Maximal violations and efficiency requirements for Bell tests with photodetection and homodyne measurements Marco Túlio Quintino, 1 Mateus Araújo, 1 Daniel Cavalcanti, 2 Marcelo França Santos, 1 and Marcelo Terra Cunha 3 1 Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil 2 Centre for Quantum Technologies, National University of Singapore, 3 Science drive 2, Singapore 117543 3 Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil (Dated: 23rd October 2018) We study nonlocality tests in which each party performs photodetection and homodyne meas- urements. The results of such measurements are dichotomized and a Clauser-Horne-Shimony-Holt (CHSH) inequality is used. We prove that in this scenario the maximal violation is attainable and fully characterize the set of maximally violating states. If we restrict our search to states composed of at most 2, 4, and 6 photons per mode, we find critical photodetection efficiencies of 0.48, 0.36, and 0.29. We also found an entangled variation of the famous cat states that has critical efficiency 0.32. These values are well within the limit of current photodetector technology, which suggests the present approach as a road for a loophole-free Bell experiment. I. INTRODUCTION Since Bell proved his theorem in 1964 [1], there has been considerable interest in experimentally ruling out local hidden variables models. Although Aspect’s 1982 experiment [2] gave a strong evidence in favor of the existence of nonlocal correlations, it relied on the fair sampling assumption, thus opening up the possibility of a local hidden variables description [3] for his experi- ment. From a fundamental point of view there is no reason to believe that nature maliciously disrespects fair sampling. However, the recent advent of device- independent protocols [4–9] gave an additional motiva- tion to search for a loophole free Bell test. In this case one may be fighting against an active opponent who can use the undetected photons to crack a given protocol. Hence, closing the detection loophole is a requirement for a demonstration of device-independent quantum information processing. In the standard Bell test using discrete variables and the Clauser-Horne-Shimony-Holt (CHSH) inequality [10], an overall detection efficiency higher than 2/3 is required to close the detection loophole [11, 12]. More recently, it was shown that the use of higher dimensional entangled states (and other inequalities) can lower this requirement [13]. However, these experimental situ- ations are still very demanding. An alternative method proposed to close the detection loophole in photonic systems is the use of homodyne measurements [14], which can be made very efficient. However, earlier results relying only on homodyne meas- urements required unfeasible states [15–19] or displayed very small violations [20, 21], indicating that homodyn- ing alone may not render the definite Bell test. More recently, Cavalcanti et al. explored a hybrid Bell test scenario that combines photodetection and homo- dyne measurements. An experimentally reasonable vi- olation of a CHSH inequality was found in a setup involving a feasible state, although with detection effi- ciencies still comparable to the best numbers found in the discrete variable cases [22]. The main goal of this paper is to show hybrid schemes that overcome Cavalcanti et al.’s result in two senses: larger violations and lower required efficiencies. First, we prove that the maximal violation of the CHSH in- equality can indeed be found within the hybrid scenario. Moreover we fully characterize the set of states that attain this maximal value. Second, we study the ro- bustness of the CHSH violations of natural classes of quantum states for several sources of errors (photodetec- tion inefficiency, transmission losses, and dark counts). We demonstrate the existence of states that achieve both a large violation of the CHSH inequality and a high resistance to detection inefficiencies. We organize our paper as follows: 1. Sec. II introduces the standard Bell test scenario and the CHSH inequality. 2. Sec. III describes the hybrid measurements scen- ario, which involves homodyne measurement and photodetection in each side of the Bell test. 3. In Sec. IV we show that maximal violation of the CHSH inequality can be obtained in the present scenario and characterize the family of states achieving such violation. 4. As the family of states achieving maximal CHSH violation is physically unreasonable, we provide, in Sec. V, a study of other families of quantum states previously discussed in the literature, such as N00N states and truncated-Fock states. We arXiv:1106.2486v3 [quant-ph] 19 Apr 2012