Bell Inequality Violations Explained

• Bell inequality violations are phenomena where measured quantum correlations exceed classical limits, demonstrating nonlocality.
• They are tested through protocols like CHSH and CGLMP, often requiring careful treatment of sampling, postselection, and timing loopholes.
• These violations not only challenge classical theories but also enable certification of higher-dimensional entanglement in advanced quantum systems.

Bell inequality violations are instances in which measured or calculated correlations exceed bounds such as the CHSH limit $S \leq 2$, the Clauser-Horne bound, the CGLMP limits $S_d \leq 2$ or $I_3 \leq 2$, or related multipartite constraints that hold for local realistic or local deterministic descriptions. In quantum mechanics, the canonical CHSH maximum is Tsirelson’s bound $2\sqrt{2}$, but the evidential force of any observed excess depends on auxiliary assumptions about fair sampling, coincidence assignment, postselection, and spacetime arrangement of the measurements (Dada et al., 2011, Larsson et al., 2013, 2002.03935).

1. Canonical inequalities and their quantum benchmarks

The standard bipartite reference point is the CHSH inequality for two parties, two settings per party, and dichotomic outcomes $\pm 1$: $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$ with, for example,

$E(a,c)=p(a=c)-p(a=-c).$

Within quantum mechanics the maximum achievable value is $S_{QM}^{max}=2\sqrt{2}$, Tsirelson’s bound; an experimental value above $2\sqrt{2}$ is therefore not evidence for “stronger-than-quantum” correlations but signals a defect in the analysis assumptions (Dada et al., 2011).

In photonic experiments that seek to avoid a fair-sampling assumption, the Clauser-Horne form is central. One such form is

$$P(A_1 = B_2 = 1) + P(A_2 = B_1 = 1) - P(A_2 = B_2 = 1) \le P(A_1 = 1) + P(B_1 = 1) - P(A_1 = B_1 = 1),$$

and is explicitly used because it relies only on realism, locality, and freedom of choice while permitting nondetection events to remain part of the tested data model (Larsson et al., 2013).

For higher local dimension, the Collins-Gisin-Linden-Massar-Popescu family provides the standard qudit generalization. In the notation used for high-dimensional systems, $S_d \leq 2$0, while for two qutrits the CGLMP expression is written as $S_d \leq 2$1 for any deterministic local realistic theory, with the Bell parameter evaluated as $S_d \leq 2$2 (Dada et al., 2011, Fabbrichesi et al., 2023).

2. Sampling, postselection, and loopholes

Fair sampling is the condition that the detected subset of events be representative of all events. In operator language, if $S_d \leq 2$3 denotes the measurement operator for setting $S_d \leq 2$4 and outcome $S_d \leq 2$5, and $S_d \leq 2$6, then fair sampling is satisfied iff

$S_d \leq 2$7

for all $S_d \leq 2$8, with $S_d \leq 2$9 an efficiency factor and $I_3 \leq 2$0 a fixed operator on the relevant Hilbert space. In high-dimensional photonic systems, including orbital-angular-momentum implementations, different settings may probe different subspaces even when detection is perfect in the subspace probed. The result can be normalization artifacts and anomalous Bell values. A key example is a four-dimensional separable state for which badly aligned settings yield $I_3 \leq 2$1, the algebraic maximum, purely because fair sampling is violated (Dada et al., 2011).

Postselection generates a distinct but related pathology. In a Bell-basis postselection scenario, Alice and Bob can independently prepare qubits and send them to a third party who performs a Bell measurement. For each Bell-state outcome, the corresponding subensemble can violate a CHSH inequality with the maximal quantum value $I_3 \leq 2$2, while the full ensemble remains uncorrelated with $I_3 \leq 2$3. If the postselecting party has access to preparation information, the same effect can be classically mimicked, even up to $I_3 \leq 2$4. A proposed remedy is to place preparation and postselection at spacelike separation so that no classical information about the preparations can reach the postselection event; in that configuration the distinction between preselection and postselection becomes foliation-dependent, which the paper terms the “relativity of pre- and postselection” (2002.03935).

The coincidence-time loophole concerns how detection events are paired into “coincidences.” Two rigorous cures are the fixed-time-slots method and the window-sum method. Applied to entangled-photon data, both remove the need for a fair-coincidence assumption and preserve the validity of the Clauser-Horne test. For continuously emitting sources, a more general construction uses Bell functions built from signed, directed distances between the parties’ entire timetag sequences over synchronized observation periods; because the inequality is defined on the full outcome strings rather than on inferred pairings, settings-dependent timing errors cannot manufacture a Bell violation (Larsson et al., 2013, Knill et al., 2014).

3. High-dimensional violations and entanglement-dimension certification

In high-dimensional systems, Bell violations can be used not only to refute local realism under appropriate assumptions but also to certify entanglement dimension. For the generalized Bell operator $I_3 \leq 2$5, the derived bound for excluding Schmidt number at most $I_3 \leq 2$6 is

$I_3 \leq 2$7

where $I_3 \leq 2$8 is the maximal eigenstate, $I_3 \leq 2$9 is the closest state with Schmidt number at most $2\sqrt{2}$0, and $2\sqrt{2}$1 are the two largest eigenvalues. Only when fair sampling is satisfied does exceeding this threshold certify genuine $2\sqrt{2}$2-dimensional entanglement (Dada et al., 2011).

For $2\sqrt{2}$3, the reported numbers are $2\sqrt{2}$4, $2\sqrt{2}$5, and $2\sqrt{2}$6. Hence any observed value $2\sqrt{2}$7 certifies at least four-dimensional entanglement. Bounds were provided up to $2\sqrt{2}$8, and for low dimensions, especially $2\sqrt{2}$9, the maximal value for maximally entangled states exceeds the $\pm 1$0-dimensional bound, making experimental discrimination feasible with moderate precision. The paper further notes that the results reported by Dada et al. for orbital-angular-momentum entanglement in $\pm 1$1 exceeded the relevant bound and thereby certified four-dimensional entanglement (Dada et al., 2011).

A complementary line of work studies whether Bell violations are typical in high dimension. For a generalized CGLMP inequality with two parties, two measurement options, and $\pm 1$2 outcomes, quantum violation is characterized by $\pm 1$3. For Haar- or Hilbert-Schmidt-random pure states, the expected value satisfies $\pm 1$4 for small $\pm 1$5 and $\pm 1$6 for large $\pm 1$7, with the transition around $\pm 1$8. Explicitly, $\pm 1$9, while $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$0, and the variance vanishes as $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$1, so Bell violations become typical (Atkin et al., 2014).

Random mutually unbiased bases provide another route to high-dimensional nonlocality. For maximally entangled qutrits and ququarts, numerical estimates indicate near-guaranteed Bell violation using only two-setting Bell inequalities when each side can measure a sufficient number of random MUBs. For maximally entangled ququints, the success probability is reported as approximately $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$2. The rare no-violation instances under two-setting tests were found to violate some more-setting Bell inequalities, suggesting that Bell nonlocality remains operationally accessible even when the parties do not share a common reference frame (Tabia et al., 2022).

4. Experimental realizations across platforms

Bell inequality violations have been reported or witnessed across photonic, atomic, solid-state, optomechanical, continuous-variable, NISQ, and high-energy platforms. Representative results are summarized below.

Platform	Tested quantity	Reported result
Photonic CH analysis (Larsson et al., 2013)	Clauser-Horne $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$3	$S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$4 with fixed-time-slots; $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$5 with window-sum
Continuous-variable homodyne test (Thearle et al., 2018)	Bell value $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$6	$S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$7
Ultracold helium atoms (Shin et al., 2018)	Bell witness	$S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$8; $S = |E(a,c) + E(a,d) + E(b,c) - E(b,d)| \leq 2,$9
Spins in silicon (Dehollain et al., 2015)	CHSH $E(a,c)=p(a=c)-p(a=-c).$0	$E(a,c)=p(a=c)-p(a=-c).$1; QND parity mapping gives $E(a,c)=p(a=c)-p(a=-c).$2
Optomechanical oscillators (Marinkovic et al., 2018)	CHSH $E(a,c)=p(a=c)-p(a=-c).$3	$E(a,c)=p(a=c)-p(a=-c).$4
Small NISQ computers (Naus et al., 2020)	CHSH sum	$E(a,c)=p(a=c)-p(a=-c).$5 on the best IBM backend reported
$E(a,c)=p(a=c)-p(a=-c).$6 decays (Fabbrichesi et al., 2023)	CGLMP $E(a,c)=p(a=c)-p(a=-c).$7	$E(a,c)=p(a=c)-p(a=-c).$8, exceeding the local bound by more than $E(a,c)=p(a=c)-p(a=-c).$9
Charmonium decays (Fabbrichesi et al., 2024)	Bell violation significance	$S_{QM}^{max}=2\sqrt{2}$0 or more in multiple channels

These results span very different measurement architectures. In the ultracold-helium experiment, only common global rotations were implemented, so a direct CHSH test was not performed, but a strong Bell correlation witness and a necessary precondition for future CHSH violation were demonstrated. In silicon, sequential electron-nuclear-spin readout and quantum non-demolition parity mapping pushed the Bell signal to $S_{QM}^{max}=2\sqrt{2}$1, while full two-qubit tomography showed fidelities above $S_{QM}^{max}=2\sqrt{2}$2. In the optomechanical experiment, the Bell test involved light-matter entanglement with the vibrational motion of two silicon oscillators, each comprising approximately $S_{QM}^{max}=2\sqrt{2}$3 atoms, and the reported violation held under the fair-sampling assumption (Shin et al., 2018, Dehollain et al., 2015, Marinkovic et al., 2018).

High-energy collider analyses extend Bell tests into regimes involving strong and weak interactions. Using LHCb helicity amplitudes for $S_{QM}^{max}=2\sqrt{2}$4, the reconstructed qutrit $S_{QM}^{max}=2\sqrt{2}$5 qutrit polarization state yielded entanglement $S_{QM}^{max}=2\sqrt{2}$6 and Bell value $S_{QM}^{max}=2\sqrt{2}$7. A separate analysis of BESIII charmonium decays reported Bell inequality violations with significance $S_{QM}^{max}=2\sqrt{2}$8 or more in multiple baryon-antibaryon and $S_{QM}^{max}=2\sqrt{2}$9 channels, and noted that the long lifetimes of strange baryons provide a natural probe of whether spin entanglement survives interactions with the beam pipe and first detector layers (Fabbrichesi et al., 2023, Fabbrichesi et al., 2024).

On the theoretical side, hybrid particle-wave optical schemes have been proposed in which photon counting is combined with homodyne detection. For the two-photon state

$2\sqrt{2}$0

the optimized CHSH value is $2\sqrt{2}$1, while for the two-mode squeezed state the optimized value is $2\sqrt{2}$2 (Cavalcanti et al., 2010).

5. Relativistic and quantum-field-theoretic formulations

Bell inequality violations also appear in explicitly relativistic analyses. One relativistic Gedankenexperiment, built entirely within Special Relativity Theory and classical probability assignments for “alive/dead” variables, derives a CHSH expression

$2\sqrt{2}$3

and reports an explicit violation

$2\sqrt{2}$4

The claimed origin is the relativity of simultaneity and the resulting impossibility of establishing a universal joint probability distribution for all four spacelike-separated events. The paper characterizes the effect as “weak nonlocality,” distinct from quantum entanglement and superposition (Belinsky et al., 2024).

In algebraic quantum field theory, Bell-CHSH violation has been established in the vacuum state of a free massive real scalar field using Weyl operators and Tomita-Takesaki theory. With Hermitian observables constructed from localized Weyl operators and Bell operator

$2\sqrt{2}$5

the vacuum expectation takes the form

$2\sqrt{2}$6

Tomita-Takesaki modular conjugation supplies the map $2\sqrt{2}$7, which transfers Alice’s operators into the commutant algebra associated with Bob’s spacelike-complementary region. Suitable test functions then produce $2\sqrt{2}$8, showing Bell-CHSH violation entirely within the field-theoretic formalism (Fabritiis et al., 2023).

6. Multipartite trade-offs and interpretive disputes

For networks of multiple qubit parties, Bell violations obey trade-off and monogamy relations rather than unrestricted accumulation. The simplest example is

$2\sqrt{2}$9

in quantum theory. More generally, for families of $$P(A_1 = B_2 = 1) + P(A_2 = B_1 = 1) - P(A_2 = B_2 = 1) \le P(A_1 = 1) + P(B_1 = 1) - P(A_1 = B_1 = 1),$$0-party full-correlation Bell expressions in a network represented by a hypergraph $$P(A_1 = B_2 = 1) + P(A_2 = B_1 = 1) - P(A_2 = B_2 = 1) \le P(A_1 = 1) + P(B_1 = 1) - P(A_1 = B_1 = 1),$$1, tight hyperspherical relations of the form

$$P(A_1 = B_2 = 1) + P(A_2 = B_1 = 1) - P(A_2 = B_2 = 1) \le P(A_1 = 1) + P(B_1 = 1) - P(A_1 = B_1 = 1),$$2

were derived in several configurations, while a generalized Svetlichny inequality was shown to exhibit monogamy for genuine multipartite nonlocality. These relations have direct applications to device-independent secret sharing, device-independent randomness extraction, and the identification of flat regions in the set of quantum correlations (Ramanathan et al., 2017).

A distinct foundational program investigates how much of Bell’s theorem survives without full Locality. Starting from Quantum Mechanics and Classical Microscopic Realism, and replacing Locality by weak assumptions such as H1 (Regularity), H2 (Transversality), H1′, and Local Dominance, one can still derive Bell- or Boole-type contradictions for suitable quadruplets of coplanar measurement directions. The conclusion is explicitly summarized as

$$P(A_1 = B_2 = 1) + P(A_2 = B_1 = 1) - P(A_2 = B_2 = 1) \le P(A_1 = 1) + P(B_1 = 1) - P(A_1 = B_1 = 1),$$3

which is presented as a strengthening of Bell’s theorem because the contradiction does not require full Locality (Faria et al., 2012).

There is, however, an active interpretive dispute about what Bell inequality violations logically exclude. One paper argues that even loophole-free Bell-CHSH violations do not disprove local realism because Bell’s factorized correlation structure is a sufficient but not necessary condition for local realism, and constructs three local-realist models that reproduce Bell-CHSH violations by using more general tensor-product vector-space structures (Zela, 2024). A separate recent critique of claims of “Bell inequality violation without entanglement” argues that the effect is “just postselection”: a classical analog reproduces essentially the same results because postselection induces setting-dependent hidden-variable distributions and thereby rejects Bell’s Statistical Independence assumption $$P(A_1 = B_2 = 1) + P(A_2 = B_1 = 1) - P(A_2 = B_2 = 1) \le P(A_1 = 1) + P(B_1 = 1) - P(A_1 = B_1 = 1),$$4 (Wharton et al., 19 Aug 2025). This suggests that Bell inequality violations are simultaneously empirical benchmarks and tests of a precisely delimited package of assumptions about locality, realism, measurement independence, and admissible data selection.