Quantum U-channels on S-spaces

If the symmetry, (an operator J satisfying J=J^*=J⁻¹) which defines the Krein space, is replaced by a (not necessarily self-adjoint) unitary, then we have the notion of an S-space which was introduced by Szafraniec. In this paper, we consider S-spaces and study the structure of completely U-positive maps between the algebras of bounded linear operators. We first give a Stinespring-type representation for a completely U-positive map. On the other hand, we introduce Choi U-matrix of a linear map and establish the equivalence of the Kraus U-decompositions and Choi U-matrices. Then we study properties of nilpotent completely U-positive maps. We develop the U-PPT criterion for separability of quantum U-states and discuss the entanglement breaking condition of quantum U-channels and explore U-PPT squared conjecture. Finally, we give concrete examples of completely U-positive maps and examples of 3 ⊗ 3 quantum U-states which are U-entangled and U-separable.