We propose an experimental scheme to implement a second-order nonlocal superposition operation a †2 + e iφb †2 and its variants by way of Hong-Ou-Mandel interference. The second-order coherent operations enable us to generate a NOON state with high particle number in a heralded fashion and also can be used to enhance the entanglement properties of continuous variable states. We discuss the feasibility of our proposed scheme considering realistic experimental conditions such as on-off photodetectors with nonideal efficiency and imperfect single-photon sources.

We show that the coherent superposition tâ râ † of photon subtraction and addition applied to each local mode of a two-mode entangled state can enhance the nonlocality manifested by the violation of a Bell inequality. A twomode squeezed state is used as an input state for this demonstration with four different Bell inequalities employed: Bell inequalities adopting displaced parity operator, pseudospin operator, homodyne measurement, and conditional entropy, respectively. We find that the coherent operation significantly enhances the nonlocality remarkably in the weak squeezing limit, compared with other possible non-Gaussian operations. It can also give a maximal Bell violation with a very small squeezing for the inequalities with pseudospin operator and conditional entropy.

We present a formalism to derive entanglement criteria beyond the Gaussian regime that can be readily tested by only homodyne detection. The measured observable is the Einstein-Podolsky-Rosen (EPR) correlation. Its arbitrary functional form enables us to detect non-Gaussian entanglement even when an entanglement test based on second-order moments fails. We illustrate the power of our experimentally friendly criteria for a broad class of non-Gaussian states under realistic conditions. We also show rigorously that quantum teleportation for continuous variables employs a specific functional form of EPR correlation.PACS numbers: 03.67. Mn, 03.65.Ud, 42.50.Dv Quantum entanglement plays a key role in making quantum-mechanical predictions distinct from their classical counterparts. Its characterization and detection have been a topic of crucial importance that has attracted a great deal of effort, both theoretically and experimentally. Despite remarkable progress, there still exists a pressing demand for further developments particularly for continuous variables (CVs). In the CV regime, Gaussian entangled states have been the subject of study, but now considerable attention has been directed to nonGaussian states. This may be attributed to the fact that non-Gaussian entangled states not only provide practical advantages over their Gaussian counterparts, e.g. in quantum teleportation [1] and dense coding [2], but also turn out to be essential, e.g. for universal quantum computation [3] and nonlocality test [4].The quantum state of a CV system is fully described in phase space by representing two quadrature variableŝ X (position) andP (momentum). To address the correlation of two CV systems, the EPR operatorsûgP B can be defined, as Einstein, Podolsky, and Rosen (EPR) originally envisioned [5]. Here g is an arbitrary real number and A and B label each subsystem. When the sum of the variances, Eg 2 , the state of the total system is inseparable [6] and such states are said to be EPR-correlated. In fact, it has been shown that the EPR correlation, after local unitary operations, is a necessary and sufficient condition for two-mode Gaussian entanglement [6] (In this Letter, we study a simple case, where we use {û,v} instead of {û ′ ,v ′ } assuming g = 1). Using the uncertainty principle as a requirement of a physical system in conjunction with the partial transposition [7], Simon found a necessary and sufficient condition for Gaussian entangled states (Simon criterion) [8]; this criterion is also concerned with the second moments of the quadrature variables [9]. These criteria are theoretically easy to calculate and readily tested experimentally since the quadrature variables can be measured using highly-efficient homodyne detectors.Beyond the Gaussian regime, some entanglement criteria have also been proposed [10][11][12][13][14][15][16][17]. Shchukin and Vogel particularly derived a hierarchy of entanglement conditions using higher-order moments [13,14]. For finitedimensional systems, a remarkable criterion based on co...