Theoretical Studies On Quantum Coherence Measures

Quantum coherence,as a fundamental feature of quantum mechanics,represents useful resources for performing various quantum information processing tasks,with broad applications in a plethora of fields ranging from quantum computation to quantum communication to quantum thermodynamics to quantum metrology and to quantum biology.With the development of quantum information science,the quantification of coherence has become a crucial problem and received much attention ever since the proposal of the seminal framework for quantifying coherence in 2014.A key issue in the field of quantifying coherence is to look for valid coherence measures and,based on them,further explore the quantitative relations between quantum coherence and other quantum features like quantum correlations,in order to gain insights into quantum information processing tasks from a resource-theoretic perspective.Aiming to address this issue,we have conducted a systematic research and obtained the following scientific achievements:First,we have studied the validity of the first family of Schatten-p-norm-based functionals as coherence measures,and found that this family of functionals does not satisfy the strong monotonicity when genuinely incoherent operations are under consideration.It is a widely-used method to obtain coherence measures from functionals that are defined as the minimal distance between a given density matrix and the set of incoherent states.The first family of Schatten-p-norm-based functionals are candidate coherence measures defined via this method.It has been proved that this family of functionals are not coherence measures under both incoherent operations and strictly incoherent operations,but it remains an open question whether they are coherence measures under genuinely incoherent operations.We answer this question in the negative by showing that the first fam-ily of Schatten-p-norm-based functionals violates the strong monotonicity under genuinely incoherent operations.Second,we have studied the validity of the second family of Schatten-p-norm-based functionals as coherence measures,and found that when p=1,the functional is a valid coherence measure under both strictly incoherent operations and genuinely incoherent operations but is not a valid coherence measure under incoherent operations,and moreover,they are not valid coherence measures under incoherent operations,strictly incoherent operations,and genuinely incoherent operations when p>1.This finding not only solves the validity issue of the second family of Schatten-p-norm-based functionals as coherence measures,and more importantly,it provides the first example demonstrating that a functional is a coherence measure under strictly incoherent operations but is not under incoherent operations,thereby offering an answer to the open question whether the set of coherence measures under incoherent operations is the same as that under strictly incoherent operations.Third,we study the maximal-value condition of coherence measures and put forward the maximal-value theorem of coherence measures.All the maximally coherent states must take the maximal value of a coherence measure,but the quantum states that can take the maximal value of a coherence measure may not be maximally coherent states.It is said that a coherence measure fulfills the maximal-value condition,if and only if maximally coherent states can achieve the maximal value of the coherence measure.We show that for any functional that displays either the non-increasing under mixing of quantum states or the monotonicity under incoherent operations,it satisfies the maximal-value condition of coherence measures in general if it satisfies the maximal-value condition of coherence measures in the pure-state case.This theorem reduces the problem of verifying the maximal-value condition of coherence measures in general to the problem of only verifying the maximal-value condition of coherence measures in the pure-state case,thereby significantly simplifying the verification process.Fourth,we have studied the relations between quantum coherence measures and quantum correlation measures,and provided an upper bound for the amount of entanglement generated from a state via an incoherent operation.Quantum coherence and quantum correlations are two related features stemming from the superposition principle and are resources for performing quantum information processing tasks.Therefore,the relations between them has been an important issue in quantum resource theories.By analyzing the second family of Schatten-1-norm-based functional,we establish a quantitative relation between coherence measure and entanglement measure and find that the amount of entanglement generated from a quantum state via an incoherent operation is bounded from above by the coherence measure.This finding reveals that coherence and entanglement can be converted into each other in a quantitative manner.