(The CCNR criterion) The computable cross-norm or realignment (CCNR) entanglement criterion, as discussed in (Gühne and Toth 2008), is based on the observation that any bipartite state $\rho$ can be decomposed as $$\rho = \sum_k \lambda_k G_k^A\otimes G_k^B,\tag1$$ for some $\lambda_k\ge0$ and Hermitian operators such that $\operatorname{Tr}(G_k^A G_\ell^A)=\operatorname{Tr}(G_k^B G_\ell^B)=\delta_{k\ell}$. Any separable $\rho$ must correspond to coefficients such that $\sum_k \lambda_k\le1$, thus $\sum_k\lambda_k>1$ certifies entanglement.

(CCNR as SVD) This decomposition can be seen as the SVD applied to the operator $$|i,j\rangle\mapsto \sum_{k,\ell}\langle k, j|\rho|\ell,i\rangle \,\,|k,\ell\rangle.$$ One way to show this is to write the SVD of such operator as $$\langle k, j|\rho|\ell,i\rangle = \sum_m p_m u^{(m)}_{k\ell} \bar v^{(m)}_{ij},$$ with $p_m\ge0$ and $\langle u^{(m)}, u^{(n)}\rangle=\langle v^{(m)},v^{(n)}\rangle=\delta_{mn}$. Rewriting this in operatorial form, we get back the decomposition in (1).

(CCNR and Choi representation) An alternative way to see this decomposition is to notice that it is the SVD of the (inverse?) Choi representation of $\rho$. To be more precise, let $\Phi$ be the channel whose Choi is $\rho$: $J(\Phi)=\rho$. Then the decomposition in (1) amounts to the SVD of the natural representation $K(\Phi)$ of $\Phi$ (this is "vectorised" version of $\Phi$: the operator such that $K(\Phi)\operatorname{vec}(X)=\operatorname{vec}(\Phi(X))$ for all $X$).

(CCNR as a statement about entanglement breaking channels) Now, we know that $\rho=J(\Phi)$ is separable iff $\Phi$ is entanglement breaking. This makes the CCNR criterion equivalent to the following statement about channels: if a channel $\Phi$ is such that $\|K(\Phi)\|_1>1$, then $\Phi$ is not entanglement breaking. Here $\|\cdot\|_1$ denotes the trace norm, which equals the sum of the singular values of its argument.

Is there any intuition behind this fact? Is there any general statement about what the trace norm of $K(\Phi)$ tells us about $\Phi$ that can be applied to this case?


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