Robust quantum state certification and uncertainty principles for total influence
Abstract
We show that nonadaptive single-qubit Pauli measurements suffice to test whether an unknown n-qubit state ρ is ε-close to or O(ε)-far from an ideal target state |ψ⟩, for all but a 2−Ω(n) fraction of target states. The test uses O(ε−2log(1/δ)) copies of ρ to achieve confidence 1−δ, which is information-theoretically optimal even among protocols with arbitrary joint measurements. The main technical innovation is an uncertainty principle for weighted generalizations of the total influence of Boolean functions. As a simple example, the unweighted variant states that Inf[f]+Inf[f̂] = Ω(n), which is a natural hypercube analogue of the Heisenberg uncertainty principle (here · ̂ denotes the 2−n/2-normalized Fourier transform). The weighted case generalizes Inf[ · ] and Inf[ · ̂ ] to Dirichlet energies associated with Glauber dynamics for certain dual measures on the cube.
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BibTeX
@misc{coladangelo2026robust,
title = {{Robust quantum state certification and uncertainty principles for total influence}},
author = {Andrea Coladangelo and Jerry Li and Joseph Slote},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.27184},
eprint = {2607.27184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.27184},
}
RIS
TY - UNPB
TI - Robust quantum state certification and uncertainty principles for total influence
AU - Andrea Coladangelo
AU - Jerry Li
AU - Joseph Slote
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2607.27184
UR - https://arxiv.org/abs/2607.27184
AB - We show that nonadaptive single-qubit Pauli measurements suffice to test whether an unknown n-qubit state ρ is ε-close to or O(ε)-far from an ideal target state |ψ⟩, for all but a 2^−Ω(n) fraction of target states. The test uses O(ε^−2log(1/δ)) copies of ρ to achieve confidence 1−δ, which is information-theoretically optimal even among protocols with arbitrary joint measurements. The main technical innovation is an uncertainty principle for weighted generalizations of the total influence of Boolean functions. As a simple example, the unweighted variant states that Inf[f]+Inf[f̂] = Ω(n), which is a natural hypercube analogue of the Heisenberg uncertainty principle (here · ̂ denotes the 2^−n/2-normalized Fourier transform). The weighted case generalizes Inf[ · ] and Inf[ · ̂ ] to Dirichlet energies associated with Glauber dynamics for certain dual measures on the cube.
ER -