bibkey: perito2026bell authors: I. Perito, R. D’Avino, M. Jung, P. Mironowicz, A. Acín, R. Augusiak year: 2026 title: “Bell inequalities tailored to optimal global randomness certification” doi: 10.48550/arXiv.2606.21362 url: https://arxiv.org/abs/2606.21362v3 claim: “Conjecture 1: the Tsirelson bound of the d-outcome Bell functional I_d = (1/2)Σ_{x<2}Σ_{y<d} ω^{xy} A_x⊗B_y + h.c. is 2/sin(π/(2d)) and is attained by A_0 = Z, A_1 = X, the maximally entangled state and B_y = Σ_k λ_{y,k} X^{k+1}Z^k; the appendix proves that these B_y are unitary with d-th-root-of-unity eigenvalues, and the bound is checked by NPA for d ≤ 6.” strata_touched:

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Perito et al., Bell inequalities for optimal global randomness certification

I. Perito, R. D’Avino, M. Jung, P. Mironowicz, A. Acín and R. Augusiak, Bell inequalities tailored to optimal global randomness certification, arXiv:2606.21362v3 (21 July 2026; quant-ph; v1 19 June 2026, v2 24 June 2026).

Verified locator

DOI: 10.48550/arXiv.2606.21362. Primary version: https://arxiv.org/abs/2606.21362v3. The TeX source main.tex of v3 supplies §II (the unitary notation $A_x={\sum }_a{\omega }^a{\Pi }_{a|x}$, $\omega =e^{2\pi i/d}$), §III.2 (the functional eq:red_ineq_gen, the strategy eq:optbob, the value eq:tsir_bound and Conjecture 1, label conj:i), Appendix app:ibounds (the bound ${\beta }_{\mathcal{Q}}(d)\le d\sqrt{2}$, eq:qbound), Appendix app:numericalbounds (NPA bounds for $d\le 6$) and Appendix app:optimalconstruction (the coefficients eq:optcoeffs and the validity proof of $B_y$).

Source statements

Functional (§III.2): “${\mathcal{I}}_d\equiv \frac{1}{2}{\sum }_{x=0}^1{\sum }_{y=0}^{d-1}{\omega }^{xy}{A}_x\otimes B_y+\text{h.c.,}$”.

Strategy (§III.2): “$A_0=Z={\sum }_k^{d-1}{\omega }^k|k⟩⟨k|,A_1=X={\sum }_k^{d-1}|k+1⟩⟨k|$”, “$B_y={\sum }_{k=0}^{d-1}{\lambda }_{y,k}{X}^{k+1}{Z}^k$”, with (Appendix, eq:optcoeffs) “${\lambda }_{y,k}=\frac{(-1{)}^k{\omega }^{\frac{k(k+1)}{2}}{\omega }^{-y(1+k)}}{d\sin \left(\frac{\pi }{d}(k+\frac{1}{2})\right)}$”, and “this strategy gives: ${\mathcal{I}}_d=2{\left[\sin \left(\frac{\pi }{2d}\right)\right]}^{-1}$”.

Conjecture 1: “The Tsirelson bound of ${\mathcal{I}}_d$ is given by Eq.~\eqref{eq:tsir_bound} … and is attained by taking $A_0=Z$, $A_1=X$; the state $|{\psi }_d⟩=\frac{1}{\sqrt{d}}{\sum }_k|kk⟩$; and $B_y$ defined by Eq. \eqref{eq:optbob}.”

Evidence (§III.2): “Using NPA, we can numerically see that the conjecture holds for $d=4,5,6$ … which explains why we could not go beyond $d=6$.”

Appendix app:ibounds: “${\beta }_{\mathcal{Q}}(d)\le d\sqrt{2}$ … This upper bound is, in general, not tight.”

Appendix app:optimalconstruction: “we show that $B_y$ defined in \eqref{eq:optbob} with coefficients \eqref{eq:optcoeffs} are proper observables, i.e. that they are unitary and with eigenvalues equal to $d$-th roots of unity. To this end we write $B_y$ as a product of $X$ and a polynomial of $XZ$”.

Scope

The validity of $B_y$ and the attained value $2/\sin (\pi /(2d))$ are proved in the paper. The open part of Conjecture 1 is the upper bound for every $d$; the paper’s analytic bound $d\sqrt{2}$ coincides with it only at $d=2$ (CHSH).