Bell theorem without inequalities for two particles. I. Efficient detectors

We consider an entangled two-particle state that is produced from two independent downconversion laser sources by the process of “entanglement swapping,” so that the particles have never met. We prove a Greenberger-Horne-Zeilinger type theorem, showing that any deterministic, local, realistic theory is inconsistent with the quantum-mechanical perfect correlations for such a state. This theorem holds for individual events, with no inequalities, for detectors of 100% efficiency.

Figure 1

Schematic diagram of the creation of the two-particle state. In this experiment, there are two downconversions, one creating the pair of photons $a\text{−}b$, and the other the pair $c\text{−}d$. Each photon undergoes a rotation through the angle ${\phi }_i$, and particles $b$ and $c$ enter a Bell state analyzer (BSA), which will annihilate them while detecting which Bell state they were in. If the angles ${\phi }_i$ are set properly, as one of the perfect correlation cases, this process forces the particles $a$ and $d$ into a two-particle Bell state. In the actual experiment, the Bell state of $a$ and $d$ is not determined, only their polarizations, but this is sufficient to rule out locally realistic, deterministic theories as an explanation of their observed properties.

Figure 2

Bell-state determination of both sets of particles. This experiment is similar to that in Fig. 1, except that the Bell states of both sets of particles, $b\text{−}c$, and $a\text{−}d$, are determined at the BSA. One set of particles is shown entering from the top, to explicitly show the symmetry of the situation. When the particles are created, in a deterministic treatment, either the situation of Fig. 1 or that of Fig. 2 can be set up by the experimenter before the particles reach their respective BSA’s, and so both possibilities must be foreseen in the original instructions to the particles.