The Determinability Of The Quantum State By Its Subsystems

As we know, random numbers play essential roles in many fields,such as the simulation of biological systems, fundamental physics test, in particular, cryptography. However, there is no true random numbers in the classical physics, we observe randomness just because we have not full knowledge of the operating mechanism of the systems.With the advent of quantum information science, it is possible to generate true randomness by the use of quantum systems. The true randomness imply that each bit occurs in same probability and is uncorrelated with any other variables. Randomness of the outputs is certified by nonlocality with respective of input-output, which is so called certified true random numbers. The generation of certified true random numbers is based on Bell test or other nonlocal tests. In practice, Bell tests can be threatened by some loopholes, such as random loophole,detection loophole. It is possible local hidden variable modes can fake nonlocal correlations by using these loopholes, which affects the security of random numbers. Therefore, the contributions of this dissertation are mainly on analyzing the security of device-independent and semi device-independent randomness expansion. Especially, the effect of these loopholes and how to close these loopholes have been investigated.Firstly, we studied the effect on randomness generation based on the generalized CHSH Bell inequalities for relaxing measurement independence in different scenarios, which gave the method to obtain certified true random numbers. Concretely, we established the relation among measurement dependence, the guessing probability and the maximal violation values of generalized CHSH Bell inequalities that an adversary (Eve) can fake in the single-run scenario, the multiple-run one and different input distributions (a general input distribution and factorizable one), respectively. To be interested, compared with the existed results, we find that it is harder for Eve to fake the violation of generalized CHSH Bell test in some special situations, which imply that the generations of certified true randomness is easier.Secondly, we gave the necessary and sufficient conditions for loophole-free violations of m-CHSH inequality and I1 inequality which show the wonderful properties in the randomness expansion. On the one side, we gave the minimum detection efficiency which is required for closing detection loophole. On the other side, if the detection efficiency is less than the minimum one we gave, for any given violation value, we constructed the corresponding optimal classic attack strategies.Thirdly, in the semi-device-independent framework, we analyzed the security of randomness expansion under the practical condition. We gave the analytical relations between the amount of the generated randomness and the degree of nonlocal correlation under the ideal condition and practical one. Based on our work, we chose a randomness extractor (i.e.,two-universal random function) and gave the security proof. Here, the practical condition indicates that device's behavior is not independent and identical in each round and there exists deviation in estimating the nonlocal behavior of devices.