Study Of Two Different Types Of Quantum Uncertainty Relations

This gradually clearness of the once vague understanding of the quantum world is progress toward science and technology.In quantum mechanics,the quantum uncertainty principle provides an upper limit for predicting the measurements accuracy of two incompatible observables of a particle.From the first relation established by Heisenberg for the incompatibility of particle positions and momentum to the subsequent work of Robertson and Schr?dinger,this relationship was extended to any observable pair.The uncertainty principle has been continuously developed and has been widely used,from quantum cryptography to quantum communication,quantum computing,and precision measurement.Moreover,entropy can be compared to the degree of chaos in the whole system,so the concept of entropy is introduced into the principle of uncertainty and has been widely used in quantum information.The relationship between entropy uncertainty and quantum entanglement is also very close,and the research significance of entanglement witness is becoming more and more prominent.The research work of this thesis mainly focuses on the following two aspects:First,the experimental verification and corresponding analysis are carried out on the uncertainty relationship characterized by variance to represent multiple observation uncertainties.In the two-dimensional and three-dimensional systems,we choose the angular momentum components with spin quantum numbers 1/2 and 1 as observations,respectively.Using single-photon detection,we have carried out experimental verification in the spin-one system from the perspectives of traditional projection measurements and Stokes parameters.At the same time,we have completed the verification experiment in the three-dimensional system by using unitary transformation and unitary decomposition.The validity of the three inequalities is verified and compared by using a linear optical optical platform.The experimental results and theoretic predictions are in a good agreement.Further analysis shows that compared with the previous uncertainty relation expressed by the lower bound based on the variance and standard deviation of the observed quantity,the uncertainty relation described by using the eigenvalue of the observed quantity and the transition probability between the eigenstate and the system state has a stricter lower bound.For ndimensional systems,we also give a similar experimental extension methods.Our results not only assist in the understanding of uncertainty relations but also facilitate potential applications to quantum technologies.Secondly,the uncertainty relation based on linear entropy is the same as the entropy uncertainty relation,which considers the uncertainty of the observed quantity from the perspective of entropy,so it can be regarded as a branch of the entropy uncertainty relation.By taking the two-qubit Ising model under Dzyaloshinskii-Moriya(DM)interaction as the research object,we investigate the effects of coupling strength and DM interaction on linear entropy uncertainty relation(EUR)of the system.Moreover,we discuss the variation of thermal entanglement with environment and compare the relationship between thermal entanglement and linear EUR.The results show that the variance trend of systemic linear entropy uncertainty and thermal entanglement depends on the selection of environmental parameters,and their overall evolution behavior is roughly anti-related,that is,they can be used as the witness of quantum entanglement,which is of great significance for the study of quantum entanglement phenomenon.Furthermore,for a complete set of mutually unbiased bases(MUBs),when we choose different combinations of measurement bases,the lower bound of the inequality will vary with the change of the number of measurement bases.Besides,the linear EUR can be transformed into an equation in special cases and its lower bound does not depend on the selection of specific observation quantity.Compared with the EUR assisted by quantum memory,it provides a useful reference for precise measurement.