Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps

TL;DR

This paper proves that tensor squares of positive qubit maps are decomposable, extending a classical result, and characterizes the structure of extremal Pauli diagonal maps in higher dimensions.

Contribution

It generalizes Størmer's decomposition result to tensor squares of qubit maps and characterizes extremal Pauli diagonal maps in ququarts.

Findings

Tensor squares of positive qubit maps are decomposable.

The cone of decomposable ququart Pauli diagonal maps is fully characterized.

A combinatorial method for extremal rays of multi-qubit maps is developed.

Abstract

It is a well-known result due to E. St{\o}rmer 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 S. Fillipov and K. Magadov. 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.