Optimal CGLMP measurements for a maximally entangled state

Problem

For every \(d\geq3\), do the standard Fourier–phase measurements optimize the CGLMP violation of the maximally entangled state among all projective \(d\)-outcome measurements? Fix the state

\begin{equation} \lvert\Phi_d\rangle =\frac{1}{\sqrt d}\sum_{j=0}^{d-1}\lvert j,j\rangle. \tag{1} \end{equation}

Equation (1) is held fixed; the shared state is not part of the optimization.

For outcomes in \(\mathbb Z_d\), let \([t]_d\in\{0,\ldots,d-1\}\) be the residue of \(t\) and define the CGLMP functional by

\begin{equation} \begin{aligned} B_d={}&\mathbb E([A_0-B_0]_d)+\mathbb E([B_0-A_1]_d)\\ &+\mathbb E([A_1-B_1]_d)+\mathbb E([B_1-A_0-1]_d). \end{aligned} \tag{2} \end{equation}

Local behaviors satisfy \(B_d\geq d-1\) under the convention in Eq. (2). The candidate measurements have bases

\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{3} \end{equation}

with \(\alpha_0=0\), \(\alpha_1=-1/2\), \(\beta_0=1/4\), and \(\beta_1=3/4\). The question is whether the bases in Eq. (3) minimize Eq. (2) on Eq. (1), up to symmetries of the state and functional.

Source

Zohren and Gill formulate the standard Fourier-phase measurements as the candidate global optimizers of the CGLMP value for a fixed maximally entangled state [ZG08].

Progress

Collins, Gisin, Linden, Massar, and Popescu introduced the high-dimensional Bell family and the Fourier–phase measurement construction underlying Eq. (3) [CGLMP02].
Zohren and Gill explicitly identify optimality of Eq. (3) for the fixed state in Eq. (1) as a longstanding conjecture and provide numerical support. They do not prove it for arbitrary \(d\) [ZG08].
For \(d=3\), Lang, Vértesi, and Navascués obtained a semidefinite upper bound on the optimum over maximally entangled states that differs from the candidate value by about \(10^{-10}\). This numerical certification is not an exact proof and does not address all dimensions [LVN14].

Comment

The unresolved statement is global measurement optimality for every finite \(d\) with the state fixed by Eq. (1). It does not ask for the unrestricted CGLMP optimum over shared states, which is a different problem.

References

[CGLMP02]: D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell Inequalities for Arbitrarily High-Dimensional Systems,” Physical Review Letters 88, 040404 (2002).DOI arXiv
[ZG08]: S. Zohren and R. D. Gill, “Maximal Violation of the Collins-Gisin-Linden-Massar-Popescu Inequality for Infinite Dimensional States,” Physical Review Letters 100, 120406 (2008).DOI arXiv
[LVN14]: B. Lang, T. Vértesi, and M. Navascués, “Closed Sets of Correlations: Answers from the Zoo,” Journal of Physics A: Mathematical and Theoretical 47, 424029 (2014).DOI arXiv

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