Entanglement swapping of noisy states: A kind of superadditivity in nonclassicality

We address the question as to whether an entangled state that satisfies local realism will give a violation of the same after entanglement swapping in a suitable scenario. We consider such a possibility as a kind of superadditivity in nonclassicality. Importantly, it will indicate that checking for violation of local realism, in the state obtained after entanglement swapping, can be a method for detecting entanglement in the input state of the swapping procedure. We investigate various entanglement swapping schemes, which involve mixed initial states. The strength of violation of local realism by the state obtained after entanglement swapping is compared with the one for the input states. We obtain a kind of superadditivity of violation of local realism for Werner states, consequent upon entanglement swapping involving Greenberger-Horne-Zeilinger-state measurements. We also discuss whether entanglement swapping of specific states may be used in quantum repeaters with a substantially reduced need to perform the entanglement distillation step.

Figure 1

Entanglement swapping between two states.

Figure 2

A star configuration swapping. A GHZ basis measurement is performed on A, C, and E. This is represented by a box.

Figure 3

A star configuration swapping. A Bell measurement is performed on A and D which is denoted by a box.

Figure 4

A star configuration swapping. $A_1,B_{11},B_{12}$, $A_2,B_{21},B_{22}$, and $A_3,B_{31},B_{32}$ share noisy GHZ states. All $B$’s are at distant locations but $A$’s are in the same laboratory. A GHZ basis measurement is performed by $A_1$, $A_2$, and $A_3$, as depicted in the figure by a box.

Figure 5

(Color online) (a) Plot of the critical visibilities $V_N^{\left(M\right)}$ for $M=2$ (the stars). This is the critical visibility required (in the parent state) for violation of local realism by the swapped state in the case when one representative from each of $N$ Werner states [Eq. (4)] come together to perform an entanglement swapping in the $N$-qubit GHZ basis (see Fig. 6). The violation of local realism in the swapped state is considered by using the MK inequalities. (b) The critical visibility $V_N^f$ (the diamonds) is that for violation of local realism by considering functional Bell inequalities in the same $N$-qubit swapped state. The figure shows the relative monotonic decrease of the two visibilities with $N$.

Figure 6

$N$ Werner states are distributed between $A_i$ and $B_i$ (for $i=1,2,\dots ,N$) and a GHZ basis measurement is performed at $A_1,A_2,\dots ,A_N$.