Fidelity Determines the Minimum Error of Quantum Recovery
Abstract
Even a recovery that maximizes worst-input entanglement fidelity can retain avoidable errors in subsequent measurements. Entanglement fidelity is the probability of passing a test of whether the state and its entanglement were preserved; infidelity is the probability of failing it. Operational error is the largest possible change in any measurement probability, quantified by the normalized diamond distance. For every finite-dimensional noise channel and fixed encoding, we prove that operational error and worst-input entanglement infidelity have the same minimum over all recovery operations. The proof constructs one final unitary (a reversible quantum operation) that makes any recovery's operational error no greater than its original worst-input entanglement infidelity. With a retained noise record, we determine the optimal error and characterize all optimal decoders subject to a Kraus-rank limit for each outcome. For a fixed decoded channel, a semidefinite program finds the optimal final unitary in the low-error regime. We characterize all optimal final corrections and show that, at a fixed error floor 0<ε<1, the operational optimizers' sensitivity to residual unitary errors of size s is at worst of order s4/3, while the canonical polar correction has quadratic sensitivity.
Related
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BibTeX
@misc{picozzi2026fidelity,
title = {{Fidelity Determines the Minimum Error of Quantum Recovery}},
author = {Dario Picozzi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07344},
eprint = {2611.07344},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07344},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fidelity Determines the Minimum Error of Quantum Recovery
AU - Dario Picozzi
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07344
UR - https://arxiv.org/abs/2611.07344
AB - Even a recovery that maximizes worst-input entanglement fidelity can retain avoidable errors in subsequent measurements. Entanglement fidelity is the probability of passing a test of whether the state and its entanglement were preserved; infidelity is the probability of failing it. Operational error is the largest possible change in any measurement probability, quantified by the normalized diamond distance. For every finite-dimensional noise channel and fixed encoding, we prove that operational error and worst-input entanglement infidelity have the same minimum over all recovery operations. The proof constructs one final unitary (a reversible quantum operation) that makes any recovery's operational error no greater than its original worst-input entanglement infidelity. With a retained noise record, we determine the optimal error and characterize all optimal decoders subject to a Kraus-rank limit for each outcome. For a fixed decoded channel, a semidefinite program finds the optimal final unitary in the low-error regime. We characterize all optimal final corrections and show that, at a fixed error floor 0<ε<1, the operational optimizers' sensitivity to residual unitary errors of size s is at worst of order s^4/3, while the canonical polar correction has quadratic sensitivity.
ER -