---
title: CHSH Bell Tests & Quantum Nonlocality
url: https://www.emergentmind.com/topics/clauser-horne-shimony-holt-chsh-bell-inequality-tests
type: topic
---

# CHSH Bell Tests & Quantum Nonlocality

The Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality tests constitute a canonical approach for probing the fundamental distinctions between classical local realism and quantum theory, with significant relevance across quantum optics, condensed matter, quantum information, and relativistic physics. The CHSH framework rigorously characterizes the maximum correlations allowable under local hidden-variable models, provides a quantitative witness for entanglement and quantum nonlocality, and serves as the basis for device-independent protocols. The technical landscape encompasses precise mathematical formulation, numerous physical realizations, loophole analysis, multipartite extensions, connections to steering and cryptography, and foundational considerations regarding probability spaces and relativity.

## 1. Mathematical Formulation and Bounds

The CHSH inequality is constructed on four dichotomic observables, $A, A'$ for the "left" subsystem and $B, B'$ for the "right" subsystem. Each measurement yields outcomes $±1$ [2405.20789, 2302.14046]. The key correlation function is
$$
E(a, b) ≔ ⟨A(a) B(b)⟩,
$$
where $a, a', b, b'$ are the respective measurement settings. The CHSH "S-parameter" is defined by
$$
S ≔ E(a, b) - E(a, b') + E(a', b) + E(a', b').
$$
If a single joint probability distribution for all four observables exists and locality holds, classical (local hidden-variable) models satisfy the absolute bound
$$
|S| ≤ 2.
$$
Quantum mechanics, exploiting noncommuting measurements and entanglement, allows for an enhanced maximal violation,
$$
|S| ≤ 2\sqrt{2},
$$
the Tsirelson bound [2405.20789, 2302.14046, 1005.4966].

Table: Classical vs Quantum bounds

| Model                   | Maximal $|S|$ |
|--------------------------|:-----------:|
| Local Hidden Variables   |     2       |
| Quantum Mechanics        |  $2\sqrt{2}$|

Violation of the CHSH inequality (i.e., $|S| > 2$) is a strong fingerprint of nonlocality, incompatible with any local hidden-variable theory.

## 2. Physical Realizations and Experimental Protocols

CHSH Bell tests are performed in diverse platforms, ranging from polarization-entangled photon pairs, optomechanical devices, and time–frequency entangled photons, to hybrid continuous-discrete optical states and quantum field theories [2512.23550, 2406.04736, 1404.0867, 1803.00331, 1609.02282, 2307.04611, 2406.20033, 2212.14407].

- In quantum optics, polarization-entangled photons are measured along arbitrary axes or elliptical bases spreading across the full Poincaré–Bloch sphere. Maximal CHSH violations are observed when both analyzers use the same basis, regardless of ellipticity, with $|S| \approx 2.828$ at optimal settings. Certain non-Bell maximally entangled states violate the CHSH inequality only in mixed-basis configurations [2512.23550].
- Hybrid measurement protocols combining photon counting and homodyne detection on weakly amplified N00N states yield loss-tolerant violations up to $|S| \approx 2.423$ at realistic efficiencies, and close both detection and locality loopholes [1404.0867].
- Entangled coherent-state schemes achieve significant Bell-CHSH violations even with asymmetric loss and nonidentical local amplitudes, allowing for robust violations at lower overall detection thresholds [1501.00807, 2406.04736].
- Frequency-bin entangled narrowband biphotons, generated via four-wave mixing, reveal nonlocal temporal correlations with $|S| = 2.52 \pm 0.48$ when measured by time-resolved coincidence, witnessing time-frequency quantum nonlocality [1609.02282].
- CHSH Bell tests have been transposed to quantum field theory, with smeared Weyl operators localized in causally disjoint regions (Rindler wedges). Violations can approach arbitrarily close to the Tsirelson bound if appropriate supports and inner product structures are engineered. Both canonical and path-integral quantizations recover the same functional structure for the test [2307.04611, 2406.20033, 2212.14407].

## 3. Entanglement and Relationship to Steering

CHSH violation is a definitive indicator of entanglement, but the converse does not strictly hold: entangled two-qubit states may not violate the CHSH inequality (e.g., after enough amplitude damping or mixing) [1306.6504]. Quantitative relations between CHSH violation and entanglement measures (negativity, concurrence, relative entropy of entanglement) have been established, with extremal states characterized for fixed CHSH violation. The Horodecki criterion frames the maximal violation for any two-qubit state as $S_{\mathrm{max}} = 2 \sqrt{M(\rho)}$, where $M(\rho)$ is the sum of the two largest eigenvalues of the correlation matrix $T^T T$ [1306.6504, 2512.23550].

EPR-steering, an intermediate form of quantum nonclassicality, admits an analogue of the CHSH inequality: in the 2x2x2 scenario (two parties, two settings each, two outcomes each), necessary and sufficient steering inequalities slightly weaker than CHSH have been derived, relying on convex hulls of parametrized ellipses [1412.8178, 1601.01703]. Notably, in two-qubit systems, all states steerable via CHSH-type correlations are also Bell-nonlocal; the maximal violations coincide [1601.01703].

## 4. Loopholes and Randomness Requirements

Practical implementation of CHSH Bell tests must contend with the locality and detection loopholes, as well as the "free-will" (input randomness) loophole [1409.7875]. The security of device-independent quantum cryptographic protocols relying on CHSH tests is contingent on unpredictability and independence in measurement settings:

- *Perfect randomness* ($P=1/4$, where $P$ is the max conditional probability of any input setting) is required to completely close the free-will loophole. Any deviation can allow an adversary to simulate violations via local deterministic strategies.
- For multiple-run scenarios, the threshold for secure randomness is significantly more stringent; correlations across runs can reduce the per-run randomness needed to $P < 0.258$ to guarantee genuine nonlocality [1409.7875].
- Device-independent cryptography, randomness amplification, and loophole-free Bell tests necessitate monitoring and estimation of the min-entropy of the random number generators; astrophysical sources of randomness have been proposed.

Table: Thresholds for randomness in secure CHSH tests

| Scenario                      | Randomness Threshold $P$ |
|-------------------------------|:-----------------------:|
| Single-run, uncorrelated      |        0.354            |
| Single-run, correlated        |        0.285            |
| Multi-run, fully correlated   |        0.258            |
| Multi-run, uncorrelated       |       ≤ 0.264           |

## 5. Generalizations, Multipartite Extensions, and Probability Spaces

Systematic methods for constructing Bell-like inequalities generalizing CHSH include recursive, polytope-facet based approaches [1005.4966, 1302.6698]. This yields families of tight inequalities for arbitrary numbers of parties and settings, with compact forms (involving only four correlation functions) available for higher-order GHZ entanglement, enabling practical multi-photon experiments. The geometry of these inequalities is mapped to facets of the correlation polytope.

The treatment of the underlying probability space—whether or not mixing data from different measurement contexts affects the CHSH bound—has been a subject of debate [1602.02628]. Explicit construction of a unified Kolmogorov probability space embedding all four measurement contexts recovers exactly the standard CHSH inequality, affirming that the experimentally observed violations are inconsistent with any local-realistic model irrespective of data mixing.

## 6. Relativistic Approaches and Weak Nonlocality

Recent theoretical advances demonstrate that CHSH inequality violations can emerge in purely relativistic (non-quantum) classical settings due to the global structure of special relativity [2405.20789]. In relativistic "Gedankenexperiments," violations are observed as a consequence of the relativity of simultaneity; no absolute present exists, precluding the construction of a single four-variable joint probability distribution. This form of "weak nonlocality" is strictly kinematic, lacking any entanglement or superluminal signaling, and does not approach the Tsirelson bound. It is weaker than quantum nonlocality but points to deep parallels between quantum mechanics and relativistic spacetime geometry.

Table: Nonlocality strengths in CHSH violations

| Source              | Max $|S|$   | Nature           |
|---------------------|:---------:|------------------|
| Quantum Entanglement|   $2\sqrt{2}$ | Dynamical, strong|
| Relativity (SRT)    |   $2.109$    | Kinematic, weak  |

Formal analogies exist: both quantum mechanics and special relativity forbid single joint probability distributions when events/measurements exceed three variables; but their origins are distinct—the algebra of noncommuting observables vs. the geometry of spacetime [2405.20789].

## 7. Foundational Impact, Open Problems, and Future Directions

CHSH Bell-inequality tests serve as standard witnesses for quantum nonlocality, device-independent entanglement estimation, and tests of multi-qubit entanglement. Recent work demonstrates efficient measurement strategies, robustness against loss, and the extension to high-dimensional, continuous-variable, and relativistic regimes [1501.00807, 2406.04736, 1306.6504, 2307.04611, 2212.14407]. Open problems include tightness and existence questions for generalized inequalities and reduction of practical complexity in multi-setting experiments [1005.4966, 1302.6698].

The CHSH framework now underpins foundational quantum protocols, including QKD schemes, steering verification, multipartite entanglement certification, quantum field theory tests, and deeper investigations into the geometry and causal structure of spacetime and quantum information.

## References

Selected representative arXiv references:

- "Bell inequalities violation in relativity theory" [2405.20789]
- "Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario" [1409.7875]
- "Bell inequality tests using asymmetric entangled coherent states in asymmetric lossy environments" [1501.00807]
- "Violation of Bell's Inequality in the Clauser-Horne-Shimony-Holt Form with Entangled Quantum States Revisited" [2302.14046]
- "Maximal violation of the Bell-Clauser-Horne-Shimony-Holt inequality via bumpified Haar wavelets" [2307.04611]
- "CHSH Inequality on a single probability space" [1602.02628]
- "Entanglement estimation from Bell inequality violation" [1306.6504]
- "Numerical approach to the Bell-Clauser-Horne-Shimony-Holt inequality in quantum field theory" [2406.20033]
- "Does CHSH inequality test the model of local hidden variables?" [1204.3437]
- "All two-qubit states that are steerable via Clauser-Horne-Shimony-Holt-type correlations are Bell nonlocal" [1601.01703]
- "Clauser-Horne-Shimony-Holt Bell-inequality Violability with the Full Poincaré-Bloch Sphere" [2512.23550]
- "Clauser-Horne-Shimony-Holt Bell inequality test in an optomechanical device" [1803.00331]
- "On the Feynman path integral formulation of the Bell-Clauser-Horne-Shimony-Holt inequality in Quantum Field Theory" [2212.14407]
- "Analog of the Clauser-Horne-Shimony-Holt inequality for steering" [1412.8178]
- "CHSH Bell Tests For Optical Hybrid Entanglement" [2406.04736]
- "Bell inequality of frequency-bin entangled photon pairs with time-resolved detection" [1609.02282]
- "Compact Bell inequalities for multipartite experiments" [1302.6698]

Source: https://www.emergentmind.com/topics/clauser-horne-shimony-holt-chsh-bell-inequality-tests