1 Random low rank mixed states are entangled

We prove that random rank r ≤ 2m − 3 mixed states in bipartite quantum systems H A ⊗ H B are entangled based on algebraicgeometric separability criterion recently proved in [1]. This also means that algebraic-geometric separability criterion can be used to detect all low rank entagled mixed states outside a measure zero set. Quantum entanglement was first noted as a feature of quantum mechanics in the famous Einstein, Podolsky and Rosen [2] and Schrodinger [3] papers. Its importance lies not only in philosophical considerations of the nature of quantum theory, but also in applications where it has emerged recently that quantum entanglement is the key ingredient in quantum computation [4] and communication [5] and plays an important role in cryptography [6,7]. A mixed state ρ in the bipartite quantum system H = H A ⊗Hn B is called separable if it can be written in the form ρ = Σjpj|ψj >< ψj| ⊗ |φj >< φj|,