Abhilash Dharmavarapu

arXiv2026

Quantum 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 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 $n = \widetildeΘ\left(\sqrt{10}^N/\varepsilon^2\right)$. 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.

quant-ph↗

arXiv2025

Pauli Measurements Are Near-Optimal for Single-Qubit Tomography

We provide the first non-trivial lower bounds for single-qubit tomography algorithms and show that at least $Ω\left(\frac{10^N}{\sqrt{N} \varepsilon^2}\right)$ copies are required to learn an $N$-qubit state $ρ\in\mathbb{C}^{d\times d},d=2^N$ to within $\varepsilon$ trace distance. Pauli measurements, the most commonly used single-qubit measurement scheme, have recently been shown to require at most $O\left(\frac{10^N}{\varepsilon^2}\right)$ copies for this problem. Combining these results, we nearly settle the long-standing question of the complexity of single-qubit tomography.

quant-ph↗

arXiv2025

Pauli measurements are not optimal for single-copy tomography

Quantum state tomography is a fundamental problem in quantum computing. Given $n$ copies of an unknown $N$-qubit state $ρ\in \mathbb{C}^{d \times d},d=2^N$, the goal is to learn the state up to an accuracy $ε$ in trace distance, with at least probability 0.99. We are interested in the copy complexity, the minimum number of copies of $ρ$ needed to fulfill the task. Pauli measurements have attracted significant attention due to their ease of implementation in limited settings. The best-known upper bound is $O(\frac{N \cdot 12^N}{ε^2})$, and no non-trivial lower bound is known besides the general single-copy lower bound $Ω(\frac{8^n}{ε^2})$, achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of $O(\frac{10^N}{ε^2})$. To complement it with a lower bound of $Ω(\frac{9.118^N}{ε^2})$, which holds under adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage over prior methods is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements. Moreover, we connect the copy-complexity lower bound to the eigenvalues of the measurement information channel, which governs the measurement's capacity to distinguish states. To demonstrate the generality of the new framework, we obtain tight-bounds for adaptive quantum tomography with $k$-outcome measurements, where we recover existing results and establish new ones.

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