Robust quantum state certification and uncertainty principles for total influence

• URL: http://arxiv.org/abs/2607.27184v1
• Date: Wed, 29 Jul 2026 17:53:23 GMT
• Title: Robust quantum state certification and uncertainty principles for total influence
• Abstract summary: We show that nonadaptive single-qubit Pauli measurements suffice to test whether an unknown $n$-qubit state $$ is $varepsilon$-close to or $O(varepsilon)$-far from an ideal target state $|rangle$.<n>The test uses $O(varepsilon-2log (1/)$ copies of $$ to achieve confidence $1-$, which is information-theoretically optimal even among protocols with arbitrary joint measurements.
• Score: 5.7531205491708945
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that nonadaptive single-qubit Pauli measurements suffice to test whether an unknown $n$-qubit state $ρ$ is $\varepsilon$-close to or $O(\varepsilon)$-far from an ideal target state $|ψ\rangle$, for all but a $2^{-Ω(n)}$ fraction of target states. The test uses $O(\varepsilon^{-2}\log(1/δ))$ 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 $\mathbf{Inf}[f]+\mathbf{Inf}[\widehat{f}] = Ω(n)$, which is a natural hypercube analogue of the Heisenberg uncertainty principle (here $\widehat{\,\cdot\,}$ denotes the $2^{-n/2}$-normalized Fourier transform). The weighted case generalizes $\mathbf{Inf}[\,\cdot\,]$ and $\mathbf{Inf}[\,\widehat{\,\cdot\,}\,]$ to Dirichlet energies associated with Glauber dynamics for certain dual measures on the cube.

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