Classical communication over quantum channels: does entanglement help in nature?
Peixue Wu
quant-ph
Oct 29, 2026 · v1
cs.IT math-ph
TL;DR
All results, including the Schur-weighted block entropy inequality and strong additivity theorems, are fully verified in Lean with an accompanying repository.
Abstract
Can entangling codewords across independent uses of a quantum channel improve the rate of classical communication? An old conjecture states that there is no such entanglement advantage for all channels. Although Hastings's result and follow-up works disproved this conjecture, it turns out that enormous dimension is required to construct a channel where entanglement enhances classical communication. This indicates that there might be no entanglement advantage for many physical channels, or for all low dimensional channels. Motivated by recent progress on (generalized) amplitude damping channels, we propose a general Schur-weighted block entropy inequality, and show that there is no entanglement advantage for all phase-covariant channels including multi-level amplitude damping channels, generalized dephasing channels, and damping-dephasing channels. As a by-product, we also recover the recently established result that there is no entanglement advantage for all qubit channels.
Problem
The question is whether entangled codewords across channel uses can raise the classical capacity of a quantum channel above its Holevo information. Additivity is known to fail in general, but only for channels of very high dimension.
Approach
The author proves a new Schur-weighted block entropy inequality comparing a positive block matrix with a smaller matrix built from its block traces. This inequality verifies a Holevo–Shirokov-type entropic criterion for strong additivity. A decomposition of qubit-output channels into an entanglement-breaking part plus a coherent remainder extends the argument. Proofs are machine-checked in Lean.
Results
Strong Holevo additivity is established for all phase-covariant channels, which include multi-level amplitude damping, generalized dephasing, and damping-dephasing channels. The known result of no entanglement advantage for all qubit channels is recovered.
