A sharp norm inequality for entanglement-breaking channels
Abstract
For a channel Φ on Md written in the generalised Bloch parameterisation r ↦ Ar+c, we prove that entanglement breaking implies |A|*2 + (d(d−1))/2 |c|2 ≤ (d−1)2 , where |·|* 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 αk|ek⟩⟨ ek| and pure outputs |ψk⟩⟨ψk| whose overlaps are an affine image of those of the measurement directions, |⟨ψj|ψk⟩|2 = P + (1−P) (d|⟨ ej|ek⟩|2 − 1)/(d−1) for a common P = Tr Φ(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 |A|*2+|c|2≤1 and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound |A|*≤1 by retaining the POVM completeness relation ∑k fk=0, which her argument discards; this recentres A from a second moment into a cross-covariance, and is the entire content of the improvement.
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BibTeX
@misc{kopel2026sharp,
title = {{A sharp norm inequality for entanglement-breaking channels}},
author = {Eran Kopel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27906},
eprint = {2609.27906},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27906},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sharp norm inequality for entanglement-breaking channels
AU - Eran Kopel
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.27906
UR - https://arxiv.org/abs/2609.27906
AB - For a channel Φ on M_d written in the generalised Bloch parameterisation r ↦ Ar+c, we prove that entanglement breaking implies |A|_*^2 + (d(d−1))/2 |c|^2 ≤ (d−1)^2 , where |·|_* 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 α_k|e_k⟩⟨ e_k| and pure outputs |ψ_k⟩⟨ψ_k| whose overlaps are an affine image of those of the measurement directions, |⟨ψ_j|ψ_k⟩|^2 = P + (1−P) (d|⟨ e_j|e_k⟩|^2 − 1)/(d−1) for a common P = Tr Φ(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 |A|_*^2+|c|^2≤1 and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound |A|_*≤1 by retaining the POVM completeness relation ∑_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.
ER -