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

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 in Problem 24.

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

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