bibkey: kopel2026sharpnorm authors: E. Kopel year: 2026 title: “A sharp norm inequality for entanglement-breaking channels” doi: 10.48550/arXiv.2609.27906 url: https://arxiv.org/abs/2609.27906v1 claim: “Theorem 1: the Bloch data (A, c) of an entanglement-breaking channel on M_d satisfy ‖A‖_*² + (d(d−1)/2)|c|² ≤ (d−1)²; open question 3 asks whether the coefficient d(d−1)/2 is optimal.” strata_touched:

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Kopel, A sharp norm inequality for entanglement-breaking channels

E. Kopel, A sharp norm inequality for entanglement-breaking channels, arXiv:2609.27906v1 (submitted 21 August 2026; quant-ph, cross-listed math-ph and math.OA).

Verified locator

DOI: 10.48550/arXiv.2609.27906. Primary version: https://arxiv.org/abs/2609.27906v1 (the only arXiv version). The TeX source of v1 supplies the Holevo form (Eq. eq:holevo, Section 1), the Bloch parameterisation (Section 2), Theorem 1 (thm:main, Section 3), the equality discussion (Section 4) and the list of open questions (Section 8); the numbering is the printed one.

Source statements

The source macros \Tr, \I and \nuc{·} are written out as $\mathrm{Tr}$, $I$ and $\Vert \cdot {\Vert }_{\ast }$.

Section 1: “A completely positive trace-preserving map $\Phi$ is \emph{entanglement breaking} (EB) if $(\mathrm{id}\otimes \Phi )(\rho )$ is separable for every input, equivalently if $\Phi$ admits a Holevo form $\Phi (\rho )={\sum }_k\mathrm{Tr}[E_k\rho ]{\sigma }_k$ with $\{E_k\}$ a POVM and ${\sigma }_k$ states”.

Section 2: “Fix a Hermitian traceless basis $\{{\lambda }_i{\}}_{i=1}^{d^2-1}$ of $M_d$ normalised by $\mathrm{Tr}[{\lambda }_i{\lambda }_j]=2{\delta }_{ij}$. […] Every state is $\rho =\frac{1}{d}I+\frac{1}{2}r\cdot \lambda$, $|r{|}^2\le \frac{2(d-1)}{d}$, with equality precisely for pure states. A channel acts affinely on the Bloch vector, $r\mapsto Ar+c$.”

Theorem 1: “Let $\Phi$ be an entanglement-breaking channel on $M_d$ with Bloch data $(A,c)$. Then $\Vert A{\Vert }_{\ast }^2+\frac{d(d-1)}{2}|c{|}^2\le (d-1{)}^2$.”

Section 4: “Whether equality with $c\ne 0$ is achievable for $d\ge 3$ is not settled here; condition (ii) becomes restrictive as the number of rank-one terms grows.”

Section 8, item 3: “Is the coefficient $d(d-1)/2$ optimal? It combines two separately sharp bounds which may not be simultaneously saturable with $c\ne 0$ in higher dimension.”

Scope

In the basis $\{{\lambda }_i\}$ the Bloch data of a channel are $A_{ij}=\frac{1}{2}\mathrm{Tr}[{\lambda }_i\Phi ({\lambda }_j)]$ and $c_i=\mathrm{Tr}[{\lambda }_i\Phi (I/d)]$, and $\Vert A{\Vert }_{\ast }$ is the nuclear (trace) norm. The paper proves Theorem 1 from the Holevo form, shows equality for the completely dephasing channel in every dimension and for a family with $c\ne 0$ at $d=2$, and relates the inequality to the covariance-matrix separability criterion.