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 $\|A\|_*^2 + \frac{d(d-1)}{2}|c|^2 \le (d-1)^2$, where $\|\cdot\|_*$ denotes the nuclear norm. The bound is attained in every dimension by the completely dephasing channel, and at $d=2$ by an explicit family with $c\neq0$. The qubit case reads $\|A\|_*^2+|c|^2\le1$ and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound $\|A\|_*\le1$ 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.

5 pages. Ancillary files: numerical verification scripts (all dimensions and qubit case)

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Operator Algebras (math.OA)

arXiv:2609.27906 [quant-ph]

(or arXiv:2609.27906v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.27906 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

verify_lemma2.py

verify_lemma2_highdim.py

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