Unbounded Holevo additivity gaps in finite dimensions
Abstract
We establish unbounded two-use Holevo additivity gaps in finite dimensions. For each sufficiently large fixed integer K and all sufficiently large n, we construct channels Tn with output dimension Kn, input dimension exp(ΘK(n2)), and χ(Tn⊗2)−2χ(Tn) ≥ n[log2K/K−2log2(1+9/K)]−O(1/n). The gap is linear in output qubits, with an explicit quadratic input-qubit cost and an explicit threshold on n. We also obtain channels whose single-use Holevo quantity tends to zero while their two-use Holevo information, and hence classical capacity, diverges. The channels arise from structured tensor products of the complementary mixed-unitary channels used in Collins's free-probabilistic proof. We establish minimum-output-entropy gaps by combining Collins--Youn's product-group Haagerup inequality with Bordenave--Collins's quantitative strong-convergence estimates. We also provide a Lean certificate for our proofs.
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BibTeX
@misc{wang2026unbounded,
title = {{Unbounded Holevo additivity gaps in finite dimensions}},
author = {Jinzhao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18222},
eprint = {2609.18222},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18222},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unbounded Holevo additivity gaps in finite dimensions
AU - Jinzhao Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.18222
UR - https://arxiv.org/abs/2609.18222
AB - We establish unbounded two-use Holevo additivity gaps in finite dimensions. For each sufficiently large fixed integer K and all sufficiently large n, we construct channels T_n with output dimension K^n, input dimension exp(Θ_K(n^2)), and χ(T_n^⊗2)−2χ(T_n) ≥ n[log_2K/K−2log_2(1+9/K)]−O(1/n). The gap is linear in output qubits, with an explicit quadratic input-qubit cost and an explicit threshold on n. We also obtain channels whose single-use Holevo quantity tends to zero while their two-use Holevo information, and hence classical capacity, diverges. The channels arise from structured tensor products of the complementary mixed-unitary channels used in Collins's free-probabilistic proof. We establish minimum-output-entropy gaps by combining Collins--Youn's product-group Haagerup inequality with Bordenave--Collins's quantitative strong-convergence estimates. We also provide a Lean certificate for our proofs.
ER -