A sharp norm inequality for entanglement-breaking channels

Abstract

For a channel $\Phi$ on $M_d$ written in the generalised Bloch parameterisation $r\mapsto Ar+c$, we prove that entanglement breaking implies $\Vert A{\Vert }_{\ast }^2+\frac{d(d-1)}{2}|c{|}^2\le (d-1{)}^2,$ where $\Vert \cdot {\Vert }_{\ast }$ denotes the nuclear norm. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient $d(d-1)/2$ cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects ${\alpha }_k|e_k⟩⟨e_k|$ and pure outputs $|{\psi }_k⟩⟨{\psi }_k|$ whose overlaps are an affine image of those of the measurement directions, $|⟨{\psi }_j|{\psi }_k⟩{|}^2=P+(1-P)(d|⟨e_j|e_k⟩{|}^2-1)/(d-1)$ for a common $P=\mathrm{Tr}\Phi (1/d{)}^2$. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at $d=2$. The qubit case reads $\Vert A{\Vert }_{\ast }^2+|c{|}^2\le 1$ and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound $\Vert A{\Vert }_{\ast }\le 1$ by retaining the POVM completeness relation ${\sum }_k{f}_k=0$, which her argument discards; this recentres $A$ from a second moment into a cross-covariance, and is the entire content of the improvement.

arXiv comment9 pages; v2: the equality cases of the inequality are characterised (Theorem 5); the constant channel onto a pure state settles the coefficient and c != 0 questions of v1 (pointed out by H. Ma and W. Zhang); equality conditions of Section 4 corrected; abstract states the relation to the covariance-matrix criterion; ancillary files updated