Quartic Certificates for Pure-State Tomography with Pauli Measurements
Abstract
Full Pauli-basis tomography of an arbitrary n-qubit state uses the expectation values of all nonidentity Pauli observables. For multiqubit systems, prior knowledge that the state is pure allows some of these observables to be omitted while retaining uniqueness. We study two resulting notions of uniqueness: whether every pure state is uniquely determined among pure states (UDP), and whether every pure state is uniquely determined among all states, including mixed states (UDA). We derive quartic uniqueness certificates from the fourth elementary symmetric polynomial of operators in the real span of the omitted Pauli observables. An exact quartic identity shows that, for n-qubit systems with n≥ 3, every pure state is UDA whenever the omitted Pauli observables contain no four distinct operators that commute pairwise and multiply to ± I. For three qubits, this condition is also necessary for UDP, yielding an exact characterization of both UDP and UDA. We further prove that UDP and UDA coincide for any number of qubits whenever the omitted Pauli observables commute pairwise. These results provide directly checkable structural criteria for certifying pure-state tomography from restricted Pauli measurements.
Related
- Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements
- Minimax Quantum State Tomography with Periodic Clifford Measurements
- Near-Optimal Mixedness Testing with Pauli Measurements
- Lower bounds on the spatial decay of the ground state of the Pauli-Fierz model
- Anticommuting Displacement Operators, Exact Pauli Algebras,and the Singlet State from the Oscillator Vacuum
BibTeX
@misc{li2026quartic,
title = {{Quartic Certificates for Pure-State Tomography with Pauli Measurements}},
author = {Hengjian Li and Baisong Sun and Bei Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12320},
eprint = {2610.12320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12320},
}
RIS
TY - UNPB
TI - Quartic Certificates for Pure-State Tomography with Pauli Measurements
AU - Hengjian Li
AU - Baisong Sun
AU - Bei Zeng
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12320
UR - https://arxiv.org/abs/2610.12320
AB - Full Pauli-basis tomography of an arbitrary n-qubit state uses the expectation values of all nonidentity Pauli observables. For multiqubit systems, prior knowledge that the state is pure allows some of these observables to be omitted while retaining uniqueness. We study two resulting notions of uniqueness: whether every pure state is uniquely determined among pure states (UDP), and whether every pure state is uniquely determined among all states, including mixed states (UDA). We derive quartic uniqueness certificates from the fourth elementary symmetric polynomial of operators in the real span of the omitted Pauli observables. An exact quartic identity shows that, for n-qubit systems with n≥ 3, every pure state is UDA whenever the omitted Pauli observables contain no four distinct operators that commute pairwise and multiply to ± I. For three qubits, this condition is also necessary for UDP, yielding an exact characterization of both UDP and UDA. We further prove that UDP and UDA coincide for any number of qubits whenever the omitted Pauli observables commute pairwise. These results provide directly checkable structural criteria for certifying pure-state tomography from restricted Pauli measurements.
ER -