Experimental test of the Greenberger-Horne-Zeilinger-type paradoxes in and beyond graph states

Quantum mechanics exhibits intrinsic nonlocality that cannot be reconciled with local realism, so local-hidden-variable (LHV) models cannot reproduce all quantum predictions. Nonlocal correlations are resources with applications in computation, device-independent cryptography, and randomness generation. Bell inequalities (e.g., CHSH and Mermin-Ardehali-Belinskii-Klyshko) statistically reveal nonlocality. Beyond statistical violations, possibilistic nonlocality shows contradictions at the level of possible events, such as the GHZ paradox and Hardy’s paradox. GHZ-type paradoxes have been explored and demonstrated in various systems, including GHZ states, cluster states, and more general graph states. However, key open issues remained: (i) whether a GHZ-type paradox can uniquely verify a graph state; (ii) whether GHZ-type paradoxes exist for states not LU-equivalent to graph states; and (iii) how to efficiently test GHZ-type paradoxes experimentally via their Hardy-type equivalents. This work addresses these: it presents a unified construction for GHZ-type paradoxes in graph states; shows GHZ-type paradoxes exist beyond graph states (including certain single-qubit Clifford-equivalent graph states, their coherent superpositions, and convex mixtures); and provides an efficient experimental method by converting GHZ-type paradoxes into perfect Hardy-type paradoxes and measuring the success probability. These results enable applications in graph-state verification, entanglement detection, and GHZ-type steering paradoxes for mixed states.

Background and prior work referenced include: foundational nonlocality (Bell, CHSH), multipartite Bell inequalities (MABK), possibilistic nonlocality hierarchy via sheaf-theoretic approach, the original GHZ paradox for three qubits and Hardy’s paradox for two qubits and its generalizations to multi-setting, multipartite, and higher-dimensional systems. Earlier GHZ-type paradoxes were shown for GHZ states, linear cluster states, and graph states. Experimental tests include three-qubit GHZ nonlocality and proposals for fault-tolerant observation using non-abelian anyons. Prior works also explored Bell inequalities tailored to graph states, Schmidt measure for multipartite entanglement, and relationships between entanglement, steering, and Bell nonlocality. This study builds upon and unifies these strands, extending GHZ-type paradox constructions and introducing efficient experimental verification via Hardy-type paradox conversion.

Theory: The authors develop a unified construction of GHZ-type paradoxes for graph states using stabilizer formalism. Consider a connected undirected graph G with m vertices; stabilizers S_i are defined from Pauli operators with adjacency given by connectivity C_{ij}. The graph state |G⟩ is the common +1 eigenstate of all S_i (ground state of H = -∑S_i). Focusing on a vertex of degree n+1 (a universal vertex connected to n others), they construct a set of observables {E_i} (Table 1; explicit construction and proof in SI) such that: (a) in LHV models, each Pauli appears an even number of times across the chosen products, forcing the product of classical expectations to be +1; (b) quantum mechanically, products include an odd number of “three-stabilizer” structures producing an overall −1, giving a GHZ-type contradiction. Theorem 1 states that a GHZ-type paradox is obtained with |{E}| = n+1 for odd n and n for even n. A worked example is given for the 4-qubit linear cluster (2–1–3–4), recovering standard GHZ-type relations on |LC4⟩. The paradox can be transformed into a Bell-type inequality I_n ≤ n−1 for LHV, with maximal quantum value n+1; from spectral properties, the violation bounds the fidelity to the target graph state and witnesses entanglement. Beyond graph states: Using the E_i from the 4-qubit example, define H′ = (E1+E2+E4)+E3. Its ground space is twofold: |LC4⟩ and |LC4′⟩ = (I⊗I⊗I⊗σ_x)|LC4⟩. Coherent superpositions |LC4(θ)⟩ = cosθ|LC4⟩ + sinθ|LC4′⟩, as well as mixtures ρ(θ) = cos^2θ|LC4⟩⟨LC4| + sin^2θ|LC4′⟩⟨LC4′|, exhibit GHZ-type paradoxes; analogous constructions hold for extended GHZ states |GHZ4(θ)⟩ and mixtures ρ′(θ) of |GHZ4⟩ and |GHZ4′⟩. Steering paradox: For ρ(θ), an EPR-steering GHZ-type paradox is constructed with two-setting measurements by Alice (σ_z⊗σ_z and σ_x⊗σ_x), yielding a steering parameter I_s with LHS bound 1 and quantum value 2, giving a sharp contradiction (details in SI). Conversion to Hardy-type paradox: The GHZ-type paradox is converted to a perfect Hardy-type paradox characterized by Hardy constraints (certain joint-outcome probabilities are zero) with success probability P_suc ≈ 1 in QM and 0 in LHV, enabling efficient experimental verification by measuring probabilities instead of nondestructive sequential stabilizer measurements. Experiment: A photonic platform encodes four qubits in the polarization and path degrees of freedom of an entangled photon pair. Starting from |Φ+⟩=(|00⟩+|11⟩)/√2, PBS/controlled operations generate |GHZ4⟩ or, with Hadamards, a CZ, and postselection, |LC4⟩. Extended states |LC4(θ)⟩ and |GHZ4(θ)⟩ are prepared using a probabilistic non-unitary rotation R(θ)=cosθ + iσ_y sinθ on a polarization qubit. The setup includes beam displacers to define path qubits, Mach-Zehnder interferometers for joint path–polarization analysis, and HOM interference to benchmark indistinguishability (visibility 97.1%). Hardy-type paradox measurements implement the specified settings to obtain P_suc and check Hardy constraints; for steering, Bob measures joint observables conditioned on Alice’s settings, and spectral bounds of measured operators estimate the dominant eigenvalues of conditional states to quantify I_s.

- Unified construction: Theorem 1 provides a general method to build GHZ-type paradoxes for graph states with a universal vertex, subsuming prior constructions. For odd n with m=n, the constructed observables single out the target graph state (quantum state verification) and enable efficient entanglement detection via a derived Bell-type inequality I_n ≤ n−1 (LHV) with quantum maximum n+1. - Fidelity bounds: From the spectra of ∑f_i E_i, the fidelity with the target graph state is bounded between (I_n−n+3)/4 and (I_n+n+1)/(2n+2). Violating I_n ≤ n−1 guarantees at least 50% fidelity. For high-Schmidt-measure states (e.g., 5-qubit ring |RC5⟩ with measure 3), even I_n as low as 2.5 certifies entanglement. - Beyond graph states: Non-graph pure states |LC4(θ)⟩ (θ≠0,π/2) and mixed states ρ(θ), as well as extended GHZ states |GHZ4(θ)⟩ and mixtures ρ′(θ), all exhibit GHZ-type paradoxes, showing existence beyond graph states. - Steering paradox: A GHZ-type steering paradox is established for the mixed ρ(θ), with LHS prediction I_s^LHS=1 and QM prediction I_s^QM=2, giving a sharp contradiction. - Experimental demonstrations via Hardy-type paradoxes: Using a photonic setup, perfect Hardy-type paradoxes corresponding to GHZ-type paradoxes were tested for representative 4-qubit states. Measured success probabilities for observing LHV-impossible events: |LC4(0)⟩: 90.1%, |GHZ4(0)⟩: 94.4%, |GHZ4(π/4)⟩: 96.2%. Each paradox comprises four contradictory predictions; detection probabilities significantly exceed 75% thresholds and falsify LHV by at least 56.0, 72.7, and 93.9 standard deviations, respectively. - Three-qubit verification: Postselecting σ_z=+1 on the first qubit of |GHZ4(0)⟩ yields |GHZ3⟩ that exhibits the GHZ paradox with I_3=3.792, indicating at least 89.6% fidelity to the ideal |GHZ3⟩ and excluding W-type entanglement. - Steering experiment: The experimentally estimated steering parameter I_s = 1.805 ± 0.014 violates the LHS bound by 59 standard deviations, demonstrating EPR steering with the mixed state.

The unified stabilizer-based construction directly addresses whether GHZ-type paradoxes can certify graph states and detect entanglement: for odd n and m=n, the paradox uniquely identifies the target graph state and, when cast as a Bell-type inequality, provides both an entanglement witness and quantitative fidelity bounds. By introducing a Hamiltonian perspective on the GHZ-type observables, the work shows that coherent superpositions and convex mixtures of certain Clifford-related graph states also support GHZ-type paradoxes, thus extending GHZ-type contradictions beyond the graph-state class and to mixed states. The conversion of GHZ-type paradoxes into perfect Hardy-type paradoxes yields an efficient and experimentally feasible verification strategy based on measuring probabilities rather than nondestructive sequential stabilizer measurements. Experimentally, high success probabilities and strong statistical significances confirm the theoretical predictions for both nonlocality and steering, and the approach supports practical tasks such as state verification and multipartite entanglement detection even at moderate correlation strengths. The results suggest broader applicability in quantum information, including pseudo-telepathy and measurement-based quantum computation scenarios, where paradox-based nonclassicality can serve as a resource.

This work presents a unified framework for constructing GHZ-type paradoxes for graph states and extends their existence to non-graph pure and mixed states. Recasting the paradoxes as Bell-type inequalities enables graph-state verification, entanglement witnessing, and fidelity estimation. A GHZ-type steering paradox is also established for mixed states, enriching the landscape of quantum nonlocality beyond Bell nonlocality. By converting GHZ-type paradoxes into perfect Hardy-type paradoxes, the authors implement efficient photonic experiments that demonstrate strong contradictions with LHV and LHS models. Future directions include leveraging local complementation to further broaden state verification (applicable to all five-qubit graph states as shown in SI), scaling to larger systems with fusion/PBS/CZ gates, and applying paradox-based verification in tasks like error correction and magic-state distillation.

- Experimental demonstrations were performed primarily on four-qubit states (with a three-qubit test via postselection); scalability to larger systems, while in principle feasible with linear-optical gates, was not experimentally shown here. - Some state preparations rely on probabilistic, non-unitary operations and postselection (e.g., cluster-state generation with ~50% success), which can limit efficiency. - Experimental imperfections such as Mach–Zehnder interferometer instability and non-unity two-photon interference visibility contribute to deviations from ideal P_suc=1. - The theoretical construction requires a graph with a universal vertex for the most direct verification guarantees; applicability to arbitrary graphs without such vertices relies on additional tools (e.g., local complementation), discussed in SI but not fully explored experimentally.