When Non-Gaussian States are Gaussian: Generalization of Non-Separability Criterion for Continuous Variables

Whether a quantum state is entangled or not represents a very important question in quantum-information theory. Such knowledge reveals whether one can take advantage of the non-local properties of the state in quantum protocols such as quantum teleportation [1] and quantum cryptography [2]. This issue has been dealt with by many authors in recent years primarily in qubit systems where the Peres-Horodecki partial transpose separability condition was the first method to figure out if a two-qubit state was separable [3]. In general, for N qubits the solution is not known. Continuous variable systems have proven to be an extremely useful setting for quantum cryptography and communication [4]. In these protocols entangled states are required and the question of separability arises naturally. For two-mode systems separability criteria for Gaussian states were established in Refs. [5, 6] which proved to be both necessary and sufficient. More recently new separability criteria based on uncertainty relations for two-mode representations of SU(2) and SU(1,1) algebras have appeared [7, 8, 9]. These criteria have particularly targeted uncovering whether non-Gaussian states are separable or not as previous criteria fail to detect relatively simple entangled states.