Theoretical Studies On Quantum Coherence And Quantum Contextuality

Quantum coherence and quantum contextuality are two of the fundamental features of quantum mechanics.Quantum coherence arises from the superposition principle of quantum mechanics,describing the capability of a quantum state in ex-hibiting quantum interference phenomenon.Quantum contextuality,as a natural generalization of Bell nonlocality,describes the contextual phenomenon in quan-tum mechanics.The study on quantum coherence and quantum contextuality not only deepens our understanding of quantum mechanics,but also reveals that both quantum coherence and quantum contextuality are crucial resources for quantum information processing.In this thesis,we focus on these two subjects.For quantum coherence,we inves-tigate the quantification of quantum coherence,including the freezing phenomenon of quantum coherence and the basic condition for coherence measures.For quantum contextuality,we investigate the relations between three main methods for revealing quantum contextuality,the Kochen-Specker theorem,the Kochen-Specker inequal-ity,and the noncontextuality inequality.The following are the main results of this thesis.First,we study the robustness of coherence in noisy environments,i.e.,the freezing phenomenon of quantum coherence.We discover for the first time the universal freezing phenomenon of quantum coherence and prove a theorem on the occurring of universal freezing of coherence.We find that in a strictly incoherent channel,if the relative entropy of coherence is not affected by the noise,then all coherence measures are not affected by the noise.Quantum coherence is a useful physical resource,but coherence of a quantum state is often destroyed by noise.A challenge in exploiting the resource is to protect coherence from the decoherence caused by noise,as the loss of coherence may weaken the abilities of a state to perform quantum information processing tasks.Our study provides a theoretical basis for protecting coherence from the decoherence caused by noise.Second,we study the basic conditions that coherence measures should satisfy.We propose a new axiom for the quantification of quantum coherence,the additivity of coherence for subspace-independent states.This axiom is an exhibition of a basic property of quantum coherence and independent of the choice of incoherent operations.With this axiom,we proposed a general framework for the quantification of quantum coherence.This framework is compatible with all known results on the quantification of quantum coherence,but much more flexible and convenient for applications.With this framework,we can not only simplify a lot of results on quantum coherence,but also solve many open problems.As an example,we use our framework to show that the trace-norm of coherence is not a legitimate coherence measure.Third,we study the inequality methods for revealing quantum contextual-ity.We prove that noncontextuality inequality always coexist with Kochen-Specker inequalities.Both Kochen-Specker.inequalities and noncontextuality inequalities are used for revealing the contextual phenomenon in quantum mechanics.As the Kochen-Specker inequalities do not require the Kochen-Specker rules,they are as-sumed to be stronger form of inequalities than the Kochen-Specker inequalities.However,we find that they are equivalent.More precisely,we prove that noncon-textuality inequalities always coexist with the Kochen-Specker inequalities.Fur-thermore,our proof is a constructive one,which provides a simple method for constructing noncontextuality inequalities.Forth,we study the relation between Kochen-Specker theorem and noncontex-tuality inequalities.We prove that every proof of Kochen-Specker theorem can be converted to a state-independent noncontextuality inequality.Kochen-Specker the-orem is the first method used for revealing the contextual phenomenon in quantum mechanics.Previous studies shows that some proofs of Kochen-Specker theorem can be converted to state-independent noncontextuality inequalities,but whether this works for all proofs of Kochen-Specker theorem is not known before our work.Our study shows that all proofs of Kochen-Specker theorem can be converted to noncontextuality inequalities.Furthermore,our proof is a constructive one,which provides us a general method to constructive state-independent noncontextuality inequalities from proofs of Kochen-Specker theorem.