| Quantum Information is the interdiscipline of quantum mechanics, information theory and computer science. As a result of this theory, many revolutionary quantum technologies surpassing classical counterparts have been developed, such as quantum computation, quantum communication, quantum cryptography and quantum metrolo-gy. Since the fundamental information carriers of all these quantum technologies are quantum states, quantum state characterization is a primitive of all quantum information technologies. To this end, quantum state tomography has been developed. Quantum state tomography is the procedure of inferring the unknown quantum state via quan-tum measurements and post data processing. There are two branches of quantum state tomography in terms of whether the dimension of a quantum system is continuous or discrete. This work focuses on quantum state tomography of discrete quantum systems.Quantum state tomography can be divided into two processes:quantum measure-ments and reconstruction algorithms. As the dimension of a quantum system grows exponentially with the size of the quantum system, both the measurement time and re-construction time suffer a similar exponential growth. Hence, efficient quantum state tomography is the first concern in our study. On the other hand, as the measurement of a quantum system leads to its collapse, thus many copies are required to reliably measure its quantum state. However, copies of a quantum system are valuable resources. Hence, the second concern in our study is how to chose quantum measurements to optimize the estimation accuracy given a finite number of copies. My PhD study mainly focuses on these two concerns.Efficient quantum state tomography includes time reduction in both reconstruc-tion algorithm and measurement. Firstly, linear regression estimation is introduced to reduce the computation complexity in the reconstruction algorithm. Then, when Pauli measurements are chosen in experiments, the linear regression estimation algorithm is further optimized and GPU parallel programming is used to speed up the algorithm, achieving the time cost of only 3.35 hours for a 14-qubit state. These works are listed as follows:1. Since the preferred maximum likelihood estimation is computationally intensive, the time cost of data processing can be much longer than that of the measurements. Hence, linear regression estimation is brought into quantum state tomography. Compared with maximum likelihood estimation, linear regression estimation re-duce the computation complexity significantly with a little sacrifice of estimation accuracy.2. Linear regression estimation with Pauli measurements are optimized by appropri-ate parametrization and further sped up by GPU parallel programming. Finally, it is capable of reconstructing a 14-qubit state with only 3.35 hours.3. Assisted by quantum teleportation, photons are measured 88 ns ahead of its cre-ation.In terms of the estimation accuracy in quantum state tomography, systematic error reduction is first considered in experiments. Given a finite number of copies, improving the estimation accuracy is then considered with adaptive and collective measurements. These works are listed as follows:1. The precision of quantum state tomography is seriously limited by the systematic error due to measurement devices. To reduce systematic error, compared with conventional combination of a quarter-wave plate and a half-wave plate, we could use only one wave plate to realize a complete set of mutually unbiased bases, achieving a two-fold systematic error reduction.2. Although one wave plate setting could reduce systematic error, it is not capable of realizing arbitrary qubit projectors and the achieved systematic error is not small enough. To this end, error-compensation measurements on polarization qubits are developed for conventional setting of quarter-wave plates and half-wave plates, which could realize arbitrary qubit projectors. This method can cancel out the first-order of systematic error due to dominant error sources, achieving a 20-fold error reduction.3. Two-step adaptive measurements are used to reach quantum precision limit in qubit state tomography experiments with only separable measurements.4. Iteratively adaptive quantum state tomography are developed on the basis of lin-ear regression estimation. This tomography protocol with the simplest product measurements can beat static quantum state tomography with nonlocal mutually unbiased bases for quantum states with a high level of purity. Iteratively adaptive quantum state tomography with nonlocal measurements available even can beat Gill-Massar bound, the quantum precision limit for separable measurements.5. Collective measurements on two copies of a quantum state encoded in the polar-ization and path freedoms of one photon can improve the estimation precision of quantum state tomography and even beat the quantum precision limit for separa-ble measurements. |
