The exact quantum value of a cyclic Bell operator

Alec Kriebel

Abstract

Perito, D’Avino, Jung, Mironowicz, Acín, and Augusiak introduced a family of cyclic Bell operators and conjectured its exact Tsirelson bound. This note supplies the missing analytic upper bound for every dimension. The proof is dimension independent and, more strongly, works for commuting unitary observables without the order-\(d\) relations or a tensor-product representation.

Exact value\(2\csc(\pi/(2d))\) for every \(d\ge2\)

Stronger scopeExact finite, approximate, and commuting-operator value

CertificateExact operator identity plus finite independent checks

1. Result

For \(\omega=e^{2\pi i/d}\), define

\[ \mathcal I_d=\frac12\sum_{x=0}^{1}\sum_{y=0}^{d-1} \omega^{xy}A_x\otimes B_y+\mathrm{h.c.} \]

Theorem. If \(A_0,A_1,B_0,\ldots,B_{d-1}\) are arbitrary commuting Alice–Bob unitaries on an arbitrary Hilbert space, then

\[\mathcal I_d\le 2\csc\!\left(\frac{\pi}{2d}\right)I.\]

Because the finite order-\(d\) strategy constructed in the originating paper attains equality, \(\beta_q=\beta_{qa}=\beta_{qc}=2\csc(\pi/(2d))\), proving its Conjecture 1.

The Bell family, formula, Weyl strategy, admissible Bob observables, and matching lower bound are all due to the originating authors. The claimed contribution is the analytic upper bound and its commuting-operator strengthening.

2. Proof in three moves

Polar factor

If \(C=V|C|\) is a polar decomposition and \(B\) is unitary, then the polar factor \(V\) may remain a partial isometry and

\[ \frac{|C|+|C^\dagger|}{2}-\operatorname{Re}(CB) =\frac12P^\dagger P,\qquad P=|C^\dagger|^{1/2}-V|C|^{1/2}B. \]

Sharp scalar extremum

Writing \(z=e^{2is}\) reduces the remaining norm sum to an equally spaced cosine grid. A parity split gives

\[ \max_{|z|=1}\sum_{y=0}^{d-1}|1+\omega^yz| =2\csc\!\left(\frac{\pi}{2d}\right), \]

with equality exactly when \(z^d=(-1)^{d-1}\). The paper includes the complete finite-sum proof, including boundary cases.

Functional calculus

Set \(U=A_0^\dagger A_1\) and \(C_y=A_0+\omega^yA_1=A_0(I+\omega^yU)\). The factor \(I+\omega^yU\) is normal, so

\[ F_d(U)=\sum_y|I+\omega^yU| \le 2\csc\!\left(\frac{\pi}{2d}\right)I. \]

Summing the polar identities yields the claimed commuting-operator inequality. The manuscript also records the complete positive-factor certificate for the residual operator.

Why the lower strategy is admissible

The full write-up does not stop at the two facts \( (Z^\dagger X)^d=(-1)^{d-1}I \) and \(\prod_r(1+a\omega^r)=1-(-a)^d\). It supplies the missing link: in the eigenbasis of \(Z^\dagger X\), the polar Bob unitary is a weighted cyclic shift. Conjugation generates its full orbit, and the product identity makes the accumulated phase exactly one. Hence every \(B_y^d=I\), with the full \(d\)-th-root spectrum.

3. Barred functional

For the related operator \(\overline{\mathcal I}_d=\mathcal I_d+\operatorname{Re}(A_0\otimes B_d)\), the same strategy aligns the extra correlator and gives

\[ \beta_q(\overline{\mathcal I}_d) =\beta_{qa}(\overline{\mathcal I}_d) =\beta_{qc}(\overline{\mathcal I}_d) =2\csc\!\left(\frac{\pi}{2d}\right)+1. \]

This determines the value only. It does not prove the separate all-dimensional randomness conjecture.

4. Verification package

verify_certificate.py — singular polar identities, the complete operator certificate, scalar equality roots, Weyl spectra, Bob order, and Bell saturation.
tests/test_certificate.py — independent regression suite.
certificate.json — machine-readable theorem, formulas, proof dependencies, and verification boundary.
MANIFEST.md — claim-to-artifact map and explicit nonclaims.

Default checks cover \(d=2,\ldots,12\), including all \(\sum_{d=2}^{12}d=77\) Bob observables, deterministic random singular matrices, direct-sum commuting representations, phase orientation, Fourier constraints, and the explicit qutrit formula. These finite checks support the analytic argument; they do not quantify over all \(d\).

5. Novelty and scope

A focused audit of the current arXiv version, its note added and appendices, recent related submissions, older qudit Bell families, general SOS methods, and public repositories found no prior proof of Conjecture 1. The closest concurrent general SOS paper treats other Bell families and does not supply this result.

Accordingly, the defensible wording is “to the best of our knowledge, the first public analytic proof.” This cannot rule out unindexed, unpublished, or simultaneous work. No claim is made that polar decomposition or the trigonometric identity is new in isolation.

In particular, the exact value does not establish uniqueness, self-testing, or maximal global randomness in every dimension. Those remain separate rigidity questions.

AI-assistance and verification disclosure

The proof architecture, independent derivations, manuscript, verification programs, and literature audit were developed with heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel’s direction. Alec Kriebel is the named human author and cannot independently validate the claims. No originating author or outside researcher was contacted.

Suggested citation

Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol, “The exact quantum value of a cyclic Bell operator,” provisional research note, 26 July 2026. Permanent page.