Research On Quantumness Of Ensembles Based On Gram Matrix

Quantum coherence is an important physical resource,which has been widely used in various information processing tasks.Quantumness of ensembles is a set of states with a priori probability distribution,which plays a fundamental role in quantum information.Since its universality and importance in quantum measurement and quantum communication,the characterization and quantification of the quantumness of ensembles has intrigued widespread interest.The Gram matrix composed of a set of ensembles contains the information of the relationship between the components of the ensemble,and the Gram matrix can also be regarded as a quantum state in some sense.Therefore,the coherence of the Gram matrix can be used to quantify the quantumness of the corresponding ensemble.In this thesis,we mainly study the quantumness of ensembles based on Gram matrix by making use of coherence measures.The main content is as follows:First,we absorb the probability of each state of the pure state ensemble into the corresponding state,define the Gram matrix of the pure state ensemble,and quantify the quantumness of the pure-state ensemble by using the generalized α-z-Rényi relative entropy of coherence.We discuss the properties of this quantumness measure.Moreover,we give explicit formulas of six typical pure state ensembles,and compare them with the results based on other quantumness quantifiers.Second,we absorb the probability of each state of the mixed-state ensemble into the corresponding state in a set of ensembles,define the Gram matrix of the mixed-state ensemble,and present a quantifier of quantumness based on generalized α-z-relative Rényi entropy of coherence.We investigate the features of the quantumness quantifier.In addition,we compute the quantumness of three mix-state ensembles and plot the corresponding graphs.Third,we consider the post-measurement ensemble,i.e.,a set of post-measured states and their corresponding probabilities,after measuring the system,define the Gram matrix of the post-measurement ensemble,and apply the skew information-based coherence to quantify the quantumness of the ensembles.As examples,we provide the concrete expressions of the quantumness of the ensembles induced by three quantum channels,and calculate the maximum as well as minimum values of the quantumness.