Direct Measurement of Density Matrices via Dense Dual Bases

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.