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 complete 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 a novel set of \(2d\) observables specifically designed to enable the complete characterization of any \(d\)-dimensional quantum state. To demonstrate the advantages of these observables, we explore two key applications. First, we show that direct measurement of density matrix elements is feasible without auxiliary systems, with any element extractable using only three selected observables. Second, we demonstrate that QST for unknown rank-\(r\) density matrices, excluding only a negligible subset, can be achieved with \(O(r \log d)\) observables. This significantly reduces the number of unitary operations compared to compressed sensing with Pauli observables, which typically require \(O(r d \log^2 d)\) operations. Each circuit is iteratively generated and can be efficiently decomposed into at most \(O(n^4)\) elementary gates for an \(n\)-qubit system. The proposed observables represent a substantial advancement in the characterization of quantum systems, enhancing both the efficiency and practicality of quantum state learning and offering a promising alternative to traditional methods.