← 研踪 PaperTrailarXiv:2608.18839 · v4
Near-Optimal Mixedness Testing with Pauli Measurements
首次提交 2026-08-19 19:58
版本更新 2026-11-04 21:05
雷达观测 2026-11-05 11:10
摘要
We consider a fundamental problem of \emph{mixedness testing}: Given $n$ copies of an $N$-qubit state $ρ$, determine whether $ρ= \mathbb{I}_d/d$ or $\|ρ-\mathbb{I}_d/d\|_1 \geq \varepsilon$ with high probability, where $d = 2^N$. In particular, we focus on performing this task with simple single-qubit measurements, i.e. measurements that are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness testing by showing that its copy-complexity is $\tildeΘ\left(\sqrt{10}^N/\varepsilon^2\right)$. To establish our lower bound, we introduce a new measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.
作者备注：Added determinstic Pauli mixedness testing results, fixed typos
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v4 · 2026-11-04 21:05 · 最新观测版本
Near-Optimal Mixedness Testing with Pauli Measurements
We consider a fundamental problem of \emph{mixedness testing}: Given $n$ copies of an $N$-qubit state $ρ$, determine whether $ρ= \mathbb{I}_d/d$ or $\|ρ-\mathbb{I}_d/d\|_1 \geq \varepsilon$ with high probability, where $d = 2^N$. In particular, we focus on performing this task with simple single-qubit measurements, i.e. measurements that are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness testing by showing that its copy-complexity is $\tildeΘ\left(\sqrt{10}^N/\varepsilon^2\right)$. To establish our lower bound, we introduce a new measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.
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