The Power of Two Bases: Robust and copy-optimal certification of nearly all quantum states with few-qubit measurements

TL;DR

This paper introduces robust quantum state certification protocols using few-qubit measurements, achieving constant robustness and optimal copy complexity for nearly all pure states, addressing scalability issues in quantum experiments.

Contribution

The authors develop new certification protocols that are robust and scalable, utilizing few-qubit measurements, and introduce a novel uncertainty principle for conditional fidelities.

Findings

Achieves constant robustness with a single O(log n)-qubit measurement.

Provides a nearly robust protocol with measurements only on single qubits.

Establishes a new uncertainty principle for conditional fidelities.

Abstract

A central task in quantum information science is state certification: testing whether an unknown state is $\epsilon_1$-close to a fixed target state, or $\epsilon_2$-far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary $n$-qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant $\epsilon_1$, they can only positively certify states within $\epsilon_1=O(1/n)$ trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated. Here we present robust certification protocols based on few-qubit measurements that apply to all but a $O(2^{-n})$-fraction of pure target states. Our first protocol achieves constant robustness, i.e. $\epsilon_1=\Theta(1)$, using a single $O(\log n)$-qubit measurement along with single-qubit measurements in the $Z$ or $X$ basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in $n$) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust: $\epsilon_1=\Omega(1/\log n)$. Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.