Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 3 / c

By linearity, the isotropic qubit map acts asIt contracts or expands every direction by the same factor and reverses the vector when $r<0$. More generally ${\mathcal{E}}_r(A)=rA+(1-r)\mathrm{tr}(A)I/2$, so the map preserves trace and Hermiticity for every real $r$.

The output of a state of Bloch radius $u$ has eigenvalues $(1\pm ru)/2$. These are nonnegative for every $0\le u\le 1$ exactly whenIf $|r|>1$, a pure input already gives a negative eigenvalue. Conversely, $|r|\le 1$ preserves the Bloch ball, hence positivity for states and, by positive scaling, for all positive operators. This is positivity of the map alone; physical implementation on part of an entangled system needs complete positivity.

Use the unnormalized Choi matrix convention $J_r={\sum }_{j,k=0}^1|j⟩⟨k|\otimes {\mathcal{E}}_r(|j⟩⟨k|)$, which has trace two. In the ordered basis $|00⟩,|01⟩,|10⟩,|11⟩$,Equivalently $J_r=2r|\Omega ⟩⟨\Omega |+(1-r)I_4/2$. The convention obtained by applying the channel to one half of the normalized Bell state gives $J_r/2$; its positivity criterion is the same.

The Bell state $|\Omega ⟩$ is an eigenvector of $J_r$ with eigenvalue $(1+3r)/2$. Its three-dimensional orthogonal complement has eigenvalue $(1-r)/2$. The Choi criterion for complete positivity therefore givesFor sufficiency one can also write a Kraus representation explicitly:In this range the four coefficients are nonnegative and sum to one, giving a physical mixture of Pauli conjugations. This is the complete positivity range of the depolarizing channel in the Bloch-scaling parameterization.