Simulation of partial entanglement with nonsignaling resources

With the goal of gaining a deeper understanding of quantum nonlocality, we decompose quantum correlations into more elementary nonlocal correlations. We show that the correlations of all pure entangled states of two qubits can be simulated without communication, hence using only no-signaling resources. Our simulation model works in two steps. First, we decompose the quantum correlations into a local and a nonlocal part. Second, we present a model for simulating the nonlocal part using only no-signaling resources. In our model partially entangled states require more nonlocal resources than maximally entangled states, but the less the state is entangled, the less frequently must the nonlocal resources be used.

Figure 1

Preliminary model. Simulating partial entanglement without communication. The model requires four PR boxes and a millionaire box (M box). The first two PR boxes create “scalar product” correlations (CGMP boxes). Then the M box “selects” the correct CGMP box, without revealing any relation between Alice’s and Bob’s measurement settings (i.e., without signaling). Finally, two additional PR boxes are required for computing the correct outputs.

Figure 2

(Color online) Sketch of the main model. (1) Alice and Bob inside the slice (see text); both CGMP boxes can be used. (2) Alice inside, Bob outside; then indeed $a_z⩽b_z$. (3) Alice outside, Bob inside. (4) Alice and Bob outside. Depending whether $a_z⩽b_z$ or $a_z⩾b_z$, the M box selects the correct CGMP box. Note that we have omitted the two additional PR boxes (${\mathrm{PR}}_3$, ${\mathrm{PR}}_4$ of Fig. 2).