Bath dimension and initial entropy for closed repeated use of a quantum channel

Abstract:We characterize the bath resources needed to supply repeated uses of a fixed finite-dimensional quantum channel in a closed device. For each horizon $T$, one bath, one initial state and one repeated unitary are fixed before the user. Each output is returned before the next input arrives; no reset, discard, fresh ancilla or uncounted controller is available. Approximation error must vanish against arbitrary adaptive users with quantum memory and references. Writing $r=\lim \log_2(R_T)/T$ for the bath dimension rate and $s=\lim S(\omega_T)/T$ for the actual initial entropy rate, we prove that the achievable region is exactly $s\ge 0$, $r+s\ge h$ and $r-s\ge\kappa$. Here $h$ is maximum entropy exchange and $\kappa$ is a smoothed independent-reference extension cost, with the zero-error limit taken before the supremum over full-rank inputs. Its exact fixed-input form is an affine transform of the zero-leakage quantum privacy funnel. The minimum dimension rate is $(h+\kappa)/2$. The proof combines entropy converses, a bath-dimension-independent support repair, and a closed adaptive implementation of encoder-only fully quantum Slepian--Wolf recycling. All seeds, clocks, workspace and retained residues are counted. Worked examples include dephasing, pure replacement and a qubit channel with $0<\kappa<h$. No computability of $\kappa$ or efficient circuit synthesis is claimed.

29 pages; includes two appendices and reproducibility documentation. Code and verification materials: this https URL

arXiv:2609.18267 [quant-ph]

(or arXiv:2609.18267v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.18267 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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