This work provides an explicit, reproducible demonstration of how theClauser-Horne-Shimony-Holt (CHSH) inequality depends jointly on thechoice of Bell state AND measurement configuration in the spin-1/2qubit convention used by gate-based quantum processors. Main result. The combination of CHSH Form I, S = | E(a,b) - E(a,b') + E(a',b) + E(a',b') |, with the commonly-cited angular configuration {0, pi/2, pi/4, -pi/4}yields S = 0 for ALL FOUR Bell states in the spin-1/2 Bloch-sphereconvention, not selectively for one. This frequently misleadspractitioners who prepare |Phi+> on gate-based hardware and applytextbook angles, obtaining S = 0 and misdiagnosing the issue as ahardware fault, code bug, or absence of entanglement. Corrections provided. For the two states whose correlation functionin the XZ plane has argument theta_a - theta_b (|Phi+> and |Psi->),two minimal one-parameter fixes recover the Tsirelson bound 2*sqrt(2): 1. Retain the angles and apply CHSH Form II: S = | E(a,b) + E(a,b') + E(a',b) - E(a',b') |. 2. Retain Form I and replace b' = -pi/4 with b' = 3*pi/4. For the remaining two Bell states (|Phi-> and |Psi+>, whose correlationargument is theta_a + theta_b), neither single-parameter fix suffices;the optimal angles b and b' must be exchanged. A complete optimal-angletable for all four states is given in Section 4 of the manuscript. Contents. - paper.pdf / paper.tex: 8-page manuscript with full derivation, tables, code listing and appendix. - chsh_verification.py: self-contained Python script (NumPy only) reproducing all numerical claims of the paper. Its standard output is the verbatim text reproduced in Section 5, and the script additionally cross-checks the closed-form correlation expressions against explicit 4x4 matrix calculations on random angles (azimuthal angles included) to machine precision. - CHANGELOG.md: complete history of corrections relative to earlier versions of this deposit. Reproducibility. Running 'python3 chsh_verification.py' produces theverification tables displayed in Section 5 of the paper, byte for byte.
This work provides an explicit, reproducible demonstration of how theClauser-Horne-Shimony-Holt (CHSH) inequality depends jointly on thechoice of Bell state AND measurement configuration in the spin-1/2qubit convention used by gate-based quantum processors. Main result. The combination of CHSH Form I, S = | E(a,b) - E(a,b') + E(a',b) + E(a',b') |, with the commonly-cited angular configuration {0, pi/2, pi/4, -pi/4}yields S = 0 for ALL FOUR Bell states in the spin-1/2 Bloch-sphereconvention, not selectively for one. This frequently misleadspractitioners who prepare |Phi+> on gate-based hardware and applytextbook angles, obtaining S = 0 and misdiagnosing the issue as ahardware fault, code bug, or absence of entanglement. Corrections provided. For the two states whose correlation functionin the XZ plane has argument theta_a - theta_b (|Phi+> and |Psi->),two minimal one-parameter fixes recover the Tsirelson bound 2sqrt(2): 1. Retain the angles and apply CHSH Form II: S = | E(a,b) + E(a,b') + E(a',b) - E(a',b') |. 2. Retain Form I and replace b' = -pi/4 with b' = 3pi/4. For the remaining two Bell states (|Phi-> and |Psi+>, whose correlationargument is theta_a + theta_b), neither single-parameter fix suffices;the optimal angles b and b' must be exchanged. A complete optimal-angletable for all four states is given in Section 4 of the manuscript. Contents. - paper.pdf / paper.tex: 8-page manuscript with full derivation, tables, code listing and appendix. - chsh_verification.py: self-contained Python script (NumPy only) reproducing all numerical claims of the paper. Its standard output is the verbatim text reproduced in Section 5, and the script additionally cross-checks the closed-form correlation expressions against explicit 4x4 matrix calculations on random angles (azimuthal angles included) to machine precision. - CHANGELOG.md: complete history of corrections relative to earlier versions of this deposit. Reproducibility. Running 'python3 chsh_verification.py' produces theverification tables displayed in Section 5 of the paper, byte for byte.
The textbook CHSH measurement configuration {a, a', b, b'} = {0, pi/2, pi/4, -pi/4} with Form I evaluates to S = 0 on all four Bell states in the spin-1/2 convention used by gate-based quantum processors. This "failure mode", documented as a state-formula-angle mismatch in a companion pedagogical note (doi:10.5281/zenodo.20146219), is shown here to have a clean algebraic origin: the corresponding Bell operator reduces to sqrt(2) * (sigma_z (x) sigma_x + sigma_x (x) sigma_z), whose spectrum is {-2*sqrt(2), 0, 0, +2*sqrt(2)}, with the kernel spanned by {|Phi+>, |Psi->} and the non-kernel pair {|Phi->, |Psi+>} mapped to each other. Main results. The note establishes three properties of the failure-mode configuration: (1) The CHSH value S is preserved at exactly zero under arbitrary symmetric local Pauli channels (depolarizing, dephasing, bit-flip) and under local Rx and Rz rotations on either qubit, for every Bell state. Verified numerically with 1000 random Pauli channels per state and across the full range of standard noise channels. (2) Local Ry(theta) rotations break S = 0 only for the pair {|Phi->, |Psi+>}, with closed form S = 2*sqrt(2)*|sin(theta)|, reaching the Tsirelson bound at theta = pi/2. (3) The asymmetry has a direct algebraic origin in the correlation tensor: the kernel pair has equal XX and ZZ moments (t_1 = t_3), while the broken pair has opposite signs (t_1 = -t_3). This connects to and gives algebraic meaning to the Bell-state taxonomy of the companion paper. Diagnostic interpretation. The same CHSH measurement that fails in textbook form becomes a no-extra-resource diagnostic for coherent Ry-type errors restricted to a specific Bell-state family. If S > 0 in this configuration, then (i) the state is in the {|Phi->, |Psi+>} family, and (ii) a coherent Ry error of magnitude theta_err = arcsin(S/(2*sqrt(2))) is present. Incoherent Pauli noise alone cannot produce this signal. Calibration of novelty. The note explicitly searches the self-testing literature (Supic-Bowles 2020 review) and the CHSH-under-noise literature (Kofman 2012, Kofman-Korotkov 2008) and reports that this specific asymmetric breaking pattern was not located in those communities. Priority is explicitly disclaimed with respect to gate-set-tomography and randomized-benchmarking literatures, which were not exhaustively surveyed. The result is presented as either a novel observation or, if previously catalogued elsewhere, an independent algebraic rederivation with closed-form expression in terms of the correlation tensor. Reproducibility. The repository contains the LaTeX source, the bibliography, both figures, and seven Python scripts that reproduce every numerical claim from scratch using only NumPy and Matplotlib. Each script is self-contained and runs in seconds. Companion paper. This work extends the pedagogical note "State-dependent CHSH inequality on gate-based quantum processors" (Perez-Eugenio 2026, doi:10.5281/zenodo.20146219), which documents the original state-formula-angle compatibility issue and the corrected configurations. The present note can be read independently.
One basis-invariant half-angle behind the T gate, CHSH, and the QTT clock Why does the same half-angle appear in a quarter-phase gate, the symmetric CHSH optimum, and the QTT two-clock projection? This paper answers the operator question exactly. Starting from the A1 clock-plane quarter-turn and the A4 real-J Pauli representation, the centered clock rotor is UA1=exp(−J8πZ). The positive symmetric CHSH analyzer is generated by UCHSH=exp(−J8πY). With the Clifford operator C=SH, for which CZC†=Y, the two rotors obey UCHSH=CUA1C†,21TrUCHSH=21TrUA1=cos(8π). The normalized character is invariant under the basis change. It joins three familiar quantum-information expressions into one exact chain: Iclk=∣⟨+∣T∣+⟩∣=PwinQM=cos(8π),Smax=8Iclk2−4=22. The paper also prints the complete angle ledger: the A1 physical angle π/4, the T-gate relative phase π/4, the spinor exponent π/8, the photon-analyzer parameter π/8, the amplitude cos(π/8), the probability cos2(π/8), and the optimal correlator 1/2 are related but are not interchangeable quantities. The exact identity is a structural cross-lock, not a claim that ordinary CHSH experiments independently validate A1. Standard quantum mechanics already contains the same spinor geometry. A laboratory test is therefore ineligible if 22.5∘, 45∘, or the target overlap is programmed into preparation, calibration, control code, or priors. The independent discriminator is the sealed, unprogrammed dual-clock reference-switch test: Rstandard=1,RQTT-A1=cos(8π). Status CLOSED: SIGMA-A1-CHSH-SPINOR-CHARACTER-IDENTITY NOT CLAIMED: SIGMA-A1-CHSH-INDEPENDENT-EMPIRICAL-VALIDATION PENDING: SIGMA-A1-UNPROGRAMMED-REFERENCE-SWITCH-TEST Version: 2.0 Concept DOI: 10.5281/zenodo.19979594 Author: Ali Attar Website: quantumtraction.org Main-book anchor: Quantum Traction Theory: Main Book v10.01, pp. 235-237, 445-449, 541-542, and 552-553 Independent test: Artian Dual-Clock Sagnac Reference-Switch Preregistration Files Reader-facing PDF, Version 2.0 Reconstruction ZIP containing LaTeX source, style file, numerical verifier, machine receipt, scientific audit, render audit, metadata, and checksums
A supplementary calculational note to the interpretational program "Spacetime as Register, Not Stage". The note reconstructs the full structure of spin-1/2 singlet correlations — the joint distribution P = 1/4(1−αβ·a·b), the correlation E(a,b) = −a·b, and CHSH S = 2√2 — from the program's picture of the "entangled" pair as one indivisible complement unit read out locally at two sites by independently oriented detector settings. The note proves sufficiency, not derivation, and keeps an explicit ledger of its epistemic status: two rules are added and underived (a half-angle spinor readout structure, and the square law for amplitudes), two are derived conditionally (antisymmetry, relative to an explicit phase bridge; normalization, fixed by completeness with no free parameter), and every remaining gap is recorded in a debt register. The model exits the factorizable (Bell) class exactly where quantum mechanics does: conditioned on the full preparation state, local marginals remain 1/2 (parameter independence holds per preparation state), while outcome independence fails — with no superluminal transport claim derived from statistics, no superdeterminism (the preparation state carries no free parameter that could co-vary with later settings), and operational no-signaling verified numerically. A deterministic hidden-orientation contrast model is included, reproducing the linear correlation −1+2θ/π and S ≤ 2, to show that the cosine requires a genuine exit from the factorizable class rather than a change of vocabulary. The English text is a full translation of the author's Polish source version (both files included; frozen 28.07.2026, changes tracked in the program's correction register, Series 33).
In this thesis, a short review on the Clauser-Horne-Shimony-Holt (CHSH) Inequality is given.
The maximization of the expectation value of the CHSH operator for pure bipartite qubit
systems is derived explicitly.
The derivation reduces the maximization of hCHSHi to an eigenvalue problem. We show that
hCHSHimax = 2p 1 + 2
where 1and 2 are the greatest two eigenvalues of a certain matrix with eigenvalues {1,D,D}
where D 4 | det c |2 1 and c is the coefficient of the expansion of the states, i.e. | i =
Pi,j cij |iji. We discover the relation between the maximum of hCHSHi and the directions for spin measurement ˆr and ˆq, and these values can be continuously deformed even when the state is not a Bell State.
We then specialize to pure classically correlated density matrices and also Bell State and arrive at results consistent with previous conclusions, i.e.
For classically correlated density matrices hCHSHi 2 ,
For pure quantum states 2 hCHSHi 2p2.
We also briefly discuss the case of Werner density matrix.
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.
This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A, B, and C, with a fixed angle θ between A and B and a range of angles θ' between B and C. For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser-Horne-Shimony-Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M. We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M, whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ'.