Bell’s experiment with intra- and inter-pair entanglement: Single-particle mode entanglement as a case study

Theoretical considerations of Bell-inequality experiments usually assume identically prepared and independent pairs of particles. Here we consider pairs that exhibit both intrapair and interpair entanglement. The pairs are taken from a large many-body system where all the pairs are generally entangled with each other. Using an explicit example based on single mode entanglement and an ancillary Bose-Einstein condensate, we show that the Bell-inequality violation in such systems can display statistical properties that are remarkably different from those obtained using identically prepared independent pairs. In particular, one can have probabilistic violation of Bell’s inequalities in which a finite fraction of all the runs result in violation even though there could be no violation when averaging over all the runs. Whether or not a particular run of results will end up being local realistically explainable is “decided” by a sequence of quantum (random) outcomes.

Figure 1

(Color online) The measurement bases used in the Bell test. The states $|\pm ⟩$ are defined as $|\pm ⟩=\left(|0⟩\pm |1⟩\right)/\sqrt{2}$. The ideal entangled state is $\left(|10⟩+|01⟩\right)/\sqrt{2}$. The measurement bases for the first mode (represented by the first symbol inside the ket) are denoted by $a$ and $a^{\prime }$, and the measurement bases for the second mode (represented by the second symbol inside the ket) are denoted by $b$ and $b^{\prime }$. Note that the measurement bases shown above, which we choose in order to simplify our numerical calculations below, differ from the ones that produce maximal violation of Bell’s inequality [7].

Figure 2

(Color online) A schematic diagram of the setup under consideration. After “passing through a beam splitter” (BS), a flying particle is in a quantum superposition of being in one of two outgoing paths. At each of the two possible final destinations of the flying particle there is a target particle that is initially in its ground state $|g⟩$ and is only driven to its excited state $|e⟩$ if the flying particle arrives at its location.

Figure 3

(Color online) A schematic diagram of a single ancillary BEC that is reused to transfer the mode entanglement of a stream of flying particles to the internal states of the target particles.

Figure 4

Probability density $P\left(C\right)$ for obtaining a given value of the correlation function $C_{a^{\prime },b^{\prime }}$ in a Bell test following our procedure of reusing the ancillary source (squares). Positive values of $C_{a^{\prime },b^{\prime }}$ correspond to a violation of Bell’s inequality. Here the ancillary source has $N=1$, i.e., a single ancillary particle shared between the two sides of the setup. For comparison we show the probability density if a new ancillary particle was used for each entangled pair (triangles). The results were obtained by constructing a histogram from ${10}^4$ runs, with $M=400$ entangled pairs generated in each run (In the calculation we have assumed that exactly one fourth of these pairs are measured in the $a^{\prime }\text{−}{b}^{\prime }$ bases). Doubling the value of $M$ reduces the width of the curve that corresponds to the case of independent pairs (triangles) by a factor of $\sqrt{2}$ but leaves the curve that corresponds to the case of correlated pairs (squares) essentially unchanged.

Figure 5

Same as in Fig. 4, but with $N=0$, i.e., no ancillary particles are used. Note that the finite violation probability in the case of independent pairs (triangles) is a result of the fact that we use a finite number of particles in the numerical calculations. This probability decreases and approaches zero if the number of particles is increased. We have verified, however, that the shape of the curve defined by the squares is essentially unchanged if we change the number of particles (provided that this number is much larger than one).