Realignment Operation and CCNR Criterion of Separability for States in Infinite-Dimensional Quantum Systems

The realignment operation and the computable cross norm or realignment (CCNR) criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let HA and HB be complex Hilbert spaces with dim HA ⊕ HB ≤ +∞. Let ρ be a quantum state acting on HA ⊕ HB and {δk} be the Schmidt coefficients of ρ as a vector in the Hilbert space C2(HA) ⊕ C2(HB). We introduce the realignment operator ρR and the computable cross norm ||ρ||CCN of ρ and show that if ρ is separable, then ||ρR||Tr = ||ρ||CCN = Σk δ ≤ 1. In particular, if ρ is a pure state, then ρ is separable if and only if ||ρR||Tr = ||ρ||CCN = Σk δk = 1. For the finite-dimensional case, this recovers the original computable cross norm criterion.