use them to calculate it. \n5. Qualitative analysis: In Table 1, but the main-text description doesn’t match the figure (it talks about the difference between $\\text{SE}_R$ and $\\text{SE}_R^+$, but $\\text{SE}_R$ isn’t in the table.\n6. Qualitative analysis: It would be nice to see a qualitative exploration of questions that get very different values of $\\text{SE}_R$ and $\\text{SE}_R^+$. Do these tend to correspond to questions the LLM is likely to have less data on? Or, some other form of additional uncertainty?\n\n\nGeneral comments: \n\n* There is a lot of space given to the quantization experiments, relative to how important it seems to the paper. I would like to see this shrunk or moved to the appendix in favor of more explanation/intuition/detail.\n* You are not very consistent in method naming (eg, $\\text{SE}_R^+$ vs SRE-UQ.\n\n\nI know that's a lot of comments and questions! But, it's because I really like the idea of the paper, and *want* to have a good in-depth understanding, but don't feel in its current form I am able to get all the information I need from it.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "* Interesting and novel way of thinking about uncertainty in LLMs\n* Small, but persistent, improvement over baselines (in a domain where the baselines are already pretty strong)", "weaknesses": "* Not enough background/intuition on QTNs and the physics-inspired approach used\n* Lack of technical detail: I am not sure how to construct all the estimators and would certainly not be able to implement the method myself.\n* Experiments do not fully tease apart the benefit of the proposed method\n\nI elaborate on all these points in my questions below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)