Entropic approach to local realism and noncontextuality

For any Bell locality scenario (or Kochen-Specker noncontextuality scenario), the joint Shannon entropies of local (or noncontextual) models define a convex cone for which the nontrivial facets are tight entropic Bell (or contextuality) inequalities. In this paper we explore this entropic approach and derive tight entropic inequalities for various scenarios. One advantage of entropic inequalities is that they easily adapt to situations such as bilocality scenarios, which have additional independence requirements that are nonlinear on the level of probabilities, but linear on the level of entropies. Another advantage is that, despite the nonlinearity, taking detection inefficiencies into account turns out to be very simple. When joint measurements are conducted by a single detector only, the detector efficiency for witnessing quantum contextuality can be arbitrarily low.

Figure 1

Some contextuality and nonlocality scenarios (the number of outcomes of each observable is arbitrary): (a) the CHSH scenario, with two parties with two measurement settings each; (b) the Klyachko scenario, with five observables arranged in a cyclic configuration such that each observable is compatible with its neighbors; and (c) a generalization of the Klyachko scenario, with $n$ observables in a cyclic configuration (the $n$-cycle [17]). The unique nontrivial entropic inequality in all these cases is given by Eq. (3).

Figure 2

(a) Violation profile of the entropic CHSH inequality for a pure state of the form $\cos \alpha |00\rangle +\sin \alpha |11\rangle$. The violation profile essentially coincides with the one of the standard CHSH inequality, although the scales are different. The curve is obtained by optimizing over the observables for each $\alpha$. (b) Maximal violation of the entropic inequality (3), achieved on a maximally entangled state of two qubits ($\alpha =\pi /4$) and $k=2,...,10$ measurement settings for each party.

Figure 3

General nonlocality distillation protocol with two boxes. The inputs for the second box may locally depend on the output of the first one and the final output of the combined boxes is a function of all inputs and outputs.

Figure 4

Slice of the no-signaling polytope corresponding to the family of boxes (25). The quantum region and ${\mathrm{CHSH}}_E$ violations are bounded respectively by the thin black and thick black curves. The regions of distillability by the wirings (22) and (23) correspond respectively to the dash-dotted red and dashed blue lines. The region of violations of the entropic CHSH inequality is strictly larger than the region of correlations distillable with the wiring (23). In contrast, in a small part of the quantum region, nonlocality can be distilled by the protocol (22) while not being detected by ${\mathrm{CHSH}}_E$.

Figure 5

Distilled nonlocality (NL) (increase in nonlocal content) of the family (27) upon application of the wiring (30) for $d=2,4$ (red solid line), $d=3$ (blue long-dashed line), and $d=5$ (black short-dashed line) as computed by linear programming. The inset shows the violation of ${\mathrm{CHSH}}_E$, which does not depend on $d$.

Figure 6

Swapping experiment scenario, where two independent entangled states are prepared and sent to three distant parts. If the central part $B$ is measured in an entangled basis, the subsystems $A$ and $C$, which had no previous contact, become nonlocally correlated.

Figure 7

Simplicial complex of jointly measurable observables in the bilocality scenario discussed in the main text. Two observables are jointly measurable whenever they share an edge; three observables are jointly measurable whenever they are the vertices of one of the four triangles.

Figure 8

Parameter space of the family (38). The red solid line delimits the region where Eqs. (39) and (40), and therefore Eq. (38), are local as witnessed by the CHSH inequality. The region above the blue dashed line contains violations of the entropic bilocality inequality 7 from Table . The black dotted line delimits the region where Eqs. (39) and (40), and therefore Eq. (38), have a quantum realization with a single three-qubit source. There is a small region (inset) in which our inequality detects that Eq. (38) is not bilocal, although the box is tripartite local.