Theoretical Studies On The Resource Theory Of Quantum Coherence

Quantum coherence is an essential feature of quantum mechanics which is re-sponsible for the departure between the classical and quantum world and plays an central role in quantum information science.It is widely used in quantum algorithm,quantum cryptography,quantum metrology,and quantum biology.In recent years,with the rapid development of quantum computing and quantum information,the qualitative description of quantum coherence dose not meet the need of researchers,which has triggered the researchers' interest in the quantitative study of quantum coherence.In this thesis,we focus on the quantification of quantum coherence.The major results of our thesis include four aspects,namely,the enhancement of coherence un-der incoherent,operations,the superadditivity of coherence measures,the ordering states with coherence measures,and the coherence measures.First,we studied the issue of enhancing coherence of a state under stochastic strictly incoherent operations.Based on the l1 norm of coherence,we obtain the maximal value of coherence that can be achieved for a state undergoing a stochastic strictly incoherent operation and the maximal probability of obtaining the maximal coherence.Our findings indicate that a pure state can be transformed into a max-imally coherent state under a stochastic strictly incoherent operation if and only if all the components of the pure state are nonzero while a mixed state can never be transformed into a maximally coherent state under a stochastic strictly incoherent operation.Second,we studied the superadditivity of convex roof coherence measures.We put forward a theorem on the superadditivity of convex roof coherence measures,which provides a sufficient condition to identify the convex roof coherence measures fulfilling the superadditivity.By applying the theorem to each of the known convex roof coherence measures,we prove that the coherence of formation and the coherence concurrence are superadditive,while the geometric measure of coherence,the convex roof coherence measure based on linear entropy,the convex roof coherence measure based on fidelity,and convex roof coherence measure based on 1/2-entropy are non-superadditive.Third,we studied the ordering of coherence measures.We give the definitions of ordering states with coherence measure,and show that the l1 norm of coherence and the formation of coherence,the relative entropy of coherence and the coherence of formation all have different ordering.Detailed calculations show that the ordering-different states include the 2-dimensional mixed states,the d(?3)-dimensional mixed states and the d(>3)-dimensional pure states,but exclude 2-dimensional pure states since all the coherence measures give the same ordering for 2-dimensional pure states.Forth,we put forward a quantitative measure of coherence by following the axiomatic definition of coherence measures.As one of its applications,we show that our measure can be used to examine whether a pure qubit state can be transformed into another pure or mixed qubit state only by incoherent operations.Meanwhile,we give a closed expression for arbitrary state of a qubit of our measure,and we then generalize the results to other convex-roof coherence measures and give its one-qubit analytical expression.