Bell inequalities holding for all quantum states

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Below are two connected problems, both asking for a better description of the possible correlation polytopes.


The setting for this problem is the same as for Problem 1: We consider correlations between [math]N[/math] parties, each of which can perform [math]M[/math] different measurements yielding one of [math]K[/math] possible outcomes each. We can reduce the number of dimensions by considering only those correlation data satisfying the no-signalling constraint, i.e., the choice of a measuring device by one party [math]A[/math] never changes the (joint) probabilities seen by all the other parties, unless results are selected with respect to the outcomes of [math]A[/math]. Only obeying no-signalling and positivity constraints, we get the \emph{no-signalling polytope} [math]P[/math]. Contained in it is the convex body [math]Q[/math] of correlations obtainable from a multipartite quantum state with \emph{quantum mechanical POVM measurements}, and inside [math]Q[/math] the polytope [math]C[/math] of correlations realizable by a \emph{classical realistic theory} (see Figure).


Problem 26.A

Consider the part of the boundary of [math]Q[/math], which is not already contained in the boundary of [math]P[/math]. Can one reach all these points by choosing each one of the local Hilbert spaces to be [math]K[/math]-dimensional, and each measurement as a complete von Neumann measurement (with [math]K[/math] orthogonal projectors) on pure states with minimal dimension?

Problem 26.B

Consider a maximal face of the polytope [math]C[/math], which is not also a face of [math]P[/math] (a blue line in the above figure). In other words, consider a "proper Bell inequality", i.\,e., a tight linear inequality for local classical correlations, which does not follow from positivity and no-signalling. Then can we find points of [math]Q[/math] outside the face? Or, phrased in terms of Bell inequalites, can every proper Bell inequality be violated by quantum correlation data? In the above figure, this asks whether or not a face like the dashed red/blue line can occur.

Partial Results

For the case [math](N,M,K) = (N,2,2)[/math], i.e. two dichotomic observables for an arbitrary number of parties, Masanes [1] has shown that it is sufficient to use pure states and projective measurements on systems of [math]N[/math] qubits.

In the case of more than two parties, Problem 26.B was decided to the negative in [2].


  1. Ll. Masanes, quant-ph/0512100 (2005)
  2. M. L. Almeida, J.-D. Bancal, N. Brunner, A. Acin, N. Gisin, S. Pironio, Phys. Rev. Lett. 104, 230404 (2010)