View source for The power of CGLMP inequalities

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'''Cite as:''' [http://qig.itp.uni-hannover.de/qiproblems/The_power_of_CGLMP_inequalities http://qig.itp.uni-hannover.de/qiproblems/27]

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

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==Problem 27. A==
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Show that every face of the local polytope <math>C</math>, which is not already contained in a face of the no-signalling polytope <math>P</math> is of CGLMP type, i.e., an inequality of the form first written out in <ref name="CGLMP"> D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Phys. Rev. Lett, 88, 040404 (2002) </ref>, but possibly lifted from lower dimensions by fusing together some outcomes.

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==Problem 27.B==
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Numerically, the observables maximally violating the CGLMP inequality on a maximally entangled state are of a very specific form <ref name="DKZ"> T. Durt, D. Kaszlikowski, and M. Zukowski, Phys. Rev. A, 64, 024101 (2001) </ref>, involving measurements in computational basis, transformed by only discrete Fourier transformation and diagonal unitaries <ref name="CGLMP"/>. 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 <ref name="Gill1"> W. van Dam, P. Grunwald, and R. Gill, quant-ph/0307125 (2003) </ref>.

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=Background=
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According to the setting <math>(N,M,K)=(2,2,d)</math>, the CGLMP inequality features two parties, <math>X</math> and <math>Y</math>, with two observables each: <math>X_1, X_2</math> and <math>Y_1, Y_2</math>,
respectively. Each observable has <math>d</math> possible outcomes. In order to simplify notation, we use the function <math>m(x) = x \mod d</math> where <math>m(x)\in \{0,1,...,d-1\}</math> for integer <math>x</math> and we denote expectation values by <math>\mathsf{E}</math>. The inequality can then be written <ref name="Gill2"> R. Gill, private communication </ref>: 

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\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.
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This statement also suggests a very elegant proof of the inequality <ref name="Gill2"/>:
Note that <math> (X_1-Y_1) + (Y_1-X_2) + (X_2-Y_2) + (Y_2-X_1-1) = -1. </math> Apply the function <math>m</math> to both sides, and use <math>m(a)+m(b)+m(c)+m(d) \ge m(a+b+c+d)</math>.

A more detailed discussion of the problem can be found in <ref name="AGG"> A. Acin, R. Gill, and N. Gisin,  Phys. Rev. Lett. 95, 210402  (2005) </ref>.

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=Partial Results=
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The CGLMP inequality is indeed a facet of the local polytope for the case <math>(N,M,K) = (2,2,d)</math>, see <ref name="Ma">Ll. Masanes, Quant. Inf. Comp. 3, 345 (2002)</ref>.

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=Literature=
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[[Category:Problem still open]]