Local realism, detection efficiencies, and probability polytopes

We present a detailed investigation of minimum detection efficiencies, below which local realism cannot be violated by any quantum system of any dimension in bipartite Bell experiments. Lower bounds on these minimum detection efficiencies are determined with the help of linear programming techniques. Our approach is based on the observation that any possible bipartite quantum correlation originating from a quantum state in an arbitrary dimensional Hilbert space is sandwiched between two probability polytopes, namely the local (Bell) polytope and a corresponding nonlocal no-signaling polytope. Numerical results are presented demonstrating the dependence of these lower bounds on the numbers of inputs and outputs of the bipartite physical system.

Figure 1

The two possible representations of a polytope by its vertexes ($\mathcal{V}$-representation) and by inequalities characterizing half-spaces ($\mathcal{H}$-representation).

Figure 2

(Color online) Lower bounds on detection efficiencies $({\eta }_1,{\eta }_2)$ for three inputs on Alice’s and Bob’s sides with the no-signal event treated as an extra output.

Figure 3

(Color online) Lower bounds on detection efficiencies $({\eta }_1,{\eta }_2)$ for three inputs on Alice’s and Bob’s sides with the no-signal event combined with the output $(a_2,b_2)$.

Figure 4

Schematic representations of identical violations $v$ of relevant Bell inequalities. Note the different distances from the polytopes.

Figure 5

Distance of a vertex of $\mathcal{P}$ from $\mathcal{L}$ as a function of $\eta$ for two inputs and outputs of both $A$ and $B$.