Quantum data processing equality
Abstract
The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.
Related
- Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity
- Equality in the Bosonic Quantum Entropy Power Inequality
- Counterexamples to the Tensorization Property of Strong Data Processing Inequalities for Quantum Divergences
- Particle-Number-Preserving Quantum Signal Processing for Thermal Properties: Canonical-Ensemble Estimation and Grand-Canonical Reconstruction
- Representation Learning with Quantum Signal Processing
BibTeX
@misc{yang2026quantum,
title = {{Quantum data processing equality}},
author = {Tai-Hsuan Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33815},
eprint = {2609.33815},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33815},
}
RIS
TY - UNPB
TI - Quantum data processing equality
AU - Tai-Hsuan Yang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.33815
UR - https://arxiv.org/abs/2609.33815
AB - The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.
ER -