Qubit Maps

H. H. Wills Physics Laboratory

Sergey N. Filippov, Kamil Yu. Magadov

Abstract We analyze positivity of a tensor product of two linear qubit maps, Φ 1 ⊗ Φ 2 . Positivity of maps Φ 1 and Φ 2 is a necessary but not a sufficient condition for positivity of Φ 1 ⊗ Φ 2 . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both Φ 1 and Φ 2 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.

Michael Berry, Nandor L. Balazs, M. Tabor +1 more

Dariusz Chruściński, Gen Kimura, Farrukh Maksutovich Mukhamedov

Abstract Unital qubit Schwarz maps interpolate between positive and completely positive maps. It is shown that the relaxation rates of the qubit semigroups of unital maps enjoying the Schwarz property satisfy a universal constraint, which provides a modification of the corresponding constraint known for completely positive semigroups. As an illustration, we consider two paradigmatic qubit semigroups: Pauli dynamical maps and phase-covariant dynamics. This result has two interesting implications: it provides a universal constraint for the spectra of qubit Schwarz maps and gives rise to a necessary condition for a Schwarz qubit map to be Markovian.

Alexander Müller‐Hermes

It is a well-known result due to Størmer [Acta Math. 110, 233–278 (1963)] that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by Filippov and Magadov [J. Phys. A: Math. Theor. 50(5), 055301 (2017)]. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show our main result, we reduce the problem to Pauli diagonal maps. We then characterize the cone of decomposable ququart Pauli diagonal maps by determining all 252 extremal rays of ququart Pauli diagonal maps that are both completely positive and completely copositive. These extremal rays split into three disjoint orbits under a natural symmetry group, and two of these orbits contain only entanglement breaking maps. Finally, we develop a general combinatorial method to determine the extremal rays of Pauli diagonal maps that are both completely positive and completely copositive between multi-qubit systems using the ordered spectra of their Choi matrices. Classifying these extremal rays beyond ququarts is left as an open problem.

David K. Ferry