Bell Inequalities with Overlapping Measurements

Which nonlocal correlations can be obtained, when a party has access to more than one subsystem? While traditionally nonlocality deals with spacelike separated parties, this question becomes important with quantum technologies that connect devices by means of small shared systems. Here, we study Bell inequalities where measurements of different parties can have overlap. This allows us to accommodate problems in quantum information such as the existence of quantum error correction codes in the framework of nonlocality. The scenarios considered show an interesting behavior with respect to Hilbert space dimension, overlap, and symmetry.

Figure 1

Left: Scenario 1. The maximal quantum value of the joint Bell inequality $J_2=I_{2,2}^{\mathcal{A}|\mathcal{B}}+I_{2,2}^{\mathcal{A}|\mathcal{C}}$ in Eq. (1) is governed by size of local physical systems. Right: Scenario 2. In the overlapping Bell inequality $K_2=I_{2,2}^{\mathcal{A}|\mathcal{B}}+I_{2,2}^{\mathcal{A}|\mathcal{C}}+I_{2,2}^{\mathcal{A}|\mathcal{B}\mathcal{C}}$ in Eq. (2) a fourth party has, independently of the measurements performed on Bob and Charlie, also access to measurements on the joint system. Here, both the dimensions of the individual subsystems as well as the overlap of $\mathcal{B}\mathcal{C}$ with $\mathcal{B}$ and $\mathcal{C}$ affect the maximal quantum value.

Figure 2

Bell inequalities with overlap. Above: Bell inequality $J_2$ in Eq. (1), where only the local dimensions play a role. Below: overlapping Bell inequality $K_2$ in Eq. (2), where both the local dimensions and the overlapping structure play a role.

Figure 3

Quantum error correction code. A code has distance $d$, if maximal entanglement can be recovered between a reference system and every code subsystem of size $n-d+1$. This can be characterized in terms of the perfect recovery of a maximally entangled state ${\varphi }^{+}$, one part of which is encoded, acted upon by noise, and decoded.