# Bell inequalities holding for all quantum states

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# Problem

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

# Background

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

## Problem 26.A

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

## Problem 26.B

Consider a maximal face of the polytope $C$, which is not also a face of $P$ (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 $Q$ 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 $(N,M,K) = (N,2,2)$, 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 $N$ qubits.

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

# Literature

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)