The Relationship Between Quantum State Order And Quantum Uncertainty Based On Quantum Coherence Metric

Quantum coherence derived from the principle of quantum superposition is one of the most basic characteristics of quantum mechanics,and is one of the fundamental rea-sons why the quantum world is different from the classical world.In 2014,Baumgratz et al.proposed a quantitative framework for quantifying quantum coherence,which requires quantum coherence measures to satisfy four conditions:nonnegativity,mono-tonicity,strong monotonicity and convexity.Based on this framework,the research about quantum coherence has been rapidly developed.Many quantum coherence measures have been proposed.In this paper,we mainly discuss quantum coherence measures as follows:l1 norm coherence,relative entropy coherence,Tsallis a relative entropy coher-ence,Renyi a relative entropy coherence,quantum coherence via skew information and geometric coherence.Moreover,many further discussions about quantum coherence have been aroused.The ordering states with quantum coherence and the uncertainty relations of quantum coherence are two important topics.Quantum coherence measure is a mathematical ap-proach to quantify coherent resources.Different physical backgrounds may need different coherence measures.A quantum coherence measure can reduces an order with quantum states.Then an interesting question is whether different quantum coherence measures will generate the same orders.Uncertainty relation is another characteristic that distinguish-es quantum world from classical world.With the development of quantum information science,uncertainty relation plays an important role in quantum information processing.Entropy uncertainty relation is an important topic of uncertainty relations.As quantum coherence is a basis dependent notion,one regards it as a quantum part of uncertainty,and introduces the concept of uncertainty relation of quantum coherence.The main work of the paper is as follows:(1)The ordering states l1 norm coherence,relative entropy coherence,Tsallis αrelative entropy coherence,and Renyi a relative entropy coherence are studied.Firstly,we show that these coherence measures generate the same ordering for single qubit pure states.However,they don’t generate the same ordering for some high dimensional states,even though these states are pure.Secondly,we consider the special case of α = 2,1,1 These coherence measures may generate different coherence order states for some single qubit mixed states.Finally,we define the concept of violation degree of coherence order states between quantum coherence measures,and intuitively measure the different degree of coherence order states between the coherence measures.(2)The necessary conditions for the same order states with quantum coherence and the effects of-the order states with quantum coherence under amplitude damping channel are studied.Firstly,we give the necessary conditions for the same order states with quantum coherence for high dimensional pure states,X States and single qubit mixed states.Secondly,the influence of order states with quantum coherence under amplitude damping channel is considered,we find that the amplitude damped channel changes the order states in some special cases,but in another special cases,the amplitude damped channel does not change the order states.(3)The uncertainty relations of quantum coherence are studied by using several dif-ferent methods,for any quantum state,the lower bounds of the sums of coherence mea-sures were given.Firstly,for any state in a d dimension system,the uncertainty relations of Tsallis a relative entropy coherence,Renyi a relative entropy coherence,relative en-tropy coherence and quantum coherence via skew information were discussed by using majorization method derived from the majorization entropy uncertainty.Secondly,for any state in a single qubit system,the uncertainty relations of Tsallis α relative entropy coherence,Renyi α relative entropy coherence,relative entropy coherence and quantum coherence via skew information were discussed by using the analytical method.Third-ly,the lower bounds obtained by the above two methods are compared in a single qubit system,and their respective advantages and disadvantages are analyzed.Finally,based on the view of optimizing reference bases,the uncertainty relations of relative entropy coherence,l1 norm coherence,quantum coherence via skew information and geometric coherence in a single qubit system are discussed,and the lower bounds related to the reference bases are given.(4)The dynamics of quantum uncertainty relations using coherence in Markov chan-nels are studied,the main channels studied in this paper contain amplitude damped chan-nel,depolarization channel,bit flipping channel and phase damped channel.For any single qubit state,the influences of sums of coherence measures and its lower bounds under Markov channels are studied respectively.These effects are closely related to the initial quantum state and the selected standard orthogonal bases.