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 log2(RT)/T for the bath dimension rate and s=lim S(ωT)/T for the actual initial entropy rate, we prove that the achievable region is exactly s≥ 0, r+s≥ h and r−s≥κ. Here h is maximum entropy exchange and κ 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+κ)/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<κ<h. No computability of κ or efficient circuit synthesis is claimed.
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BibTeX
@misc{douglas2026bath,
title = {{Bath dimension and initial entropy for closed repeated use of a quantum channel}},
author = {Seth Douglas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18267},
eprint = {2609.18267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18267},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bath dimension and initial entropy for closed repeated use of a quantum channel
AU - Seth Douglas
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18267
UR - https://arxiv.org/abs/2609.18267
AB - 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(ω_T)/T for the actual initial entropy rate, we prove that the achievable region is exactly s≥ 0, r+s≥ h and r−s≥κ. Here h is maximum entropy exchange and κ 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+κ)/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<κ<h. No computability of κ or efficient circuit synthesis is claimed.
ER -