Novel Approaches to Quantum State Tomography

Quantum state tomography (QST) aims at reconstructing the representation of a quantum state from the measurement of a sufficiently large number of observables. Traditionally, in QST the number of required observables grows exponentially with the dimension of the Hilbert space associated with the system under consideration, unless a priori knowledge of the state is available. In practice, QST becomes infeasible for systems of limited size. Thus, it is necessary to develop novel methods that are easy to implement and allow one to reconstruct high-dimensional states. The focus of this PhD research is to improve the efficiency of QST protocols. The main goal is to develop QST approaches that do not make any assumptions about the state to be reconstructed, reduce the number of required observables, are computationally efficient, and are experimentally feasible. The core idea is to introduce a threshold parameter that allows an efficient trade-off between the number of observables and the accuracy of the state reconstruction. Threshold quantum state tomography (tQST) is the first protocol based on this concept, tailored to systems composed of qubits that can be measured via projective measurements, and the state is reconstructed via maximum likelihood estimation. In many cases, including those of interest for technological applications, tQST works and accurately reconstructs the density matrix. We leveraged the feasibility of the protocol to numerically validate it on computer-generated data and experimentally test it on two different platforms, a superconducting and a photonic one. We show that tQST can drastically reduce the resources required for state reconstruction, but it can also be used to obtain an approximate density matrix by further reducing the number of measurements and the experimental efforts. State reconstruction can also be performed using tools other than maximum likelihood estimation. Furthermore, the tQST protocol might allow us to estimate quantities that are functions of the density matrix elements, such as purity, without reconstructing the entire state. We investigated this possibility through an original deep learning model that leverages the specific symmetries of the density matrix to enhance its performance. We then develop the enhanced compressive threshold QST (ECT-QST) protocol, specifically designed for systems composed of qudits where one can implement measurement settings. As an extension of tQST, ECT-QST is tailored for multiplexing platforms where one can perform projective measurements on all the basis vectors of a given setting. We numerically validate and experimentally test ECT-QST on the same platforms as tQST. Finally, moving beyond application matters, we turn to deeper questions and examine a problem at the crossroads of information theory and energetics, which are at the core of modern physics. We explore ECT-QST and QST through the lens of the quantum Maxwell demon, directly comparing their roles in the fundamental challenge of quantum work extraction.