Geometric Rényi divergence and RLD Fisher information under quantum processes without definite causal order
Abstract
Quantum experiments without a definite causal order offer new ways to distinguish devices, estimate physical parameters, and communicate. We prove a common limit on these tasks: the geometric Rényi divergence at the output of a deterministic completely positive process cannot exceed the sum of the divergences of its individual channel uses. The inequality holds for every Rényi order in (0,1)∪(1,2] and any number of independently replaceable channels. Parallel experiments with separate reference-assisted inputs approach the bound, giving an exact equality after optimization over processes. We also prove an additive bound on the right logarithmic derivative Fisher information of parameter-dependent channels under every fixed process in this class. A resulting Cramér--Rao bound limits locally unbiased estimation precision without assuming a definite causal order, and an erasure-probability example attains the bound. The proofs for the geometric Rényi divergence and RLD Fisher information follow from the requirement that every independent choice of channels produce a normalized output. We show how this requirement controls positive local operations, and how small channel perturbations convert the divergence inequality into a Fisher-information bound, including for singular channels. Further applications give finite-use bounds on channel discrimination and exclusion, together with communication and private-state-generation converses when the processes preserve the relevant comparison conditions.
Related
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BibTeX
@misc{huang2026geometric,
title = {{Geometric Rényi divergence and RLD Fisher information under quantum processes without definite causal order}},
author = {Zixin Huang and Mark M. Wilde},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39217},
eprint = {2610.39217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39217},
}
RIS
TY - UNPB
TI - Geometric Rényi divergence and RLD Fisher information under quantum processes without definite causal order
AU - Zixin Huang
AU - Mark M. Wilde
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39217
UR - https://arxiv.org/abs/2610.39217
AB - Quantum experiments without a definite causal order offer new ways to distinguish devices, estimate physical parameters, and communicate. We prove a common limit on these tasks: the geometric Rényi divergence at the output of a deterministic completely positive process cannot exceed the sum of the divergences of its individual channel uses. The inequality holds for every Rényi order in (0,1)∪(1,2] and any number of independently replaceable channels. Parallel experiments with separate reference-assisted inputs approach the bound, giving an exact equality after optimization over processes. We also prove an additive bound on the right logarithmic derivative Fisher information of parameter-dependent channels under every fixed process in this class. A resulting Cramér--Rao bound limits locally unbiased estimation precision without assuming a definite causal order, and an erasure-probability example attains the bound. The proofs for the geometric Rényi divergence and RLD Fisher information follow from the requirement that every independent choice of channels produce a normalized output. We show how this requirement controls positive local operations, and how small channel perturbations convert the divergence inequality into a Fisher-information bound, including for singular channels. Further applications give finite-use bounds on channel discrimination and exclusion, together with communication and private-state-generation converses when the processes preserve the relevant comparison conditions.
ER -