Near-Optimal Mixedness Testing with Pauli Measurements

Abstract

We consider a fundamental problem of \emph{mixedness testing}: Given $n$ copies of an $N$-qubit state $\rho$, determine whether $\rho = \mathbb{I}_d/d$ or $\|\rho-\mathbb{I}_d/d\|_1 \geq \varepsilon$ with high probability, where $d = 2^N$. In particular, we focus on performing this task in the practical setting of single-qubit measurements, where measurements are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness tesing by showing $n = \tilde{\Theta}\left(\sqrt{10}^N/\varepsilon^2\right)$. To establish our lower bound, we introduce a 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.