We analyze positivity of a tensor product of two linear qubit maps, Φ1 ⊗ Φ2. Positivity of maps Φ1 and Φ2 is a necessary but not a sufficient condition for positivity of Φ1 ⊗ Φ2. We find a nontrivial sufficient condition for positivity of the tensor product map beyond the cases when both Φ1 and Φ2 are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for n-tensor-stable positive qubit maps, i.e. such qubit maps Φ that Φ ⊗n is positive. Particular cases of 2-and 3-tensor-stable positive qubit maps are fully characterized, and the decomposability of 2-tensor-stable positive qubit maps is discussed. The case of non-unital maps is reduced to the case of appropriate unital maps. Finally, n-tensor-stable positive maps are used in characterization of multipartite entanglement, namely, in the entanglement depth detection.I.

Absolutely separating quantum maps and channels AbstractAbsolutely separable states ñ remain separable under arbitrary unitary transformations  † U U . By example of a three qubit system we show that in a multipartite scenario neither full separability implies bipartite absolute separability nor the reverse statement holds. The main goal of the paper is to analyze quantum maps resulting in absolutely separable output states. Such absolutely separating maps affect the states in a way, when no Hamiltonian dynamics can make them entangled afterwards. We study the general properties of absolutely separating maps and channels with respect to bipartitions and multipartitions and show that absolutely separating maps are not necessarily entanglement breaking. We examine the stability of absolutely separating maps under a tensor product and show that F ÄN is absolutely separating for any N if and only if Φ is the tracing map. Particular results are obtained for families of local unital multiqubit channels, global generalized Pauli channels, and combination of identity, transposition, and tracing maps acting on states of arbitrary dimension. We also study the interplay between local and global noise components in absolutely separating bipartite depolarizing maps and discuss the input states with high resistance to absolute separability.

We introduce a spin polarization-scaling map for spin-j particles, whose physical meaning is the decrease of spin polarization along three mutually orthogonal axes. We find conditions on three scaling parameters under which the map is positive, completely positive, entanglement breaking, 2-tensor-stable positive, and 2-locally entanglement annihilating. The results are specified for maps on spin-1 particles. The difference from the case of spin-1 2 particles is emphasized.

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.