Bell inequalities: many questions, a few answers

Cite this problem as Problem 32.

Problem

The title was taken from a recent paper by Nicolas Gisin [1] where many open questions concerning Bell inequalities are presented. They are organized in three categories: fundamental, linked to experiments, and exploring nonlocality as a resource. In the following, we list the ones which are not already listed on this page.

A. Is hidden non-locality, as defined in [2], generic for all entangled states? In other words: is it true that all entangled states admit a local filtering operation after which they become nonlocal? In the same line, is there an example of hidden nonlocality that requires a sequence of local filters rather than a single one (the local model should reproduce all intermediate results)?

B. Is there a local quantum state $\rho$ such that $\rho^{\otimes n}$ violates some Bell inequality?

C. Find genuine n-party inequalities violated by all n-party pure entangled states.

D. Find a Bell inequality that requires POVMs for optimal violation on some quantum states.

E. Are there Bell inequalities which allow one to distinguish between real quantum mechanics and complex quantum mechanics?

F. Is there a bound entangled state that violates some Bell inequality?

G. Given a multi-partite state $\rho$ , how can one know whether $\rho$ is non-local?

H. Find Bell inequalities easier to test experimentally with today’s technology, while avoiding all known loopholes.

I. Find inequalities suitable for a Bell test with simple quantum optics states and homodyne detectors.

J. Is there a Bell inequality valid for all correlations one can simulate with a single bit of communication and violated by some partially entangled 2-qubit states?

K. Find inequalities satisfied by all correlations which can be simulated by two PR-boxes.

L. Find any non-signalling box with finitely many inputs and outputs with which one can simulate partially entangled states.

M. Can a secret key be distilled out of any nonlocal correlation?

Partial progress

Problem A was partly solved by in [3], where Hirsch et al. show that two-qubit Werner states $\rho_W=p|\psi^-\rangle\langle \psi^-|+(1-p)\frac{{\mathbb I}}{4}$ with $p\leq 0.3656$ remain local after a filtering operation. This overlaps with the entanglement region $p>\frac{1}{3}$ .
Problem B was solved by C. Palazuelos in [4], by arguing that multiple copies of high-dimensional Bell states can be used to beat the maximum classical score of the Khot and Visnoi game [5].
Problem D is solved by Vértesi and Bene in [6]. They consider a Bell scenario where both Alice and Bob can effect two different measurements with two outcomes each, but Alice has a third measurement setting with three outcomes. Then they define a uniparametric family of Bell functionals $I_{CH^3}(c)$ and prove that, for some values of $c$ , the maximum values of $I_{CH^3}(c)$ attainable in two-qubit systems via projective measurements or general POVMs differ.
Problem E is solved for the bipartite case in [7], where Pál and Vértesi prove that the sets of correlations achievable in real and complex quantum mechanics are identical. The extension to the multipartite case is proven in [8].
Problem F was solved in by Vértesi and Brunner, by giving explicit examples of PPT states which violate tripartite [9] and bipartite [10] Bell inequalities.
Problem G is essentially solved in [11], where Hirscht et al. show give an algorithmic procedure that, in bounded time, determines whether the state is non-local or $\delta$ -close to a local state for arbitrary $\delta>0$ .
As it turns out, problem H was solved by Clauser, Horne, Shimony and Holt in 1971 [12], because the first report of a loophole-free Bell experiment [13] used the CHSH Bell inequality.
Problem I is solved in [14], using a combination of homodyne measurements and single-detection apparatuses.
Problem J is advanced in [15], where Renner et al. prove that two-qubit states of the form $\sqrt{p}|00\rangle+\sqrt{1-p}|11\rangle$ can be simulated with one bit of communication if $p\geq 0.835$ . The authors leave open whether the remaining states can be simulated with one bit as well. Nonetheless, they prove that a trit of communication suffices to simulate the statistics of any two-qubit state.