The power of CGLMP inequalities – Open Quantum Problems

Cite this problem as Problem 27.

Problem

In the setting of [[Bell inequalities holding for all quantum states|Problem 26]], consider especially the case $(N,M,K)=(2,2,d)$ .

Problem 27. A

Show that every face of the local polytope $C$ , which is not already contained in a face of the no-signalling polytope $P$ is of CGLMP type, i.e., an inequality of the form first written out in [1], but possibly lifted from lower dimensions by fusing together some outcomes.

Problem 27. B

Numerically, the observables maximally violating the CGLMP inequality on a maximally entangled state are of a very specific form [2], involving measurements in computational basis, transformed by only discrete Fourier transformation and diagonal unitaries [1]. Show that this is necessarily the case. Show also that these measurements realize the highest resistance of violation to noise, and the best discrimination against classical realism in the sense of Kullback-Leibler divergence [3].

Background

According to the setting $(N,M,K)=(2,2,d)$ , the CGLMP inequality features two parties, $X$ and $Y$ , with two observables each: $X_1, X_2$ and $Y_1, Y_2$ , respectively. Each observable has $d$ possible outcomes. In order to simplify notation, we use the function $m(x) = x \mod d$ where $m(x)\in \{0,1,...,d-1\}$ for integer $x$ and we denote expectation values by $\mathsf{E}$ . The inequality can then be written [4]

$\mathsf{E}(m(X_1-Y_1)) + \mathsf{E}(m(Y_1-X_2)) + \mathsf{E}(m(X_2-Y_2)) + \mathsf{E}(m(Y_2-X_1-1)) \ge d-1.$

This statement also suggests a very elegant proof of the inequality [4]:

Note that $(X_1-Y_1) + (Y_1-X_2) + (X_2-Y_2) + (Y_2-X_1-1) = -1.$ Apply the function $m$ to both sides, and use $m(a)+m(b)+m(c)+m(d) \ge m(a+b+c+d)$ .

A more detailed discussion of the problem can be found in [5].

Partial Results

The CGLMP inequality is indeed a facet of the local polytope for the case $(N,M,K) = (2,2,d)$ , see [6].

In [7], using a semidefinite programming relaxation, it is shown that the maximum violation of the CGLMP inequality for $d=3$ on maximally entangled states of arbitrarily high dimensions differs at most by $10^{-10}$ from the local maximum found in [2] for maximally entangled states of dimension $3$ . This proves, up to computer precision, the optimality of the observables identified in [2].

It has been noticed numerically that, for $d=3,4,5,6,7,8$ , one can achieve higher violations of the CGLMP inequality using the same measurements and the eigenvector corresponding with minimum eigenvalue of the corresponding Bell operator [8]. In [9], it was shown that the maximum violation of CGLMP achievable by quantum systems coincides (up to computer precision) with the quantum values identified in [8].

By exploiting the symmetries of the CGLMP inequalities, the authors of [10] use the techniques of [9] to find tight analytic upper bounds on the maximum violation of the CGLMP inequalities for $d=3,4$ . More specifically, they find analytic Sum Of Hermitian Squares (SOS) decompositions of the form:

$\lambda_d^\star\mathbb{I}-B_d=\sum_iF_iF_i^\dagger.$ (*)

Here $B_d, \{F_i\}_i$ denote polynomials of Alice and Bob’s measurement operators, with the particularity that the state average of $B_d$ corresponds to the CGLMP value for the case $(N,M,K) = (2,2,d)$ . Since the right-hand side of the above equation is a positive semidefinite element in all operator representations, it follows that $\lambda_d^\star\geq \langle CGLMP_d\rangle$ . For $d=3,4$ , the authors of [10] are able to find analytic SOS decompositions of the CGLMP polynomial for which $\lambda_d^\star$ corresponds to the lower bound identified in [8]. Thus, as conjectured, measurements on the computational and Fourier bases bring about the maximum violation of CGLMP in the $(N,M,K) = (2,2,d)$ case.

References

[1] D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Phys. Rev. Lett, 88, 040404 (2002).

[2] T. Durt, D. Kaszlikowski, and M. Zukowski, Phys. Rev. A, 64, 024101 (2001).

[3] W. van Dam, P. Grunwald, and R. Gill, quant-ph/0307125 (2003).

[4] R. Gill, private communication.

[5] A. Acin, R. Gill, and N. Gisin, Phys. Rev. Lett. 95, 210402 (2005).

[6] Ll. Masanes, Quant. Inf. Comp. 3, 345 (2002).

[7] B. Lang, T. Vértesi and M. Navascués, J. Phys. A 47, 424029 (2014).

[8] A. Acín, T. Durt, N. Gisin and J. I. Latorre, Phys. Rev. A 65, 052325 (2002).

[9] M. Navascués, S. Pironio and A. Acín, Phys. Rev. Lett. 98, 010401 (2007).

[10] M. Ioannou and D. Rosset, arXiv:2112.10803 (2021).