Entanglement Detection: Complexity and Shannon Entropic Criteria
Entanglement plays a key role in quantum information. The separability problem arises: how to prove that a given quantum state is entangled? Covariance-matrix-based entanglement criteria are the most important and widely used approach to this problem in continuous variable systems. Here, the separability problem is systematically studied in the more experimentally realistic setting that the questioned state is only partly known. A simple general formalism is proposed for this variant of the separability problem. It builds relations between certain classes of criteria and bounds the power of these criteria. In particular, it shows the equivalence of many seemingly different covariance-matrix-based criteria; it offers guidance on the study of and reveals the inherent limitations of covariance-matrix-based criteria. Also, a Shannon entropic criterion is proposed for m circle plus n-mode states. It unifies and improves many known covariance-matrix-based criteria and Shannon entropic criteria; it is necessary and sufficient for 1 circle plus n-mode Gaussian states and m circle plus n-mode bisymmetric Gaussian states.
complexity theory covariance matrix entropy quantum entanglement quantum mechanics