Chapter Three. Completely Positive Maps

Ingemar Bengtsson, Karol Życzkowski

Good mathematicians see analogies. Great mathematicians see analogies between analogies. Stefan Banach We have already discussed the static structure of our ‘Quantum Town’ – the set of density matrices – on the one hand, and the set of all physically realizable processes which may occur in it on the other hand. Now we are going to reveal a quite remarkable property: the set of all possible ways to travel in the ‘Quantum Town’ is equivalent to a ‘Quantum Country’ – an appropriately magnified copy of the initial ‘Quantum Town’! More precisely, the set of all transformations which map the set of density matrices of size N into itself (dynamics) is identical to a subset of the set of density matrices of size N 2 (kinematics). From a mathematical point of view this relation is based on the Jamiołkowski isomorphism , analysed later in this chapter. Before discussing this intriguing duality, let us leave the friendly set of quantum operations and pay a short visit to a neighbouring land of maps, as yet unexplored, which are positive but not completely positive. Positive and decomposable maps Quantum transformations which describe physical processes are represented by completely positive (CP) maps. Why should we care about maps which are not CP? On the one hand it is instructive to realize that seemingly innocent transformations are not CP, and thus do not correspond to any physical process. On the other hand, as discussed in Chapter 15, positive but not completely positive maps provide a crucial tool in the investigation of quantum entanglement.

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.

W.A. Majewski

Sixia Yu

The concept of the half density matrix is proposed. It unifies the quantum states that are described by density matrices and physical processes that are described by completely positive maps. With the help of the half-density-matrix representation of the Hermitian linear map, we show that every positive map that is not completely positive is a difference of two completely positive maps. A necessary and sufficient condition for a positive map that is not completely positive is also presented, which is illustrated by some examples.