As discussed in Watrous' book, quantum-to-classical channels are CPTP maps whose output is always fully depolarised. These can always be written as $$\Phi_\mu(X) = \sum_a \langle X,\mu(a)\rangle E_{a,a}$$ for some set of positive operators $\mu(a)\ge0$ satisfying $\sum_a \mu(a)=I$.

What is the (a) Kraus representation of these maps?


We start from the defining form of the channel as $\Phi_\mu(X)=\sum_a \operatorname{tr}(\mu(a)X)E_{a,a}$.$\newcommand{\PP}{\mathbb{P}}\newcommand{\tr}{\operatorname{tr}}\newcommand{\calX}{\mathcal X}\newcommand{\calY}{\mathcal Y}\newcommand{\calZ}{\mathcal Z}\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\bs}[1]{\boldsymbol{#1}}$

(Natural representations) To derive the natural representation of the map, note that $$\Phi_\mu(E_{k,\ell})=\sum_a\mu(a)_{\ell,k} E_{a,a}.$$ It follows that $$K(\Phi_\mu)_{ij,k\ell} = \langle i\rvert \Phi_\mu(E_{k,\ell})\lvert j\rangle = \sum_a \mu(a)_{\ell,k} \langle i\rvert E_{a,a}\lvert j\rangle=\delta_{ij} \mu(i)_{\ell,k},$$ where $E_{a,b}\equiv\lvert a\rangle\!\langle b\rvert$ and $K(\Phi)$ denotes the natural representation of $\Phi$. As an operator, this reads $$K(\Phi_\mu) %= \sum_a \lvert a,a\rangle \langle \mu(a)^T\rvert \equiv \sum_a \ket{a,a}\!\operatorname{vec}(\mu(a)^*)^T.$$

(Choi representation) Consider now the Choi operator, defined as $J(\Phi)\equiv \sum_{i,j}\Phi(E_{i,j})\otimes E_{i,j}$. From this we get $$J(\Phi_\mu) = \sum_{a,i,j} \mu(a)_{j,i} E_{a,a}\otimes E_{i,j} = \sum_a E_{a,a}\otimes \mu(a)^T.$$ We can also get this from $K(\Phi)$, using the relation $\langle i,j\rvert J(\Phi)\lvert k,\ell\rangle = \langle i,k\rvert K(\Phi)\lvert j,\ell\rangle$.

(Kraus representation from Choi) One way to get the Kraus representation is via the spectral decomposition of the Choi. From the relations above, we see that the spectral decomposition of the Choi is in this case quite easy: define $\ket{v_{a,j}}\equiv \ket a\otimes \ket{p_{a,j}^*}$ with $\ket{p_{a,j}}$ the eigenvector of $\mu(a)$ with eigenvalue $p_{a,j}$, and using $\ket{p_{a,j}^*}$ to denote the complex conjugate of $\ket{p_{a,j}}$.

From this we get the Kraus operators as the maps $A_{a,j}$ of the form: $$ A_{a,j} = \sqrt{p_{a,j}} \lvert a\rangle\!\langle p_{a,j}\rvert \Longleftrightarrow (A_{a,j})_{ik} = \sqrt{p_{a,j}}\langle i,k\ket{v_{a,j}} = \sqrt{p_{a,j}} \delta_{a,i}\langle k\rvert p_{a,j}^*\rangle. \tag1 $$ With these operators, we can write $$\Phi_\mu(X) = \sum_{a,j} A_{a,j} X A_{a,j}^\dagger.$$

(Direct derivation) For a direct route that doesn't require passing through the Choi representation, let us write down the explicit form of $\Phi_\mu(X)$: $$\Phi_\mu(X) = \sum_{a,\ell k} \mu(a)_{k,\ell}X_{\ell,k} E_{a,a}.$$ Because, by hypothesis, $\mu(a)\ge0$, we can find some operator $M_a$ such that $\mu(a)=M_a^\dagger M_a$. Componentwise, this reads $\mu(a)_{k,\ell} = \sum_j(M_a^*)_{j,k}(M_a)_{j,\ell}.$ Using this in the expression above we get $$\Phi_\mu(X) = \sum_{a,jk\ell} E_{a,a}(M_a^*)_{j,k} X_{\ell,k} (M_a)_{j,\ell} E_{a,a}.$$ The corresponding Kraus operators thus have the form $$A_{a,j}= \lvert a\rangle\!\langle j\rvert M_a.\tag2$$ Of course, this now begs the question: are the Kraus operators in (2) compatible with those previously derived in (1)? The answer is: not necessarily. Equation (2) is more general, due to the freedom in the choice of $M_a$, and in particular doesn't necessarily lead to orthogonal Kraus operators, like (1) does. To see this, notice that we can generally express $M_a$ in terms of the eigendecomposition of $\mu(a)$ as $$M_a = \sum_\ell \sqrt{p_{a,\ell}} \lvert u_{a,\ell}\rangle\!\langle p_{a,\ell}\rvert,$$ for any choice of orthonormal vectors $\lvert u_{a,\ell}\rangle$. In particular, we can choose $\lvert u_{a,\ell}\rangle=\lvert \ell\rangle$ to retrieve (1).

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    $\begingroup$ Can't you also construct those by writing $\Phi(\rho)=\mathrm{tr}(\rho F_i)\sigma_i$ and writing the trace explicitly as a sum and taking your favorite root $M_i^\dagger M_i=F_i$? $\endgroup$ – Norbert Schuch Jul 17 at 23:12
  • $\begingroup$ @NorbertSchuch indeed, I was interested in the route passing through the Choi because I wanted orthogonal Kraus ops as well, but I edited the post adding the direct route as well $\endgroup$ – glS Jul 18 at 18:23

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