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

Abstract

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 H A and H B be complex Hilbert spaces with dim H A ⊕ H B ≤ +∞. Let ρ be a quantum state acting on H A ⊕ H B and {δ k } be the Schmidt coefficients of ρ as a vector in the Hilbert space C 2 ( H A ) ⊕ C 2 ( H B ). We introduce the realignment operator ρ R and the computable cross norm ||ρ|| CCN of ρ and show that if ρ is separable, then ||ρ R || Tr = ||ρ|| CCN = Σ k δ ≤ 1. In particular, if ρ is a pure state, then ρ is separable if and only if ||ρ R || Tr = ||ρ|| CCN = Σ k δ k = 1. For the finite-dimensional case, this recovers the original computable cross norm criterion.