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

A central task in quantum information science is state certification: testing whether an unknown state is $ε_1$-close to a fixed target state, or $ε_2$-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 $ε_1$, they can only positively certify states within $ε_1=O(1/n)$ 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. $ε_1=Θ(1)$, 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: $ε_1=Ω(1/\log n)$. Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.

💡 Research Summary

The paper tackles a fundamental limitation of recent quantum state certification protocols that rely only on single‑qubit measurements. While works such as Huang‑Preskill‑Soleimanifar (2025) and Gupta‑He‑O’Donnell (2025) showed that almost any n‑qubit pure state can be certified using only local measurements, their robustness is only O(1/n). In other words, the tests can only positively certify states that are already vanishingly close to the target, making them unsuitable for large‑scale experiments where a constant error tolerance is required.

The authors present two new families of certification algorithms that achieve constant‑order robustness for “almost all” pure target states (all but an exponentially small 2⁻ⁿ fraction). The first algorithm uses a hybrid measurement strategy: it measures n − O(log n) qubits in a single basis—either the computational (Z) basis or the Hadamard (X) basis—with equal probability, and then performs a conditional measurement on the remaining O(log n) qubits. Concretely, after measuring the first block, the outcome z determines a conditional target state |ψ_z⟩ on the leftover qubits; the lab state’s reduced part is then projected onto |ψ_z⟩ using the two‑outcome POVM {|ψ_z⟩⟨ψ_z|, I − |ψ_z⟩⟨ψ_z|}. The same procedure is repeated in the X basis. The test guarantees:

• Completeness: if the lab state ρ has fidelity ≥ 1 − ε with the target |ψ⟩, it is accepted with probability at least 1 − ε.
• Soundness: if the test accepts with probability 1 − ε, then the fidelity of ρ with |ψ⟩ is at least 1 − 2ε + o(1). Thus the robustness constant c equals 2, i.e., ε₁ = Θ(1).

Because the robustness is constant, the required number of copies of ρ to amplify the test to any desired confidence δ scales as O(ε⁻² log (1/δ)), independent of n. This matches the optimal copy‑complexity for state certification.

The second algorithm eliminates the O(log n)‑qubit measurement entirely by replacing it with the single‑qubit test of Gupta‑He‑O’Donnell. Since that test is not robust, the robustness degrades by a factor proportional to the number of qubits on which it is applied, namely O(log n). Consequently the algorithm achieves ε₁ = Ω(1/ log n) robustness while still using only single‑qubit measurements. The copy complexity becomes O(ε⁻² log n log (1/δ)), which is nearly optimal.

A central theoretical contribution is an “uncertainty principle for conditional fidelities.” For any pure states |ψ⟩ (target) and |ϕ⟩ (lab), let μ_std and μ_had denote the distributions over measurement outcomes when measuring the first n − C log n qubits in the Z and X bases, respectively. Let |ψ_z⟩, |ϕ_z⟩ (resp. |ĥψ_z⟩, |ĥϕ_z⟩) be the post‑measurement states on the remaining C log n qubits. The authors prove that, for all but an exponentially small fraction of Haar‑random |ψ⟩, the average squared inner products satisfy

E_z