Hierarchy of Rényi Coherent Information in Stabilizer Codes
Abstract
Rényi coherent information, a computable proxy for the von Neumann coherent information, is widely used to study mixed-state phases of matter and decodability transitions in noisy quantum error-correcting codes. However, being a difference of two Rényi entropies, it need not be monotonic in the Rényi index, and lacks the operational meaning of its von Neumann counterpart. Here we address both issues for stabilizer codes. First, for Pauli noise generated by independent Bernoulli events, we prove that the Rényi-n coherent information is nondecreasing in n ∈ ℤ+. This follows from a general theorem: if independent random bits are mapped linearly to a fine label T and a coarse label C, then the Rényi entropy difference Hn(C)−Hn(T) is nondecreasing in n. For stabilizer codes, T is the joint syndrome--logical class and C is the syndrome, and the difference is the Rényi-n coherent information up to a constant. The same theorem covers classical linear codes and independent detector error models. Second, for arbitrary stochastic Pauli noise, we give the Rényi-n coherent information an operational meaning via postselection on matching syndromes between one data block and n−1 auxiliary blocks. We determine when this defines a quantum channel and show that saturation of the Rényi-n coherent information is equivalent to asymptotically perfect recovery of the postselected channel. Moreover, the Rényi-n coherent information also upper-bounds the ordinary coherent information achievable after any syndrome-conditioned recovery.
Related
- Fault-tolerant quantum error correction for constant-excitation stabilizer codes under coherent noise
- Rényi Multi-entropy for Stabilizer States: A Finite Field Gauss Sum Problem
- Stabilizer Rényi moments beyond entanglement
- Real-Space Renormalization of Stabilizer Rényi Entropies in Spin Chains
- Fast and exact approach for stabilizer Rényi entropy via the XOR-FWHT algorithm
BibTeX
@misc{vijay2026hierarchy,
title = {{Hierarchy of Rényi Coherent Information in Stabilizer Codes}},
author = {Akash Vijay and Luis Colmenarez and Jong Yeon Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11930},
eprint = {2609.11930},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11930},
}
RIS
TY - UNPB
TI - Hierarchy of Rényi Coherent Information in Stabilizer Codes
AU - Akash Vijay
AU - Luis Colmenarez
AU - Jong Yeon Lee
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11930
UR - https://arxiv.org/abs/2609.11930
AB - Rényi coherent information, a computable proxy for the von Neumann coherent information, is widely used to study mixed-state phases of matter and decodability transitions in noisy quantum error-correcting codes. However, being a difference of two Rényi entropies, it need not be monotonic in the Rényi index, and lacks the operational meaning of its von Neumann counterpart. Here we address both issues for stabilizer codes. First, for Pauli noise generated by independent Bernoulli events, we prove that the Rényi-n coherent information is nondecreasing in n ∈ ℤ^+. This follows from a general theorem: if independent random bits are mapped linearly to a fine label T and a coarse label C, then the Rényi entropy difference H_n(C)−H_n(T) is nondecreasing in n. For stabilizer codes, T is the joint syndrome--logical class and C is the syndrome, and the difference is the Rényi-n coherent information up to a constant. The same theorem covers classical linear codes and independent detector error models. Second, for arbitrary stochastic Pauli noise, we give the Rényi-n coherent information an operational meaning via postselection on matching syndromes between one data block and n−1 auxiliary blocks. We determine when this defines a quantum channel and show that saturation of the Rényi-n coherent information is equivalent to asymptotically perfect recovery of the postselected channel. Moreover, the Rényi-n coherent information also upper-bounds the ordinary coherent information achievable after any syndrome-conditioned recovery.
ER -