Strong violations of Bell-type inequalities for path-entangled number states

We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path-entangled number state may be performed. They lead to a violation of a Clauser-Horne Bell inequality for any finite photon number $N$. We also present an analytical expression for the two-mode Wigner function of a maximally path-entangled number state and investigate a Clauser-Horne-Shimony-Holt Bell inequality for such a state. We test other Bell-type inequalities. Some are violated by a constant amount for any $N$.

Figure 1

Unbalanced homodyne detection scheme for a Bell experiment with $N00N$ states. Here $\mid \Psi ⟩=\frac{1}{\sqrt{2}}(\mid N{⟩}_a\mid 0{⟩}_b-\mid 0{⟩}_a\mid N{⟩}_b)$ and $a$ and $b$ label the modes.

Figure 2

(Color online) Violation of the Clauser-Horne Bell inequality as a function of $N$.

Figure 3

The marginal $Q$ function $Q_m(y,v)$ for $N=1$.

Figure 4

The marginal $Q$ function $Q_m(y,v)$ for $N=2$.

Figure 5

The marginal $Q$ function $Q_m(y,v)$ for $N=3$.

Figure 6

The marginal Wigner function $W_m(y,v)$ for $N=1$.

Figure 7

The marginal Wigner function $W_m(y,v)$ for $N=2$.

Figure 8

The marginal Wigner function $W_m(y,v)$ for $N=3$.

Figure 9

(Color online) Violation of the inequality (22) as a function of $N$.

Figure 10

(Color online) Violation of the inequality (24) as a function of $N$.

Figure 11

(Color online) Violation of the inequality (25) as a function of $N$.

Figure 12

Alice and Bob apply a local unitary operation on her/his mode $a$ and $b$.