Quantum channel tomography: optimal bounds in the near-boundary regime
Abstract
Quantum channel tomography (also known as quantum process tomography) is a fundamental task in quantum information and quantum computing, aiming to learn the full description of an unknown quantum channel from black-box queries. For channels with input dimension d1, output dimension d2, and Kraus rank at most r, [CGO+26] identifies the dilation rate τ=rd2/d1≥ 1 as a key parameter governing a Heisenberg-to-classical transition in query complexity. While the optimal bounds in the boundary regime (τ=1) and the away-from-boundary regime (τ≥ 1+Ω(1)) are now known, the precise behavior in the near-boundary regime (1<τ<1+o(1)) has remained unresolved. In this paper, we close this gap, thereby fully settling the query complexity of quantum channel tomography. Specifically, we show that the query complexity is - Ω(rd1d2/ε2) for diamond norm error ε, matching the upper bound by [MB26, CGO+26], - Θ(d12/ε+min{(τ−1)d12,(τ−1)2d12d22}/ε2) for Choi trace norm error ε. Our results reveal two qualitatively different forms of the Heisenberg-to-classical transition. For diamond norm tomography, the Heisenberg scaling disappears suddenly as soon as τ leaves the boundary regime. In contrast, Choi trace norm tomography exhibits a gradual transition from Heisenberg to classical scaling as τ moves from the boundary regime to the away-from-boundary regime.
Related
- Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements
- Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank
- Joint Source-Channel Coding of Gaussian Sources over Block Erasure Channels: Nonasymptotic Bounds and Channel-Uniform Normal Approximations
- Tight Lower Bounds for State Tomography with Limited Entanglement
- Information Lower Bound for Regime Boundary Estimation — Optimal Precision Limit at the n² Scale and Universal Constant c=1/8 in the Fixed-Breakpoint Model
BibTeX
@misc{chen2026quantum,
title = {{Quantum channel tomography: optimal bounds in the near-boundary regime}},
author = {Kean Chen and Aadil Oufkir},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03639},
eprint = {2611.03639},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03639},
}
RIS
TY - UNPB
TI - Quantum channel tomography: optimal bounds in the near-boundary regime
AU - Kean Chen
AU - Aadil Oufkir
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03639
UR - https://arxiv.org/abs/2611.03639
AB - Quantum channel tomography (also known as quantum process tomography) is a fundamental task in quantum information and quantum computing, aiming to learn the full description of an unknown quantum channel from black-box queries. For channels with input dimension d_1, output dimension d_2, and Kraus rank at most r, [CGO+26] identifies the dilation rate τ=rd_2/d_1≥ 1 as a key parameter governing a Heisenberg-to-classical transition in query complexity. While the optimal bounds in the boundary regime (τ=1) and the away-from-boundary regime (τ≥ 1+Ω(1)) are now known, the precise behavior in the near-boundary regime (1<τ<1+o(1)) has remained unresolved. In this paper, we close this gap, thereby fully settling the query complexity of quantum channel tomography. Specifically, we show that the query complexity is - Ω(rd_1d_2/ε^2) for diamond norm error ε, matching the upper bound by [MB26, CGO+26], - Θ(d_1^2/ε+min{(τ−1)d_1^2,(τ−1)^2d_1^2d_2^2}/ε^2) for Choi trace norm error ε. Our results reveal two qualitatively different forms of the Heisenberg-to-classical transition. For diamond norm tomography, the Heisenberg scaling disappears suddenly as soon as τ leaves the boundary regime. In contrast, Choi trace norm tomography exhibits a gradual transition from Heisenberg to classical scaling as τ moves from the boundary regime to the away-from-boundary regime.
ER -