Local Hidden-Variable Models

• Local hidden-variable models are classical frameworks that simulate quantum correlations using shared variables and local response functions.
• They underpin foundational tests of Bell nonlocality, distinguishing entanglement from genuine quantum correlations and challenging classical interpretations.
• Recent studies employ analytic, SDP, and machine-learning methods to delineate quantum-local boundaries and construct explicit LHV decompositions.

A local hidden-variable (LHV) model is a classical probabilistic framework intended to reproduce quantum-mechanical predictions for measurement outcomes based on the assumption that all correlations are mediated by shared classical variables, with measurement outcomes determined locally conditioned on these variables. The LHV paradigm is central to the questions of quantum nonlocality, device-independent quantum information processing, and the precise distinction between quantum entanglement and Bell nonlocality. LHV models are rigorously formulated both for standard Bell scenarios and a wide range of generalizations, including multipartite, contextual, and network settings. Not every quantum state admits an LHV model for all possible measurements; the delineation of the quantum-local boundary and the explicit construction of LHV decompositions are active areas of research.

1. Formal Structure of LHV Models

Let $N$ denote the number of spatially separated parties, each able to perform local measurements. In the prototypical Bell scenario, a quantum state $\rho$ is distributed among these parties. Each party $j$ chooses a local setting $x_j$ (e.g., measurement axis or POVM label) and registers outcome $a_j$.

An LHV model asserts that the joint distribution of outcomes $P(a_1,\dots,a_N|x_1,\dots,x_N)$ can be decomposed as

$$P(a_1,\dots,a_N|x_1,\dots,x_N) = \int_\Lambda d\lambda\, \rho(\lambda)\prod_{j=1}^N q_j(a_j|x_j,\lambda)$$

where:

• $\Lambda$: the hidden-variable space, typically a compact topological or finite set;
• $\rho(\lambda)\geq0$, $\int_\Lambda d\lambda\,\rho(\lambda)=1$: normalized hidden-variable distribution;
• $q_j(a_j|x_j,\lambda)$: stochastic local response function, satisfying $\sum_{a_j}q_j(a_j|x_j,\lambda)=1$ for all $x_j,\lambda$.

In bipartite scenarios this reduces to the form given in (Augusiak et al., 2014, Bowles et al., 2014, Hirsch et al., 2015), while multipartite and network generalizations involve more intricate indexings and possible source dependencies (Silva et al., 2023). For general measurements, these response functions can define, e.g., outcome probabilities for POVMs (Cavalcanti et al., 2015).

The key locality constraint is that $q_j$ may depend only on the local setting $x_j$ and $\lambda$, not on other parties' settings or outcomes.

2. Historical and Conceptual Significance

The LHV model concept arises from attempts to attribute measurement statistics to pre-existing properties unaffected by spacelike-separated events, in line with Einstein locality. Bell's theorem (Augusiak et al., 2014) demonstrated that certain quantum states (notably the spin singlet) produce correlations that cannot be reproduced by any LHV model; specifically, quantum correlations can violate convex polytope bounds (Bell inequalities) satisfied by all LHV models.

Werner's model (Augusiak et al., 2014) marked a paradigm shift by showing that mixed entangled states (Werner states) can admit LHV models, i.e., are entangled but local—a crucial distinction from Bell's original results. Subsequent work extended these observations to multipartite, higher-dimensional and generalized-measurement settings.

3. Explicit Construction and Algorithmic Realization

Several families of LHV models have been constructed using analytic, geometric, and optimization-based methods:

• Werner's Model: For $d$-dimensional Werner states $\rho_W(d,p)$, a model exists for projective measurements when $p\leq (d-1)/d$; Alice's response is the outcome minimizing $|\langle\lambda|P_a|\lambda\rangle|$; Bob's is linear in $|\lambda\rangle$ (Augusiak et al., 2014, Bowles et al., 2014).
• Finite Shared Randomness: Bowles et al. (Bowles et al., 2014) quantifies the amount of randomness needed for LHV models (e.g., $3.58$ bits for two-qubit Werner states at visibility $\eta \leq 0.43$ via an icosahedral sampling).
• SDP/LP Hierarchies: Hirsch et al. (Hirsch et al., 2015) and Cavalcanti et al. (Cavalcanti et al., 2015) formulate the search for LHV models as a series of linear or semidefinite programs over finite measurement sets, converging to the full quantum set as the measurement set is made dense.
• Machine Learning Approaches: Modern methods use gradient-descent to optimize large parameterized families of response functions and hidden-variable assignments, enabling explicit LHV decompositions for general states and measurements, and numerically estimating locality thresholds (critical visibilities) (Selzam et al., 2024).

4. Structure in Generalized and Network Scenarios

LHV models extend to multipartite and network settings, where the factorization structure may involve arbitrary graphs of shared sources (Silva et al., 2023). For network-locality (bilocality, triangle network), the decomposition is

$$P(a_1,\dots,a_n|x_1,\dots,x_n) = \sum_{\lambda_1,\dots,\lambda_k} P(\lambda_1)\cdots P(\lambda_k) \prod_{i=1}^n P(a_i|x_i,\Lambda_i)$$

where each party $i$'s response may depend on several locally distributed sources $\Lambda_i$.

In generalized contextuality scenarios, the hierarchy of local sets depends subtly on the constraints imposed (contextuality, non-disturbance) (Mazzari et al., 2023). This leads to nested sets of behaviors ranging from strictly noncontextual LHV to more general (possibly "disturbing") hidden-variable models.

5. Notable Variants and Operational Aspects

• Detection-loophole and Data-rejection LHV Models: Pearle's model (Gill, 2015) achieves singlet correlations by restricting attention to detected pairs, using hidden variables to model both the outcomes and the detection events.
• Local Quasi Hidden-Variable Models (LqHV): Dropping the positivity constraint on the hidden-variable measure, every quantum correlation scenario can be expressed as a "local quasi hidden-variable" model (real—possibly negative—measure) (Loubenets, 2011). LHV models are a strict subclass with strictly classical measures.
• Finite Local Models for GHZ: In the context of the GHZ paradox, explicitly local finite models can reproduce the quantum predictions by embedding the detection inefficiency as a preassigned hidden property, as constructed systematically in the ESR framework (Garola et al., 2012).
• Optimal Detection-Efficiency LHV Models: Wang's geometric approach (Wang, 2014) constructs LHV models tailored to the minimal detection efficiency needed to close the detection loophole in Bell experiments, showing full equivalence to Bell-inequality thresholds.

6. Limitations, No-Go Results, and Open Problems

• Maximally Entangled States: Any HV model for maximally entangled bipartite states must have local averages coinciding with the quantum ones; "local parts" in a crypto-nonlocal decomposition must vanish (Ghirardi et al., 2012).
• Dynamical LHV Models: While static (instantaneous) LHV models may exist for families of states, constructing a deterministic, state-independent dynamical evolution for the underlying hidden variables consistent with all quantum dynamics is generally impossible for interacting systems, as shown by rigorous dimensionality constraints (Selzam et al., 18 Dec 2025). This reveals a new form of "dynamical" nonlocality even when all instantaneous states are local.
• Multipartite Nonlocality: There exist genuinely multipartite entangled (GME) states that admit fully local LHV models for all non-sequential measurements, although their nonlocality may be "activated" by sequences of measurements (hidden nonlocality) (Bowles et al., 2015). Determining the boundary between GME and genuinely multipartite nonlocality remains an open problem (Augusiak et al., 2014).
• Computational Barriers: Exhaustive construction of LHV models is algorithmically hard for large systems due to the exponential growth of deterministic strategies. SDP/LP and machine-learning-based frameworks provide tractable hierarchies, but exact theoretical characterization in high-dimensional and multipartite cases remains out of reach (Hirsch et al., 2015, Selzam et al., 2024).

7. Applications and Broader Implications

LHV models delimit the field of device-independent quantum information protocols, inform the design of Bell tests, and elucidate the relationship between quantum entanglement, classical simulability, and nonlocality. The search for LHV models in noisy, mixed, or restricted-measurement regimes is essential for certifying resources in quantum communication and for probing the foundational boundaries of quantum mechanics.

The development of explicit, efficient, and even simulatable LHV models for certain highly entangled states under restricted measurement sets demonstrates that nonlocality is a strictly stronger resource than entanglement (Anwar et al., 2014, Augusiak et al., 2014). Furthermore, the quantification of resources (e.g., amount of shared randomness, detection efficiency) required for LHV simulation has both physical and algorithmic importance (Bowles et al., 2014, Wang, 2014).

The ongoing refinement of constructive methods, the analysis of generalized and dynamic extensions of locality, and the pursuit of tight locality/nonlocality thresholds remain central in the theoretical and applied study of quantum correlations.