Experimental Violation of Multipartite Bell Inequalities with Trapped Ions

We report on the experimental violation of multipartite Bell inequalities by entangled states of trapped ions. First, we consider resource states for measurement-based quantum computation of between 3 and 7 ions and show that all strongly violate a Bell-type inequality for graph states, where the criterion for violation is a sufficiently high fidelity. Second, we analyze Greenberger-Horne-Zeilinger states of up to 14 ions generated in a previous experiment using stronger Mermin-Klyshko inequalities, and show that in this case the violation of local realism increases exponentially with system size. These experiments represent a violation of multipartite Bell-type inequalities of deterministically prepared entangled states. In addition, the detection loophole is closed.

Figure 1

Graph states that find application in measurement-based quantum computation. Red circles represent qubits, connecting lines relate to the states’ generation method, as described in the text. (a) Box graph $|B{C}_4⟩$, (b) linear graph $|L{C}_4⟩$. Error correction graphs: (c) $|E{C}_1⟩$, (d) $|E{C}_3⟩$, (e) $|E{C}_5⟩$.

Figure 2

Examples of (fully connected) graph states that are (up to local unitary operations) equivalent to $n$-qubit GHZ states with $n=2,\dots ,5$, [(a), $\dots ,$ (d)]. Red circles represent qubits, connecting lines relate to the states’ generation method, as described in the text.

Figure 3

Multipartite Bell inequality violations for GHZ states of different sizes. Data are taken from Ref. [15]. $\mathcal{R}$ is the relative violation via the ratio of the quantum mechanical expectation value of the Bell observables and the maximal reachable value in a LHV model. It is given by $\mathcal{R}=⟨{\mathcal{B}}_n(G)⟩/\mathcal{D}(G)$ for the graph inequalities (red lines) and $R=⟨B_n⟩/D=2^{(n-1)/2}⟨B_n⟩$ for the MK inequalities (black lines). In each case, solid (dashed) lines show the ideal (experimental) case. Error bars in experimental results are all smaller than the point sizes. Any value larger than $\mathcal{R}=1$ corresponds to a Bell violation. Note the logarithmic scaling of the axis.