Statistical strength of CGLMP measurements

Problem

For every \(d\geq3\), do the standard CGLMP Fourier–phase measurements maximize the relative-entropy statistical strength against local realism among all projective \(d\)-outcome measurements on the fixed state \(\lvert\Phi_d\rangle=d^{-1/2}\sum_{j=0}^{d-1}\lvert j,j\rangle\), when the setting distribution is also optimized? For a behavior \(p(a,b\mid x,y)\), a distribution \(\mu(x,y)\) on the four setting pairs, and the local polytope \(\mathcal L\), define

\begin{equation} S(p;\mu) :=\inf_{\ell\in\mathcal L} \sum_{a,b,x,y}\mu(x,y)p(a,b\mid x,y) \log_2\!\frac{p(a,b\mid x,y)}{\ell(a,b\mid x,y)}, \tag{1} \end{equation}

Set \(S^\star(p):=\sup_{\mu\in\Delta(\{0,1\}^2)}S(p;\mu)\); by Eq. (1), this is the optimized asymptotic evidence rate against the best local model. The candidate bases are

\begin{equation} \begin{aligned} \lvert a;x\rangle &=\frac{1}{\sqrt d}\sum_{j=0}^{d-1} \exp\!\left(\frac{2\pi i}{d}j(a+\alpha_x)\right)\lvert j\rangle,\\ \lvert b;y\rangle &=\frac{1}{\sqrt d}\sum_{j=0}^{d-1} \exp\!\left(\frac{2\pi i}{d}j(-b+\beta_y)\right)\lvert j\rangle, \end{aligned} \tag{2} \end{equation}

where \(\alpha_0=0\), \(\alpha_1=-1/2\), \(\beta_0=1/4\), and \(\beta_1=3/4\). Let \(p_{\mathrm{CGLMP}}\) denote the behavior produced on \(\lvert\Phi_d\rangle\) by the bases in Eq. (2), and write \(p_M\) for the behavior produced by any other measurement choice \(M\) on \(\lvert\Phi_d\rangle\). The conjectured optimality of Eq. (2) is

\begin{equation} S^\star(p_{\mathrm{CGLMP}}) =\sup_M S^\star(p_M), \tag{3} \end{equation}

where the supremum is over two projective \(d\)-outcome measurements per party. Equation (3) is the question to be resolved.

Source

Gill explicitly proposes the global statistical-strength optimality of the CGLMP measurement construction for a fixed maximally entangled state [Gil07].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

Van Dam, Gill, and Grünwald established the operational interpretation of Eq. (1) as the asymptotic evidence rate of a Bell experiment and formulated the joint optimization problem [DGG05].
Acín, Gill, and Gisin numerically found the CGLMP measurement bases for their relative-entropy searches at \(d=3\) and \(d=4\). Their globally best state was not maximally entangled, so this does not resolve the fixed-state equality in Eq. (3) [AGG05].
Gill reports that numerical searches with the maximally entangled state fixed found only the CGLMP measurements for both Euclidean and relative-entropy criteria, but treats optimality as conjectural; the search does not prove Eq. (3) for arbitrary \(d\) [Gil07].

Comment

The remaining gap is a global proof or counterexample to Eq. (3) for arbitrary \(d\). The shared state is fixed; this differs from joint state–measurement optimization and from maximizing the linear CGLMP violation.

References

[DGG05]: W. van Dam, R. D. Gill, and P. D. Grünwald, “The Statistical Strength of Nonlocality Proofs,” IEEE Transactions on Information Theory 51, 2812–2835 (2005).DOI arXiv
[AGG05]: A. Acín, R. Gill, and N. Gisin, “Optimal Bell Tests Do Not Require Maximally Entangled States,” Physical Review Letters 95, 210402 (2005).DOI arXiv
[Gil07]: R. D. Gill, “Better Bell Inequalities (Passion at a Distance),” in Asymptotics: Particles, Processes and Inverse Problems, IMS Lecture Notes–Monograph Series 55, 135–148 (2007).DOI arXiv

Statistical strength of CGLMP measurements

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