标题： 通过密集双基直接测量密度矩阵 显示英文标题

标题： Direct Measurement of Density Matrices via Dense Dual Bases

摘要： 高效理解量子系统根本上依赖于可观测量的选择。 泡利可观测量和相互无偏基（MUBs）在实践中被广泛使用，并通常被视为量子态层析成像（QST）的理论最优选择。 然而，泡利可观测量需要大量测量来进行全态层析成像，并且无法用固定数量的可观测量直接测量密度矩阵元素。 对于MUBs，所有维度中完整集合的\(d+1\)基的存在性仍未解决，突显了需要替代的可观测量。 在本工作中，我们引入了密集对偶基（DDB），一种新的\(2d\)可观测量集，专门设计用于完全表征任何\(d\)维量子态。 这些可观测量提供了两个关键优势。 首先，它们无需辅助系统即可直接测量密度矩阵元素，仅使用三个选定的可观测量即可提取任何元素。 其次，对于未知的秩为\(r\)的密度矩阵——除了一个可以忽略的子集之外——可以使用\(O(r \log d)\)个可观测量实现QST，显著提高了测量效率。 至于电路实现，每个可观测量是迭代生成的，并且可以高效地分解为\(O(n^4)\)个基本门，用于\(n\)量子比特系统。 这些进展使DDB成为传统方法的实用且可扩展的替代方案，为提高量子系统表征的效率和可扩展性提供了有前景的机会。 显示英文摘要

摘要： Efficient understanding of a quantum system fundamentally relies on the selection of observables. Pauli observables and mutually unbiased bases (MUBs) are widely used in practice and are often regarded as theoretically optimal for quantum state tomography (QST). However, Pauli observables require a large number of measurements for full-state tomography and do not permit direct measurement of density matrix elements with a constant number of observables. For MUBs, the existence of complete sets of \(d+1\) bases in all dimensions remains unresolved, highlighting the need for alternative observables. In this work, we introduce Dense Dual Bases (DDB), a novel set of \(2d\) observables specifically designed to enable the complete characterization of any \(d\)-dimensional quantum state. These observables offer two key advantages. First, they enable direct measurement of density matrix elements without auxiliary systems, allowing any element to be extracted using only three selected observables. Second, QST for unknown rank-\(r\) density matrices--excluding only a negligible subset--can be achieved with \(O(r \log d)\) observables, significantly improving measurement efficiency. As for circuit implementation, each observable is iteratively generated and can be efficiently decomposed into \(O(n^4)\) elementary gates for an \(n\)-qubit system. These advances establish DDB as a practical and scalable alternative to traditional methods, offering promising opportunities to advance the efficiency and scalability of quantum system characterization.