Near-Optimal Mixedness Testing with Pauli Measurements
Jayadev Acharya, Abhilash Dharmavarapu, Yuhan Liu, Nengkun Yu
原文摘要
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.
quant-ph · cs.CC · cs.IT