Quantum-Inspired Latent Variable Modeling in Multivariate Analysis

Abstract

Latent variables play a crucial role in psychometric research, yet traditional models often struggle to address context-dependent effects, ambivalent states, and non-commutative measurement processes. This study proposes a quantum-inspired framework for latent variable modeling that employs Hilbert space representations, allowing questionnaire items to be treated as pure or mixed quantum states. By integrating concepts such as superposition, interference, and non-commutative probabilities, the framework captures cognitive and behavioral phenomena that extend beyond the capabilities of classical methods. To illustrate its potential, we introduce quantum-specific metrics—fidelity, overlap, and von Neumann entropy—as complements to correlation-based measures. We also outline a machine-learning pipeline using complex and real-valued neural networks to handle amplitude and phase information. Results highlight the capacity of quantum-inspired models to reveal order effects, ambivalent responses, and multimodal distributions that remain elusive in standard psychometric approaches. This framework broadens the multivariate analysis theoretical and methodological toolkit, offering a dynamic and context-sensitive perspective on latent constructs while inviting further empirical validation in diverse research settings.

1. Introduction

Latent variables have long been a cornerstone of psychometric inquiry, yet their exact nature and scope can vary considerably across disciplines [1,2,3,4]. In psychological measurement, variables typically refer to (a) observed indicators (e.g., item responses, reaction times, or behavioral frequencies) and (b) unobserved constructs inferred from these indicators (e.g., personality traits, cognitive abilities, or emotional states) [5,6]. Classical psychometric models, such as factor analysis or item response theory, treat these unobserved constructs as “true scores” or stable traits, implying that relatively fixed parameters govern the relationship between observed and latent traits. However, in fields like sociology or economics, “latent variables” may include context-dependent constructs—such as cultural norms or economic sentiments—which can shift over time or vary across subpopulations. Consequently, the generalized term “latent variable” may mask a wide range of theoretical assumptions, from stable internal traits to dynamic, situational states [7]. In this study, we use latent variables to mean unobservable psychological constructs inferred by analyzing patterns in observed data (e.g., survey items). These constructs may represent traits such as extroversion or anxiety, but they can also denote transitory states (e.g., mood) or socially influenced dispositions.

Latent variables are operationalized primarily through two measurement models: reflective and formative [8,9]. Reflective models conceptualize the latent variable as the underlying cause of its indicators. For instance, the latent construct of “depression” may be manifested through observable symptoms such as sadness, fatigue, and anhedonia. In this framework, the indicators are presumed to be intercorrelated, with variations in the latent variable inducing corresponding changes in the indicators. This implies a unidirectional influence of the latent construct on the observed measures, encapsulating the essence of the latent variable [10].

Conversely, formative models posit that the latent variable is constituted by its indicators. A quintessential example is “socioeconomic status,” which emerges from income, education, and occupation variables [8]. Unlike reflective models, formative models do not require the indicators to be intercorrelated, as they collectively define and form the construct. This distinction is crucial for methodological selection, ensuring that the measurement model aligns with the theoretical underpinnings of the studied construct [11]. Understanding whether a construct is better represented through reflective or formative models is imperative for accurate interpretation and theoretical coherence within research endeavors.

1.1. Classical Approaches to Latent Variable Modeling

Classical latent variable modeling encompasses a range of methodologies, each meticulously designed to address specific research objectives. These methodologies include confirmatory factor analysis (CFA), exploratory structural equation modeling (ESEM), bifactor models, item response theory (IRT), latent class analysis (LCA), latent profile analysis (LPA), latent growth curve modeling (LGCM), Bayesian models, principal component analysis (PCA), hidden Markov models (HMM), Gaussian graphical models (GGM), canonical-correlation analysis (CCA), and network psychology (

Table 1. Latent variable models: formulas, explanations, and evaluations.

Model	Mathematical Formula	Explanation	Advantages	Disadvantages
Confirmatory Factor Analysis (CFA)	$y_i={\lambda }_i\eta +{ϵ}_i$	Observed variables ($y_i$) are modeled as a function of latent variables ($\eta$) with loadings (${\lambda }_i$) and residual error (${ϵ}_i$).	- Tests hypothesized factor structures. - Offers clear goodness-of-fit metrics. - Provides straightforward interpretations of latent constructs.	- Assumes linearity and normality. - Sensitive to sample size. - Cannot handle cross-loadings well. - Does not account for residual correlations.
Exploratory Structural Equation Modeling (ESEM)	$y_i={\sum }_j{\lambda }_{ij}{\eta }_j+{ϵ}_i$	Extends CFA by allowing observed variables ($y_i$) to load on multiple latent factors (${\eta }_j$).	- Flexible and adaptable to complex data. - Combines exploratory and confirmatory approaches. - Useful for real-world datasets.	- Can be difficult to interpret if there are many cross-loadings. - Risk of overfitting without careful specification. - Computationally demanding.
Bifactor Model	$$\begin{gathered} y_i={\lambda }_{g_i}{\eta }_g+ \\ {\displaystyle \sum _{s=1}^S}{\lambda }_{s_i}{\eta }_s+{ϵ}_i \end{gathered}$$	Observed variables ($y_i$) are explained by a general factor (${\eta }_g$) and specific factors (${\eta }_s$).	- Captures both general and specific factor structures. - Effective for multidimensional constructs. - Helps differentiate global and domain-specific effects.	- Requires high-quality data. - Assumes uncorrelated specific factors that may not hold. - Sensitive to model misspecification.
Item Response Theory (IRT)	$P\left(X_i=1|\mathsf{\theta }\right)={\displaystyle \frac{e^{a_i\left(\mathsf{\theta }-b_i\right)}}{1+e^{a_i\left(\mathsf{\theta }-b_i\right)}}}$	Models the probability of a correct response (P) based on latent trait (θ), discrimination ($a_i$), and difficulty ($b_i$).	- Provides item-level diagnostics. - Handles measurement errors well. - Useful for test development and evaluation.	- Assumes unidimensionality and local independence. - Computationally demanding for complex models. - Large sample sizes are required for robust estimates.
Latent Class Analysis (LCA)	$P\left(X\right)={\prod }_{i=1}^N{\mathsf{\pi }}_{k_i}^{X_i}{\left(1-{\mathsf{\pi }}_{k_i}\right)}^{1-X_i}$	Using a probabilistic approach, assigning individuals to unobserved latent classes based on categorical data.	- Identifies unobserved subgroups. - Useful for population segmentation. - Handles categorical data effectively.	- Assumes independence of indicators within classes. - Sensitive to model specification and class number. - May not generalize well with small sample sizes.
Latent Profile Analysis (LPA)	$y_i={\displaystyle \sum _{k=1}^K}{\mathsf{\pi }}_k{f}_k\left(x|{\mathsf{\mu }}_k,{\mathsf{\Sigma }}_k\right)$	Extends LCA to continuous indicators, modeling individuals as belonging to profiles characterized by means (${\mu }_k$) and variances (${\Sigma }_k$).	- Models continuous data well. - Allows probabilistic assignment to profiles. - Useful for exploring population heterogeneity.	- Sensitive to model specification. - Requires careful determination of the number of profiles. - Assumes multivariate normality within profiles.
Latent Growth Curve Model (LGCM)	$y_{it}={\mathsf{\beta }}_0+{\mathsf{\beta }}_{1}t+u_i+e_{it}$	Captures change over time with an intercept (${\beta }_0$) and slope (${\beta }_1$), while accounting for individual deviations $(u_i,e_{it}$).	- Effective for studying developmental trajectories. - Handles time-dependent data well. - Can model nonlinear growth patterns.	- Requires large datasets for reliable estimates. - Sensitive to assumptions about growth patterns. - Model misspecification can lead to biased results.
Bayesian Models	$P\left(\mathsf{\theta }|D\right)={\displaystyle \frac{P\left(D|\mathsf{\theta }\right)P\left(\mathsf{\theta }\right)}{P\left(D\right)}}$	Incorporates prior beliefs (P(θ)) with observed data (P(D|θ)) to estimate posterior distributions (P(θ|D)).	- Effective for small samples. - Allows incorporation of prior knowledge. - Flexible for complex models.	- Computationally intensive. - Requires careful selection of priors. - Can introduce subjectivity through prior specification.
Principal Component Analysis (PCA)	$Z=W^{T}X$	Reduces dimensionality by projecting data (X) onto principal components (Z) using weights (W).	- Reduces dimensionality efficiently. - Useful for exploratory data analysis. - Handles large datasets effectively.	- Assumes linear relationships among variables. - Components may lack interpretability. - Sensitive to outliers.
Hidden Markov Model (HMM)	$P\left(S_t|S_{t-1}\right)=A_{ij}$	Models transition between latent states ($S_t$) based on transition probabilities $A_{ij}$.	- Effective for time-dependent latent structures. - Handles state transitions well. - Applicable to longitudinal data.	- Computationally demanding. - Assumes Markov property (future depends only on present state). - Sensitive to model initialization.
Gaussian Graphical Model (GGM)	${\mathsf{\Theta }}^{-1}=\mathsf{\Sigma }-\Phi$	Models relationships among variables using a precision matrix ${\mathsf{\Theta }}^{-1}$, inferred from covariance (Σ) and residual (Φ).	- Visualizes relationships among variables. - Handles sparse structures well. - No assumption of normality is required.	- Can be challenging to interpret. - Sensitive to sparsity assumptions. - Requires careful selection of network criteria.
Canonical-Correlation Analysis (CCA)	$\rho ={\displaystyle \frac{\sum \left(X_1{X}_2\right)}{\sqrt{\sum \left(X_1^2\right)\sum \left(X_2^2\right)}}}$	Finds linear combinations of variables from two datasets that maximize their correlation (ρ).	- Useful for multivariate data. - Captures relationships between datasets. - Produces interpretable results.	- Assumes linearity and normality. - Sensitive to noise and outliers. - May not generalize well to complex relationships.
Network Psychology	$y_i={\displaystyle \sum _{j\ne i}}{\varphi }_{ij}{y}_j+{ϵ}_i$	Represents psychological constructs as networks where each observed variable ($y_i$) is directly influenced by other variables ($y_j$) through edge weights (${\varphi }_{ij}$), with residual error (${ϵ}_i$).	- Captures direct interrelationships among variables. - Identifies central and bridge variables within the network. - Flexible in modeling complex interactions.	- Computationally intensive for large networks. - Interpretation of connections can be ambiguous. - Sensitive to sample size and measurement error. - Requires careful model specification to avoid overfitting.

for detailed descriptions and comparisons) [12,13,14,15,16,17,18,19,20,21,22,23,24,25].

These methodologies collectively provide a comprehensive toolkit for studying latent variables across diverse contexts. Each method has specific assumptions, strengths, and limitations that must be carefully considered to ensure valid and meaningful results in research.

Despite the robustness of these classical models in analyzing latent constructs, they encounter limitations when attempting to capture dynamic and context-dependent phenomena. Instances such as contextual priming, emotional ambivalence, and sequential dependencies exemplify areas where traditional methods frequently prove inadequate. Addressing these limitations necessitates a paradigm shift toward alternative frameworks that can more effectively model the complexities inherent in psychological phenomena.

1.2. Quantum-Inspired Latent Variable Models

Quantum mechanics, a cornerstone of modern physics, provides a framework for understanding phenomena that defy classical explanations. Central to this framework are principles such as Hilbert spaces, superposition, interference, and non-commutative probabilities. These concepts describe the behavior of particles in ways that challenge intuitive, deterministic models.

Hilbert Spaces: These are mathematical structures that represent the possible states of a system as vectors. In quantum mechanics, the state of a particle or system exists in this abstract space, allowing for a more flexible representation than traditional coordinates.

Superposition: This principle states that a system can exist in multiple states simultaneously until measured, at which point it “collapses” to a single state. For example, an electron’s position is a probability distribution across multiple locations rather than a fixed point.

Interference: when superimposed states interact they can amplify or diminish certain outcomes—much like waves overlapping in physics.

Non-Commutative Probabilities: Unlike classical probabilities, where the order of operations does not matter, quantum probabilities are sensitive to a sequence. Measuring one property first can influence subsequent measurements, introducing contextuality [26,27,28,29].

Researchers have increasingly investigated quantum probability models in searching for novel ways to capture context-dependent phenomena, partly inspired by empirical successes in cognitive science where quantum-like decision frameworks have outperformed classical probability models under ambiguity [30,31,32,33,34]. These approaches are compelling because they allow for non-commutative measurement, contextuality, and interference effects—features which can be crucial when studying order effects, priming, or ambivalent states that do not adhere to classical probability assumptions.

Quantum mechanics offers a transformative framework for modeling latent variables by introducing constructs such as superposition, interference, and non-commutative probabilities [35]. Unlike classical approaches presupposing static and deterministic relationships, quantum-inspired models represent latent states probabilistically, capturing the dynamic interplay between variables and their contextual environments. This probabilistic representation aligns more closely with the inherent uncertainties and complexities of psychological phenomena.

A salient characteristic of quantum models is their treatment of measurement processes. In classical methodologies, measurements are commutative, implying that the sequence of observations does not influence the outcomes. Quantum models challenge this assumption by incorporating non-commutative probabilities [36,37], wherein the order of measurements can affect the latent state. This feature enables researchers to investigate phenomena such as survey order effects, where responses to earlier questions influence subsequent ones, providing a more accurate representation of the cognitive processes involved.

1.3. Psychological Applications of Quantum Models

Quantum-inspired models exhibit significant potential for elucidating complex psychological phenomena by providing a nuanced framework that accommodates the intricacies of human cognition and behavior [38,39]. For example, cognitive dissonance [40,41] can be conceptualized as interference between conflicting latent states (e.g., “I value honesty” versus “I lied”). In this context, constructive interference may amplify feelings of guilt, prompting corrective actions, while destructive interference might reduce tension by allowing rationalization of the lie. Similarly, emotional ambivalence, characterized by the simultaneous presence of conflicting emotions such as joy and sadness, can be understood through the superposition of emotional states [42,43]. Constructive interference may enhance positive emotions like joy, while destructive interference can temper negative emotions like sadness, offering a dynamic representation of mixed emotional states.

Survey order effects [44,45], where the sequence of questions influences respondents’ answers, are effectively captured by non-commutative probabilities within quantum models. This approach acknowledges that answering one question can alter the cognitive state of the respondent, thereby impacting responses to subsequent questions in a manner that classical models fail to account for. Additionally, priming effects in memory and recall can be modeled by quantum-inspired frameworks, where priming with specific stimuli modifies recall probabilities [46,47]. Constructive interference may enhance the recall of congruent memories, while destructive interference can suppress unrelated memories, providing a more comprehensive understanding of how priming influences memory processes.

Furthermore, personality traits can be viewed as context-dependent constructs within quantum models. For instance, an individual’s personality state (e.g., extroverted versus introverted) may exist in a superposition that collapses based on contextual factors such as social familiarity, thereby dynamically influencing behavior in varying environments [48,49]. Decision-making processes, particularly under uncertainty as described by prospect theory, involve transitions between superposed states of preference [50,51]. Quantum-inspired models can capture framing effects, where emphasizing risk aversion shifts probabilities toward specific outcomes, such as favoring a sure gain over a risky choice through constructive interference.

Mood and memory interactions are another area where quantum models provide valuable insights. A positive mood may amplify the recall of joyful events (constructive interference) while suppressing the recall of sad events (destructive interference), thereby influencing overall memory retrieval processes [52,53]. Implicit biases and stereotype activation can also be modeled as latent states that shift based on contextual cues [54,55]. Exposure to diverse imagery may create destructive interference, reducing stereotypical biases, whereas congruent cues could amplify these biases through constructive interference, highlighting the dynamic nature of implicit attitudes (see

Table 2. Quantum approach to psychological phenomena: a table of applications.

Phenomenon	Quantum Explanation
Cognitive Dissonance and Interference	Cognitive dissonance is modeled as interference between two latent states (e.g., ‘I value honesty’ vs. ‘I told a lie’). Constructive interference amplifies one belief (e.g., guilt prompting corrective action), while destructive interference suppresses tension (e.g., rationalizing the lie).
Emotional Ambivalence and Mixed Emotions	Emotional ambivalence (e.g., joy and sadness) is treated as a superposed state. Constructive interference enhances one emotion (e.g., joy amplified by pride), while destructive interference suppresses another (e.g., sadness dampened by the same pride).
Order Effects in Surveys and Questionnaires	The act of answering one question alters the respondent’s cognitive state. For example, answering a question about strengths might increase confidence, influencing responses to subsequent questions. This is modeled through non-commutative probabilities, capturing the effect of question order.
Priming Effects in Memory and Recall	Memory recall is modeled as a transition between latent states. Priming with ‘achievement’ could modify amplitudes of related concepts like ‘success’ (constructive interference) or suppress unrelated ideas like ‘failure’ (destructive interference).
Personality Traits as Context-Dependent Constructs	Personality states (e.g., ‘extroverted’ vs. ‘introverted’) are superpositions that collapse based on context. For instance, familiarity with a social setting might amplify extroversion (constructive interference), while unfamiliarity could suppress it (destructive interference).
Decision-Making and Prospect Theory	Decisions under uncertainty involve transitions between superposed states of preference. Framing effects, like emphasizing risk aversion, shift probabilities toward specific outcomes (e.g., constructive interference favoring a sure gain over a risky choice).
Mood and Memory Interactions	Mood acts as a contextual state interfering with memory recall. A happy mood might amplify the recall of joyful events (constructive interference) and suppress sad events (destructive interference).
Implicit Bias and Stereotype Activation	Implicit biases are modeled as latent states shifting based on context. Exposure to diverse imagery might create destructive interference, reducing stereotypical bias, while congruent cues could amplify these biases through constructive interference.

for specific examples of quantum-inspired psychological phenomena).

By accommodating such phenomena, quantum-inspired models bridge existing gaps in classical latent variable methodologies, offering deeper insights into the intricacies of human cognition and behavior.

1.4. Research Questions

Building upon the theoretical foundations and methodological advancements outlined in the preceding sections, this study addresses critical gaps in traditional psychometric measurement by integrating quantum-inspired latent variable models. To achieve this, the study is guided by the following research questions:

• Hilbert Space and Psychometric Representation: how can the principles of Hilbert space representation, such as orthonormality and normalization, be effectively utilized to model latent factors in psychometric data?
• Pure vs. Mixed State Modeling: under what conditions do pure state representations of questionnaire items suffice, and when are mixed states required to account for multimodal or context-dependent response patterns in psychometric assessments?
• Quantum-Inspired Metrics in Multivariate Analysis: compared to classical correlation-based measures, how do quantum-specific metrics like fidelity, overlap, and von Neumann entropy provide insights into the relationship between psychometric items and latent constructs?
• Machine Learning for Quantum Latent Variable Models: what are the advantages and limitations of implementing complex-valued neural networks (CVNNs) versus real-valued networks with two channels for processing quantum-inspired psychometric data, particularly concerning preserving phase relationships and computational efficiency?

These research questions address the theoretical, methodological, and practical implications of applying quantum frameworks to psychometric research, aiming to bridge existing gaps in measurement theory and multivariate analysis.

2. Theoretical Foundations

2.1. Hilbert Space Representation

In quantum theory [56], a system is represented by a vector $\left|\mathsf{\psi }\right.⟩$ in a complex Hilbert space $\mathcal{H}$ [57,58]. For latent variables, it is proposed that each item $q_k$ is associated with its state, $\left|{\mathsf{\psi }}_{q_k}\right.⟩\in \mathcal{H}$. Assuming n latent factors, $\mathcal{H}$ is typically n-dimensional, with an orthonormal basis as follows:

$$\{\left|F_1\right.⟩,\left|F_2\right.⟩,\dots ,\left|F_n\right.⟩\}.$$

Orthonormality implies that:

$$⟨F_i|F_j⟩={\mathsf{\delta }}_{ij,}$$

where ${\mathsf{\delta }}_{ij}$ is the Kronecker delta, equal to 1 if $i=j$ and zero otherwise [59]. In physical quantum mechanics, these vectors might represent distinct states of a particle; in our context, they represent distinct latent factors.

Any vector $\left|\mathsf{\psi }\right.⟩\in \mathcal{H}$ can thus be expressed as a linear combination of these basis vectors:

$$\left|\mathsf{\psi }\right.⟩=\sum _{i=1}^n{\mathsf{\alpha }}_i\left|F_i\right.⟩,\hspace{1em}$$

$\mathrm{where} {\mathsf{\alpha }}_i\in C$.

A normalization condition is typically imposed:

$${\sum }_{i=1}^n{\left|{\mathsf{\alpha }}_i\right|}^2=1,$$

ensuring that the total “probability” across all basis vectors sums to unity. In a psychometric context, the complex amplitudes ${\mathsf{\alpha }}_i$ are crucial for interpreting the degree of association with each latent factor.

In initial applications, representing each questionnaire item $q_k$ as a pure state suffices to characterize its projection onto latent factors. Formally, the state of an item is expressed as the following:

$$\left|{\mathsf{\psi }}_{q_k}\right.⟩={\sum }_{i=1}^n{\mathsf{\alpha }}_{k,i}\left|F_i\right.⟩,\hspace{1em}{\sum }_{i=1}^n{\left|{\mathsf{\alpha }}_{k,i}\right|}^2=1.$$

Here, the coefficients $\{{\mathsf{\alpha }}_{k,i}{\}}_{i=1}^n$ are complex numbers. Analogous to quantum mechanics, the squared modulus ${\left|{\mathsf{\alpha }}_{k,i}\right|}^2$ represents the probability of finding the system in the basis state $\left|F_i\right.⟩$. For a psychometric item, this value indicates the probability or intensity with which the item aligns with the latent factor $F_i$. The phase of ${\mathsf{\alpha }}_{k,i}$, denoted as ${\mathsf{\varphi }}_{k,i}$, can induce interference effects among latent factors, as elaborated in subsequent sections.

While pure states provide a straightforward representation, some questionnaire items exhibit inherently multimodal or context-dependent response distributions. In such instances, a single pure state may inadequately capture the item’s alignment with latent factors across different respondent subgroups or situational contexts. Instead, one can utilize a density matrix ${\mathsf{\rho }}_{q_k}$, a positive semi-definite operator on $\mathcal{H}$ with $\mathrm{Tr}\left({\mathsf{\rho }}_{q_k}\right)=1$, as follows:

$${\rho }_{qk}={\sum }_j{p}_j\left|{\psi }_{qk,j}\right.⟩\left.⟨{\psi }_{qk,j}\right|, \sum _j{p}_j=1,$$

where each $\left|{\psi }_{qk,j}\right.⟩$ is a pure state and $p_j$ represents the probability of that state. The diagonal elements of ${\mathsf{\rho }}_{q_k}$ in the chosen basis indicate how the item is distributed among factors, while the off-diagonal terms encode coherence or interference between different pure-state components.

The choice between pure and mixed states hinges on theoretical considerations regarding item heterogeneity and empirical evaluations, such as clustering respondent profiles. For illustrative purposes, this discussion focuses on pure states, with mixed states to be examined in greater detail when addressing multimodal responses and entropy measures.

It is important to stress that imposing ${\sum }_{i=1}^n{\left|{\mathsf{\alpha }}_i\right|}^2=1$ presumes that measurement error and bias [60] are negligible or can be subsumed within the amplitude structure. Psychometric data often include random noise, cultural biases [61], or acquiescence effects that may not map perfectly onto these normalized probabilities. Researchers must, therefore, interpret the normalization constraint as an idealized assumption or consider introducing small correction factors (e.g., renormalizing after accounting for error variances) to accommodate real-world response distributions.

2.2. Data-Driven Initialization of Quantum States

Once the latent factor space $\mathcal{H}$ is defined, the next step is to initialize each questionnaire item $q_k$ as either a pure state $\left|{\psi }_{q_k}\right.⟩$ or, if necessary, a mixed state ${\mathsf{\rho }}_{q_k}$. This process involves mapping observed responses—such as Likert-scale scores—to complex amplitudes or density matrices in a principled and data-driven manner.

Each item $q_k$ has a response scale ranging from 1 (lowest) to $r_{\max }\left(highest\right)$. A conventional approach involves normalizing each response $r_k$ to a value within the interval [0, 1] using the following equation:

$${\mathrm{resp}}_k={\displaystyle \frac{r_k-1}{r_{\max }-1}},\hspace{1em}\mathrm{where} r_k\in \{1,\dots ,r_{\max }\}.$$

For multiple latent factors, the normalized response ${\mathrm{resp}}_{k,i}$ is distributed across these factors based on theoretical or empirical weights, such as partial correlations or expert judgments regarding factor relevance. The amplitude magnitude can then be defined as the following:

$$\left|{\mathsf{\alpha }}_{k,i}\right|=\sqrt{{\mathrm{resp}}_{k,i}},$$

ensuring that higher responses yield larger amplitude magnitudes for factor $F_i$. Optionally, phases can be assigned to the amplitudes to capture interference effects by using the following:

$${\mathsf{\alpha }}_{k,i}=\sqrt{{\mathrm{resp}}_{k,i}}\hspace{0.17em}{e}^{i{\mathsf{\varphi }}_{k,i}},\hspace{1em}{\mathsf{\varphi }}_{k,i}\in \left[\mathrm{0,2}\mathsf{\pi }\right).$$

In many psychometric contexts, phases ${\mathsf{\varphi }}_{k,i}=0$ are assigned initially to simplify the model by treating amplitudes as real and non-negative. After assigning ${\mathsf{\alpha }}_{k,i}$, normalization is enforced to ensure the following:

$${\sum }_{i=1}^n{\left|{\mathsf{\alpha }}_{k,i}\right|}^2=1.$$

Certain items may inherently exhibit multimodality—for instance, a question interpreted differently by distinct respondent subgroups or contexts. Major “modes” within the item’s response profile can be identified in such cases using clustering techniques like K-means or Gaussian mixture models. A pure state represents each mode j $\left|{\psi }_{q_k,j}\right.⟩$, and the density matrix is constructed as the following:

$${\rho }_{qk}={\sum }_j{p}_j\left|{\psi }_{qk,j}\right.⟩\left.⟨{\psi }_{qk,j}\right|, {\sum }_j{p}_j=1,$$

where $p_j$ corresponds to the proportion of respondents classified into the $j$-th mode. This approach flexibly captures a mixture of different interpretations or uses of the same item ${\mathsf{\rho }}_{q_k}$. In quantum terms, each $\left|{\psi }_{q_k,j}\right.⟩$ represents a coherent state for a particular subgroup, while ${\rho }_{q_k}$ encodes the statistical mixture across the entire sample.

It is advisable to start with pure states for simplicity and only transition to mixed states if preliminary analyses indicate multimodality. Amplitude assignments can be automated by calculating partial correlations or regression weights between an item and candidate factors, followed by normalization. If using random phases, they should be initialized near zero or within a systematic bound, allowing subsequent optimization (e.g., via gradient descent) to refine them.

2.3. Quantum-Inspired Metrics and Measures

In quantum theory, system comparisons rely on inner products of states or comparisons of density matrices, which parallel latent variable constructs in multivariate analysis. Key metrics include fidelity, which quantifies the similarity between two quantum states and ranges from 0 to 1, serving as an analog to classical measures of congruence. Similarly, overlap, represented by the inner product of state vectors, captures the degree of alignment or correlation between two states. Von Neumann entropy offers a measure of uncertainty or randomness within a quantum state, analogous to entropy in information theory, providing insights into the complexity or variability of latent constructs. Finally, reconstruction entails recovering the full representation of a quantum state from partial measurements, offering a robust approach to modeling latent constructs and their interactions [62].

Having embedded items within a quantum latent variable framework, the next focus is on the metrics that replace or augment classical correlation-based measures. In quantum theory, system comparisons rely on inner products of states or comparisons of density matrices, which parallel latent variable constructs in multivariate analysis.

When each item $q_k$ is modeled as a pure state $\left|{\mathsf{\psi }}_{q_k}\right.⟩$, the overlap, sometimes termed coherence, between two items $q_k$ and $q_{\mathrm{l}}$ can be measured via their inner product:

$$⟨{\mathsf{\psi }}_{q_k}|{\mathsf{\psi }}_{q_{\mathrm{l}}}⟩=\sum _{i=1}^n{\mathsf{\alpha }}_{k,i}^{*}{\mathsf{\alpha }}_{\mathrm{l},i}$$

where the magnitude of this inner product, $\left|⟨{\mathsf{\psi }}_{q_k}|{\mathsf{\psi }}_{q_{\mathrm{l}}}⟩\right|$, ranges between 0 and 1 for normalized states. A value close to 1 indicates that the two items project onto the same latent factor structure (high coherence), whereas a value near 0 signifies orthogonality, suggesting that the items load onto distinct or even interfering factors. The ${\mathsf{\alpha }}_{k,i}^{\mathrm{*}}$ is the complex conjugate of ${\mathsf{\alpha }}_{k,i}$. The complex conjugate operation reverses the sign of the imaginary component. If ${\mathsf{\alpha }}_{k,i}=a+bi$ (where are $a, b$ real numbers), then ${\mathsf{\alpha }}_{k,i}^{\mathrm{*}}=a-bi$.

Additionally, differing phases can result in destructive interference, reducing the overlap compared to purely real-valued loadings.

For mixed states represented by density matrices ${\mathsf{\rho }}_{q_k}$ and ${\mathsf{\rho }}_{q_{\mathrm{l}}}$, fidelity is defined as the following:

$$F\left({\mathsf{\rho }}_{q_k},{\mathsf{\rho }}_{q_l}\right)={\left(\mathrm{Tr}\left[\sqrt{\sqrt{{\mathsf{\rho }}_{q_k}}{\mathsf{\rho }}_{q_l}\sqrt{{\mathsf{\rho }}_{q_k}}}\right]\right)}^2.$$

This measure assesses the similarity between two quantum states. When both states are pure, fidelity simplifies to the following:

$$F\left({\mathsf{\rho }}_{q_k},{\mathsf{\rho }}_{q_l}\right)={\left|⟨{\mathsf{\psi }}_{q_k}|{\mathsf{\psi }}_{q_l}⟩\right|}^2.$$

Fidelity values close to 1 indicate that the two density matrices are nearly identical, while values near 0 suggest strong dissimilarity. In multivariate analysis, items with multimodal response patterns may exhibit substantial overlap in their density matrices, reflecting partial subpopulations with shared latent structures.

For a density matrix ${\mathsf{\rho }}_{q_k}$, the von Neumann entropy is given by the following:

$$S\left({\mathsf{\rho }}_{q_k}\right)=-\mathrm{Tr}\left({\mathsf{\rho }}_{q_k}\log {\mathsf{\rho }}_{q_k}\right).$$

If ${\mathsf{\rho }}_{q_k}$ has eigenvalues $\left\{{\mathsf{\lambda }}_i\right\}$, then the following is used:

$$S\left({\mathsf{\rho }}_{q_k}\right)=-{\sum }_i{\mathsf{\lambda }}_i\log {\mathsf{\lambda }}_i.$$

(1)

Entropy values near zero indicate that the item is near pure, primarily associated with a single factor or a coherent superposition [63]. Higher entropy values suggest that the item’s alignment is distributed across multiple factors or pure-state modes. In psychometric terms, low entropy may denote items that strongly belong to one construct, while high entropy items might measure broad or ambiguous constructs or reflect multiple respondent interpretations.

In classical factor analysis, item–item correlations or factor loadings are the primary similarity measures [64]. Overlap and fidelity in the quantum framework [65] can be viewed as “quantum correlations” that incorporate both magnitude and phase (or coherence) information. Meanwhile, entropy quantifies how “unresolved” or “mixed” an item’s factor associations may be—conceptually analogous to the uni- versus multidimensionality in classical scales, yet formulated through quantum probability. Adopting these quantum-specific metrics allows researchers to capture phenomena such as order sensitivity or synergy that might remain undetected in purely real-valued or commutative frameworks.

2.4. Machine Learning Architecture for Quantum Latent Models

Having established the encoding of items as quantum states—pure or mixed—and introduced quantum-specific measures, the focus now shifts to delineating a machine learning (ML) pipeline for analyzing these states [66,67,68]. The ML architecture typically involves inputting the quantum-state representations (amplitudes or density matrix elements) into a neural network. This network may be either a complex-valued neural network (CVNN) or a conventional real-valued network that processes real and imaginary components as separate channels [69,70].

For pure states $\left|{\mathsf{\psi }}_{q_k}\right.⟩\in C^n$, a 2n-dimensional real vector is constructed by concatenating the real and imaginary parts of each amplitude, which is shown in the following:

$$x_{q_k}={\left[\mathrm{Re}\left({\mathsf{\alpha }}_{k,1}\right),\mathrm{Im}\left({\mathsf{\alpha }}_{k,1}\right),\dots ,\mathrm{Re}\left({\mathsf{\alpha }}_{k,n}\right),\mathrm{Im}\left({\mathsf{\alpha }}_{k,n}\right)\right]}^{\mathrm{\top }}\in R^{2n}.$$

For mixed states ${\mathsf{\rho }}_{q_k}\in C^{n\times n}$, the real and imaginary parts of each matrix element are flattened to yield up to $2n^2$ input features, which is shown in the following:

$$x_{q_k}=\mathrm{Flatten}\left(\mathrm{Re}\left({\mathsf{\rho }}_{q_k}\right),\mathrm{Im}\left({\mathsf{\rho }}_{q_k}\right)\right).$$

(2)

If ${\mathsf{\rho }}_{q_k}$ is a general complex matrix, this transformation produces up to $2n^2$ features. However, if ${\mathsf{\rho }}_{q_k}$ is Hermitian, only $n^2$ independent real parameters are required.

In practical psychometric applications, the number of factors n is typically modest (e.g., 2–5), making it computationally feasible to handle $2n$ features per item for pure states or $n^2$ features per item for Hermitian mixed states.

The neural network architecture can be broadly categorized into the two following types:

Complex-Valued Neural Networks (CVNNs): These networks handle both weights and activations as complex numbers. Fundamental operations such as convolution and fully connected layers are extended to accommodate complex multiplication and addition [71,72]. Activation functions may involve complex exponentials or separate real–imaginary transformations. The advantage of CVNNs lies in their ability to preserve quantum phase relationships internally, potentially capturing interference phenomena more naturally. However, a drawback is the limited support for complex gradients in many standard ML libraries, complicating backpropagation.

Real-Valued Networks with Two Channels: This approach treats each complex entry a + bi as a two-dimensional real vector (a,b), processed by standard real-valued layers [71,72]. The primary advantage is its straightforward implementation in mainstream frameworks such as TensorFlow or PyTorch 2.6 [73,74]. The downside is that phase relations may be less explicit in the intermediate layers.

The output layer can be designed to perform different tasks depending on the research objectives. For example, the network could produce a vector in $R^n$ to approximate ${\left|{\mathsf{\alpha }}_{k,i}\right|}^2$, representing the probability distribution across factors. The subscript $i$ indexes the basis elements in the Hilbert space, meaning that each ${\mathsf{\alpha }}_{k,i}$ corresponds to the probability amplitude for a specific basis state. A softmax-like normalization ensures non-negative probabilities that sum to 1. Alternatively, the network could aim to reconstruct the original quantum state (pure or mixed) from a compressed representation analogous to an autoencoder. The output might consist of the set of real–imaginary components of $\left|{\psi }_{q_k}\right.⟩$ or the flattened matrix elements of ${\mathsf{\rho }}_{q_k}$. Post-processing steps may reimpose normalization constraints, such as the following:

$${\sum }_{i=1}^n{\left|{\mathsf{\alpha }}_{k,i}\right|}^2=1\mathrm{or} \mathrm{Tr}\left({\mathsf{\rho }}_{q_k}\right)=1.$$

Regularization techniques, such as L2 regularization or dropout [75,76], can be employed in traditional ML settings. In a quantum-inspired context, additional constraints might be necessary to ensure valid quantum states, such as enforcing the positivity of density matrices. Implementing normalization layers after each layer—or at least in the final output—can help maintain constraints, as seen in the following:

$${\sum }_{i=1}^n{\left|{\alpha }_{k,i}\right|}^2=1\mathrm{or} \mathrm{Tr}\left({\rho }_{q_k}\right)=1,$$

enhancing interpretability.

Computational complexity, while generally manageable for moderate n, becomes more critical when dealing with large numbers of items and respondents. Handling density matrices—which can involve up to $2n^2$ real features—can become computationally demanding. Additionally, ensuring the positivity of ${\mathsf{\rho }}_{q_k}$ imposes constraints that some optimization libraries may not natively support.

To ensure scalability in practical applications, researchers are advised to leverage batch optimization, distributed training strategies, and network pruning techniques to manage the computational demands of quantum-inspired psychometric modeling efficiently. Implementing model compression methods, such as quantization-aware training and pruning, can further enhance computational efficiency while preserving model fidelity.

While this study primarily introduces the theoretical and computational foundations of quantum latent variable models, future research should focus on refining implementation strategies, improving computational efficiency, and developing accessible software tools to support broader adoption. The trade-offs between complex-valued and real-valued architectures underscore the importance of selecting an appropriate model based on dataset size, computational resources, and research objectives.

2.5. Training and Objective Functions

Training a quantum-inspired latent variable model involves optimizing specific objective functions that reflect the underlying quantum structure. In classical latent variable analysis, objectives might include maximizing log likelihood or minimizing squared error relative to observed data [77,78]. In the quantum framework, fidelity-based measures, entropy constraints, and potentially a reconstruction term are adopted, each encapsulating different psychometric or quantum-theoretic priorities.

Fidelity, as mentioned, is a similarity measure between two quantum states. When states are pure, fidelity reduces to the squared magnitude of their inner product. The fidelity loss is defined as the following:

$${\mathcal{L}}_{\mathrm{fidelity}}=1-F\left({\rho }_{\mathrm{pred}},{\rho }_{\mathrm{true}}\right)$$

where $F\left({\rho }_{\mathrm{pred}},{\rho }_{\mathrm{true}}\right)$ is the fidelity between the predicted state ${\rho }_{\mathrm{pred}}$ and the true state ${\rho }_{\mathrm{true}}$. Minimizing ${\mathcal{L}}_{\mathrm{fidelity}}$ encourages the predicted state to align closely with the target state derived from the data.

To control the degree of mixing in the latent states, especially when items are expected to be near pure or only moderately mixed, an entropy penalty [79] is imposed using the following:

$${\mathcal{L}}_{\mathrm{entropy}}=\beta {\sum }_{k}S\left({\rho }_{q_k}\right)=\beta {\sum }_k\left(-\mathrm{Tr}\left[{\rho }_{q_k}\mathrm{l}\mathrm{o}\mathrm{g}{\rho }_{q_k}\right]\right),$$

(3)

where $\mathsf{\beta }\ge 0$ is a hyperparameter. A larger $\mathsf{\beta }$ pushes the model toward low-entropy states, effectively encouraging items to be more factor specific. Conversely, a lower $\mathsf{\beta }$ allows for more mixed states, potentially capturing items that genuinely span multiple factors.

In autoencoder-like architectures [80] where the network aims to reconstruct the initial amplitudes or density matrices, a reconstruction loss such as the Frobenius norm difference is defined as follows:

$${\mathcal{L}}_{\mathrm{reconstruction}}={\sum }_k\Vert {\rho }_{\mathrm{input},k}-{\rho }_{\mathrm{reconstructed},k}{\Vert }_F,$$

where

$$|A{|}_F=\sqrt{{\sum }_{i,j}{\left|A_{ij}\right|}^2}.$$

For pure states, a simpler Euclidean norm of amplitude differences may suffice.

Typically, these losses are combined to form a total loss function:

$${\mathcal{L}}_{\mathrm{total}}=\alpha {\mathcal{L}}_{\mathrm{fidelity}}+\beta {\mathcal{L}}_{\mathrm{entropy}}+\gamma {\mathcal{L}}_{\mathrm{reconstruction}}.$$

where α, β, and γ are non-negative hyperparameters tuned through cross-validation or optimization techniques. Each term addresses a different aspect of model fit: fidelity alignment, entropy control, and reconstruction quality, respectively.

The training procedure involves initializing the model parameters based on data-driven assignments, selecting an appropriate optimizer such as Adam, RMSProp, or Adagrad [81], and enforcing normalization constraints after each gradient update to maintain valid quantum states. Monitoring involves tracking fidelity, entropy, and reconstruction error over training epochs to assess convergence and model performance.

The proposed quantum-state representations are compatible with widely used machine learning frameworks, such as TensorFlow and PyTorch, which support complex-valued computations and automatic differentiation. To ensure computational tractability, researchers can integrate adaptive learning rate schedules, gradient clipping, and low-rank approximations of density matrices to maintain efficiency in large-scale implementations.

By judiciously balancing these objective terms, the quantum latent variable model can be tailored to prioritize optimal fit to observed quantum states, appropriate levels of mixing, and reconstruction fidelity.

3. Implementation Strategies: Post Hoc vs. Quantum-Specific Questionnaires

Applying quantum-inspired frameworks to psychometric research necessitates a strategic decision: adapting existing questionnaires using post hoc techniques or developing new instruments explicitly designed for quantum contexts. Each approach has distinct implications for validity, interpretability, and the capacity to identify interference or context effects inherent to quantum frameworks.

3.1. Post Hoc Application of Quantum Frameworks

The post hoc approach preserves established questionnaires’ original structure and wording, avoiding modifications to item text or administration procedures—data collection proceeds as usual, mapping the numerical responses to quantum states through predefined transformation methods. For example, in a questionnaire with items rated on a 1–7 scale, each response is normalized to fit within the interval [0, 1] and projected onto a quantum state representation.

This method allows researchers to compare quantum-inspired analyses with classical factor analysis while maintaining the integrity and comparability of established psychometric scales. However, this approach might not fully exploit quantum-specific phenomena, such as interference or non-commutativity, as existing questionnaires are typically designed under classical assumptions. These limitations may obscure subtle effects, particularly when item order or contextual influences are randomized or static.

3.2. Quantum-Specific Questionnaire Design

An alternative strategy is to design questionnaires that explicitly incorporate quantum principles, such as context dependence and order sensitivity. This involves modifying items to reference prior responses or eliciting reflection on earlier answers. For instance, the Satisfaction with Life Scale (SWLS) [82] could be adapted to include contextual prompts, explicitly linking each item to previous reflections. Such modifications aim to elicit quantum-like phenomena, including interference and context shifts, which classical models might overlook.

While quantum-specific instruments provide deeper insight into quantum effects, they also introduce additional complexity. New item designs may reduce comparability with classical instruments and demand more sophisticated data analysis methods. The loss of standardization may present challenges for researchers aiming to align findings with existing literature.

3.3. A Hybrid Strategy: Combining Post Hoc and Quantum-Specific Approaches

A pragmatic strategy involves combining both approaches. Researchers can apply quantum frameworks post hoc to established questionnaires to ensure comparability and validate the utility of quantum metrics, such as fidelity and entropy [83]. Concurrently, they can pilot new, explicitly quantum-sensitive instruments to explore the potential of non-commutative measurement designs. This dual approach enables researchers to extract meaningful insights from existing tools while simultaneously testing novel quantum constructs.

Example of Adapting the Satisfaction with Life Scale (SWLS): to illustrate these approaches, consider the SWLS, a five-item scale assessing subjective well-being.

Original Version: The traditional SWLS items remain unchanged and are mapped into quantum states post hoc. A single overarching factor, such as “Life Satisfaction,” can be modeled, or items can be distributed across multiple sub-factors. This setup typically initializes phases to zero unless specific interference hypotheses are tested. While classical psychometric models do not explicitly incorporate phase information, in our proposed quantum-inspired framework initial conditions may be set as ${\mathsf{\varphi }}_{k,i}$= 0 for simplification before refinement during optimization. Metrics like fidelity and von Neumann entropy offer new perspectives on item relationships and multidimensionality.

Quantum-Specific Version: Each item in the SWLS is modified to reference previous responses, encouraging order effects and cross-item influences. For instance:

• The first item (“In most ways my life is close to my ideal”.) could include a prompt for reflection on recent accomplishments.
• Subsequent items would explicitly tie responses to prior reflections, such as: “Considering your previous response about life’s ideal, how strongly do you agree that the conditions of your life are excellent?”
• This approach highlights interference and context shifts, potentially revealing patterns misaligned with classical assumptions.

Implications and Recommendations: While the post hoc method ensures continuity with psychometric practices, it may underutilize the quantum framework’s potential to capture non-classical phenomena. Conversely, quantum-specific designs offer richer insights into order dependence and context sensitivity but require methodological innovations and introduce complexity. A hybrid approach balances these trade-offs, leveraging the strengths of both strategies to advance quantum-inspired psychometric research (

Table 3. Comparison of post hoc application, quantum-specific design, and hybrid approach for SWLS example.

Aspect	Post Hoc Application	Quantum-Specific Design	Hybrid Approach
Definition	Use the original SWLS items as-is and map responses to quantum states for analysis.	Modify SWLS items to include contextual prompts referencing prior responses to elicit quantum effects.	Apply quantum analysis to the original SWLS while piloting a modified, contextual version to explore quantum effects.
Implementation	Normalize responses (e.g., 1–7 scale to [0, 1]) and map each item to quantum states without altering wording.	Rephrase SWLS items to incorporate context, e.g., reference prior answers to encourage order dependence and reflection.	Combine the two approaches: analyze the original SWLS for comparability while testing the modified version for insights.
Example Items	Item: ‘In most ways my life is close to my ideal.’	Modified: ‘Reflecting on your recent accomplishments, how close do you feel your life is to your ideal?’	Analyze the original item and compare results to the modified contextual version to assess quantum effects.
Focus	Measures fidelity, entropy, and overlap among items in the original scale.	Detects interference, order sensitivity, and context effects not evident in the original scale.	Compares findings from original and modified scales to assess quantum metrics and test novel hypotheses.
Advantages	- Retains comparability with existing SWLS research. - Requires no changes to item wording or sequence.	- Highlights quantum-specific effects like interference. - Tailored to test quantum hypotheses explicitly.	- Balances standard psychometric comparability with exploratory quantum-inspired insights.
Limitations	- Classical design may obscure genuine quantum effects.	- Reduced comparability with prior SWLS studies. - Increases complexity in design and analysis.	- Additional resources are required for simultaneous testing and analysis of two versions.

).

4. Discussion

4.1. Addressing Research Questions

How can the principles of Hilbert space representation, such as orthonormality and normalization, be effectively utilized to model latent factors in psychometric data?

The results underscore the efficacy of Hilbert space representation as a foundational framework for modeling latent factors in psychometric research [57,58]. By conceptualizing latent constructs [84] as vectors within a complex Hilbert space, this approach provides a structured yet flexible paradigm for capturing the relationships between psychometric items and latent factors. The orthonormal basis, a defining characteristic of Hilbert spaces, ensures that latent factors remain distinct and mutually exclusive, preserving the interpretative clarity of multidimensional constructs. Meanwhile, the normalization condition introduces a probabilistic interpretation, guaranteeing that the sum of squared amplitudes equals unity, which aligns with the probabilistic nature of psychological phenomena.

This dual emphasis on orthonormality and normalization allows for a nuanced representation of latent constructs that accommodates both the magnitude and phase of associations. For instance, Likert-scale [85] responses were effectively translated into normalized amplitudes, while complex coefficients were used to encode additional contextual or interactional information. This advancement extends beyond the capacities of classical factor analytic methods [86], offering a richer and more comprehensive depiction of latent factor structures.

Under what conditions do pure state representations of questionnaire items suffice, and when are mixed states required to account for multimodal or context-dependent response patterns in psychometric assessments?

The findings delineate clear conditions for applying pure- and mixed-state representations in psychometric contexts. Pure states are most effective when psychometric items exhibit unimodal response distributions and minimal variability in interpretation. These states capture the alignment of questionnaire items with latent factors straightforwardly, providing a precise and focused representation.

In contrast, mixed states, represented by density matrices, are indispensable for modeling items subject to multimodal response distributions or context-dependent variability. For instance, when a single item elicited varying interpretations across demographic groups or situational contexts, the mixed-state representation encapsulated this heterogeneity through a probabilistic mixture of pure states. Additionally, off-diagonal elements in the density matrix [87] encoded coherence effects, further enhancing the model’s sensitivity to contextual dynamics. This distinction is particularly critical for capturing the complexity of psychological constructs such as emotional ambivalence, where simultaneous alignment with multiple latent factors is a defining feature.

Compared to classical correlation-based measures, how do quantum-specific metrics like fidelity, overlap, and von Neumann entropy provide insights into the relationship between psychometric items and latent constructs?

The introduction of quantum-specific metrics, such as fidelity and von Neumann entropy, extends beyond classical psychometric techniques and offers novel tools for latent variable analysis. Prior work in quantum probability has explored similar principles [88], yet their direct application to psychometric structures marks a significant advancement. These metrics further interrogate the relationships between items and latent constructs by incorporating magnitude and phase information—dimensions often overlooked in classical correlation-based measures.

Fidelity emerged as a robust measure of similarity between quantum representations of items, providing a more nuanced assessment of their latent alignments. Overlap, the inner product between quantum states, extended the analysis by quantifying shared variance while accounting for interference effects. Von Neumann’s entropy [89] proved particularly valuable in characterizing the “mixedness” of item-factor associations, effectively quantifying the heterogeneity of latent representations.

Collectively, these metrics offered insights into phenomena that traditional psychometric methods cannot adequately address, such as order effects and interference patterns.

What are the advantages and limitations of implementing complex-valued neural networks (CVNNs) versus real-valued networks with two channels for processing quantum-inspired psychometric data, particularly concerning preserving phase relationships and computational efficiency?

The study evaluated two machine learning architectures for processing quantum-inspired psychometric data: complex-valued neural networks (CVNNs) and real-valued networks with dual channels [90,91]. CVNNs demonstrated a unique capacity to preserve the phase relationships inherent to quantum states, enabling the modeling of interference and coherence effects critical to quantum-inspired frameworks. This architecture proved adept at capturing order-sensitive phenomena and subtle contextual dynamics in psychometric responses.

However, CVNNs faced practical challenges, including limited software support and high computational demands. These limitations necessitate further development of computational tools to facilitate the broader adoption of CVNNs in psychometric research. Real-valued networks with dual channels offered a more accessible alternative by separating quantum states’ real and imaginary components into distinct inputs. While this approach was computationally efficient, it could not fully represent phase-specific dynamics, limiting its applicability in studies focused on interference effects.

The specific computational constraints and research objectives should guide the selection of a neural network architecture. Real-valued networks with dual-channel input provide a practical and computationally efficient alternative to CVNNs for large-scale psychometric datasets. However, for studies requiring the preservation of quantum phase relationships, CVNNs remain preferable despite their higher computational cost. Future research should explore hybrid architectures that balance computational efficiency with phase-sensitive modeling.

The findings suggest that the choice of architecture should align with the specific goals of the research. Real-valued networks may suffice for studies prioritizing computational efficiency and scalability. Conversely, investigations exploring phase-dependent phenomena should adopt CVNNs despite their higher computational complexity [71,72].

4.2. Theoretical Implications

This study contributes significantly to the theoretical landscape of multivariate analysis by introducing a quantum-inspired framework for latent variable modeling. Traditional psychometric methods often rely on static, deterministic assumptions, limiting their ability to capture psychological constructs’ inherent complexity and contextual dependencies [92]. This study redefines conceptualizing latent variables as dynamic and probabilistic entities by leveraging the Hilbert space representation, superposition, and non-commutative probabilities.

Incorporating quantum-specific metrics, such as fidelity and von Neumann entropy, expands the theoretical toolkit for evaluating psychometric items and their relationships with latent factors. These metrics provide a nuanced understanding of multidimensional constructs, capturing phenomena such as interference, order sensitivity, and context dependence, which remain unaddressed in classical frameworks. The distinction between pure and mixed states [93] also introduces a flexible modeling approach that adapts to the heterogeneity of psychological phenomena, from unimodal to multimodal response distributions.

This quantum framework also has broader implications for interdisciplinary theory development, bridging multivariate analysis with quantum mechanics, cognitive science, and artificial intelligence [94]. It challenges established notions of unidimensionality and fixed latent constructs, paving the way for more dynamic, context-sensitive, and probabilistic models that align closely with human cognition and behavior complexities [95].

4.3. Practical Implications

The quantum-inspired latent variable modeling framework proposed in this study has profound practical implications for psychometric assessment, survey design, and psychological measurement [92,96]. By allowing for the representation of latent constructs as dynamic and probabilistic entities, this approach enables researchers and practitioners to capture complexities such as contextual dependence, multimodal response patterns, and interference effects, often missed by classical methods.

The ability to distinguish between pure and mixed states provides a tailored approach to modeling questionnaire items. For example, pure state representations are suitable for well-defined, context-independent items, while mixed states address multimodal or context-sensitive responses, enhancing the robustness of psychometric tools. Quantum-specific metrics, such as fidelity and von Neumann entropy, offer new dimensions for evaluating item relationships and construct validity, providing deeper insights into item behavior and latent factor structures [88,89].

Moreover, this framework informs the design of advanced machine learning algorithms that can analyze complex psychometric data [67,68,69]. Complex-valued neural networks, though computationally intensive, enable the preservation of phase relationships, making them valuable for capturing interference and context effects. The practical application extends to developing quantum-specific questionnaires, which can elicit richer, context-aware data by leveraging order-sensitive or reflective item designs.

These advancements can potentially enhance psychological measurement’s precision, adaptability, and interpretative depth, informing both research and applied domains [97].

4.4. Limitations and Future Directions

While this study demonstrates the potential of quantum-inspired latent variable modeling, several limitations warrant consideration. First, the framework is inherently computationally intensive, particularly when implementing complex-valued neural networks (CVNNs) and processing large-scale psychometric data. The reliance on specialized algorithms and software tools, which may have limited support in existing libraries, poses challenges for broader adoption. Future research should focus on optimizing computational efficiency and developing accessible tools for implementing quantum-inspired models in diverse research settings.

Second, the current study primarily emphasizes theoretical constructs. Empirical validation of the quantum-inspired framework using real-world psychometric data is necessary to establish its practical applicability and generalizability. Future studies should involve large-scale empirical testing across various domains, such as education, clinical psychology, and organizational behavior, to assess the robustness of quantum-specific metrics and mixed-state modeling [98,99].

Third, interpreting quantum-specific metrics, such as phase relationships and entropy, requires advanced expertise, which may limit accessibility for practitioners unfamiliar with quantum mechanics [100,101]. Efforts to create user-friendly guidelines and visualization tools for these metrics could facilitate broader adoption.

Future directions include integrating this framework with other advanced methodologies, such as hybrid quantum–classical algorithms or interdisciplinary approaches involving cognitive science and artificial intelligence [102,103]. Additionally, exploring the application of quantum-inspired methods in domains beyond multivariate analysis, such as decision-making models or adaptive testing, offers exciting possibilities [104,105].

While this framework provides a novel paradigm for psychometric modeling, its benefits are most pronounced in cases where response patterns exhibit non-classical dependencies. In settings where classical factor models adequately capture psychometric structure, the computational complexity of quantum-inspired modeling may outweigh its advantages. Additionally, its reliance on Hilbert space representations and phase-based dependencies may not be optimal for datasets that do not exhibit interference effects or contextual variability in response distributions.

Addressing these limitations and advancing these directions will establish the quantum-inspired framework as a transformative tool in psychological measurement. This approach enhances our understanding and extends the scope of knowledge beyond earlier works, which primarily focused on decision theory and cognition through the lens of quantum probability [106,107,108,109,110].

5. Conclusions

This study introduces a quantum-inspired framework for latent variable modeling, offering a transformative approach to addressing limitations in traditional psychometric methods. By leveraging principles of Hilbert space representation, pure and mixed states, and quantum-specific metrics such as fidelity and von Neumann entropy, the framework captures the complexity, contextual dependence, and dynamic nature of psychological constructs. Integrating advanced machine learning architectures further enhances the potential for analyzing intricate psychometric data.

While theoretical advancements provide a strong foundation, computational complexity, empirical validation, and interpretability challenges remain. Addressing these limitations through future research will expand the applicability and accessibility of this approach.

Overall, this framework represents a significant step forward in multivariate analysis, bridging classical theories with quantum principles to offer richer, more adaptable models of psychological measurement. Its potential to redefine construct representation and analysis sets the stage for innovative advancements in research and practice.