arXiv:quant-ph/0610180v3 4 Jan 2007 Maximally Path-Entangled Number States Violate a Bell’s Inequality Christoph F. Wildfeuer, 1, ∗ Austin P. Lund, 2 and Jonathan P. Dowling 1 1 Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA 2 Centre for Quantum Computer Technology, Department of Physics, University of Queensland, Brisbane 4072, QLD, Australia We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path- entangled number state may be performed and lead to a violation of a Clauser-Horne Bell inequality for any finite photon number N. We present also 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 states. PACS numbers: 42.50.Xa, 03.65.Ud, 03.65.Wj, 03.67.Mn Maximally path-entangled number states of the form |Ψ〉 = 1 √ 2 (|N〉 a |0〉 b + e iϕ |0〉 a |N〉 b ) , (1) (often abbreviated to NOON states) have important applica- tions to quantum imaging [1], metrology [2, 3], and sens- ing [4]. Characterizing their quantum mechanical proper- ties is therefore a valuable task for improving upon sug- gested schemes. Entanglement is the most profound prop- erty of quantum mechanical systems. NOON states are non- separable states and hence are entangled. But do they also show nonlocal behavior when we perform a correlation exper- iment on the modes? The amount of nonlocality demonstrated by a Bell-type experiment provides an operational meaning of entanglement. It distinguishes between the class of states that are entangled but do admit a local hidden variable model and those which do not and so may be called EPR correlated [5]. Several publications [6] address the question whether the NOON states are EPR correlated for the case N = 1. Gisin and Peres have shown that for any nonfactorable pure state of two quantum systems it is possible to find pairs of ob- servables whose correlations violate a Bell’s inequality [7]. This result was later extended to more than two systems by Popescu and Rohrlich [8]. Recent experiments [9] have re- ported strong evidence that NOON states violate a Bell’s in- equality for N = 1, leaving open the question as to what ex- periments might show EPR correlations for N > 1. We pro- pose for the first time a specific experiment that shows that NOON states are EPR correlated for any finite N. We inves- tigate two measurement schemes using the unbalanced homo- dyne tomography setup described in [10] and compare the results. The correlation functions we calculate can be re- lated to well-known phase space distributions, the two-mode Q function and the two-mode Wigner function. Banaszek and W´ odkiewicz first pointed out the operational meaning of the Q and Wigner function [10]. We modify this approach and cal- culate the distribution functions for the NOON states entirely from these phase space distributions, and thereby construct a Clauser-Horne and a Clauser-Horne-Shimony-Holt Bell’s in- equality. Experimentally this can be implemented with an unbal- anced homodyne tomography setup as given, for example, in [10] and shown in Fig. 1. For simplicity we choose ϕ = π for FIG. 1: Unbalanced homodyne tomography setup for a Bell experi- ment with NOON states. Here |Ψ〉 = 1 √ 2 (|N〉 a |0〉 b −|0〉 a |N〉 b ) and a and b label the modes. the states in Eq. (1). It is now understood that the introduction of a reference frame is required in any Bell test [11]. Claims that NOON states do not violate Bell-type experiments have not properly appreciated this point. In the number basis, a shared local oscillator acts as the required reference frame. The beam splitters in this approach are assumed to operate in the limit where the transmittivity T → 1. We further assume that a strong coherent state |γ〉, where |γ|→ ∞, is incident to one of the two input ports. The beam splitter then acts as the displacement operator ˆ D(γ √ 1 − T ) on the second input port [12, 13, 14]. We introduce complex parameters α = γ a √ 1 − T and β = γ b √ 1 − T . The phase space parameterization with re- spect to these is then analogous to a correlation experiment with polarized light and different relative polarizer settings, where the nonlocality of polarization entangled states such as |Ψ〉 =(|H 〉 a | V 〉 b −| V 〉 a |H 〉 b )/ √ 2 is well established. In the first experimental setup we consider a simple non- number resolving photon-detection scheme. In the case of the homodyne tomography setup under consideration, the local positive operator valued measures (POVM’s) are given by, ˆ Q(α)= ˆ D(α)|0〉〈0| ˆ D † (α) , (2) ˆ P(α)= ˆ D(α) ∞ ∑ n=1 |n〉〈n| ˆ D † (α) . (3)