Nonadditivity in fermionic quantum Shannon theory
Abstract
We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state τ, we obtain the exact entanglement of formation, EFphys(τ⊗ m)=⌈ m/2⌉log2 for every m≥1. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order α∈(0,∞]. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For τ, Bell violation under local parity-preserving measurements first appears at three copies.
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BibTeX
@misc{salek2026nonadditivity,
title = {{Nonadditivity in fermionic quantum Shannon theory}},
author = {Farzin Salek and Amir-Reza Negari and Graeme Smith and Jens Eisert and Debbie Leung},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06758},
eprint = {2610.06758},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06758},
}
RIS
TY - UNPB
TI - Nonadditivity in fermionic quantum Shannon theory
AU - Farzin Salek
AU - Amir-Reza Negari
AU - Graeme Smith
AU - Jens Eisert
AU - Debbie Leung
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06758
UR - https://arxiv.org/abs/2610.06758
AB - We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state τ, we obtain the exact entanglement of formation, E_F^phys(τ^⊗ m)=⌈ m/2⌉log2 for every m≥1. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order α∈(0,∞]. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For τ, Bell violation under local parity-preserving measurements first appears at three copies.
ER -