Quantum Coherence Measurement And Related Problems

The superposition of quantum states is the fundamental feature of quantum mechanic-s.Quantum coherence,arising from quantum superposition,is one of important properties of quantum mechanics.Quantum entanglement and quantum correlation are essential re-sources in quantum information processing tasks(such as:quantum teleportation and super-dense coding,etc.),and the existence of the quantum coherence is a necessary condition for the existence of quantum entanglement and quantum correlation,which illustrate the impor-tance of quantum coherence in quantum information and computation.Therefore,search-ing physically meaningful and mathematically rigorous coherence quantifiers are the urgent tasks in quantum information theory.To this end,Baumgratz et.al.introduced the character-ization and quantification of quantum coherence from a resource-theoretic perspective and proposed the constraint conditions(nonnegativity,monotonicity,strong monotonicity and convexity)for a bona fide measure of quantum coherence.Recently,it has been reported that quantum coherence plays a vital role in quantum computation,quantum phase transition and quantum biology,etc.Following this theoretical framework,the purpose of this paper is to study whether different distance measures for quantum information meet the constraint condition for quan-tifying coherence and the basic properties of different coherence quantifiers.The main con-tributions of this thesis are listed as follows:In Chapter 3,we study the strong monotonicity of coherence quantifiers based on fi-delity and trace norm.Using the Bloch sphere representation of qubit,we show that the fidelity of coherence does not satisfy condition strong monotonicity by presenting an exam-ple.For the trace norm of coherence,we show that the qubit states and some special qutrit states can satisfy strong monotonicity with some restrictions for the incoherent operators.Our results show that the trace norm of coherence is equivalent to l1 norm of coherence for qubits and special qutrits.In Chapter 4,we revisit quantum coherence quantifiers based on the Tsallis relative a entropies and investigate the fundamental properties for the interesting case a = 2.We show that l1 norm of coherence is equal to Tsallis relative 2 entropy of coherence for all pure states and give the additivity property for Tsallis relative 2 entropy of coherence.Using the additivity property,we construct Bell-type inequalities to demonstrate the nonlocal proper-ties in quantum coherence.Some interesting inequalities are given connecting l1 norm of coherence,relative entropy of coherence and Tsallis relative 2 entropy of coherence in qubit system.Defining the decay of coherence measures,we consider the decay of l1 norm of coherence and Tsallis relative 2 entropy of coherence under the bit flip,the phase flip,the depolarizing and the amplitude damping channels.In Chapter 5,we revisit quantum coherence quantifiers based on the Renyi a relative entropies and study the strong monotonicity.We show that the Renyi a-relative entropy of coherence does not satisfy strong monotonicity and extension of strong monotonicity by using two dififerent methods.Due to the quantum system inevitably interacting with the environment that affect the purity of quantum states,we examine the trade-off relations between coherence and mixedness.Some properties are further exemplified with a single qubit for a = 2.In Chapter 6,we provide a pictorial interpretation of l1 norm and relative entropy of coherence for Bell-diagonal states and observed its evolution under decoherence quantum channels.There has some close relationships among quantum coherence,quantum discord and one-way quantum deficit.Due to the complex process of optimization,it makes the calculation of one-way quantum deficit difficult.We use the parameter substitution method to reduce the number of unknown parameters and provide a method to estimate the one-way quantum deficit for general two qubit states.