The Study Of Mutually Unbiased Bases And Mutually Unbiased Measurements

The paper study mutually unbiased bases in which all the bases are unextendible maxi-mally entangled ones and study the separability of quantum states by using mutually unbiased measurements.We first choose a unextendible maximally entangled base and add two separa-ble states to this unextendible maximally entangled base in C2(?)C4,by using the properties of unitary matrix and mutually unbiased bases,present a necessary and sufficient condition of constructing a pair of mutually unbiased bases in C2 ? C4.And an analytical necessary con-dition for constructing mutually unbiased bases is given.Moreover we illustrate our approach by some detailed examples in C2 ? C4.The results are generalized to C2 ? Cd(d? 3)and a detailed example in C2 ? C8 is given.Secondly,we can well study the separability of quantum states by using mutually unbiased measurements.We analyze the separable problem in diago-nal symmetry states.By using the relationship between the diagonal symmetry states and the partial transpose positive of density matrix,combined with mutually unbiased measurements,we study the separability of two kinds of diagonal symmetric states in the bipartite quantum systems,and give the necessary and sufficient conditions for separability.And we study sepa-rability of the quantum states in the bipartite quantum systems with 2-dimensional subsystems Using the Bloch representation of the density matrix,and the relationship between the mutually unbiased measurements and the group generators,we give the necessary condition for separa-bility.We study the separability of general quantum states in bipartite quantum systems.By using the relationship between the trace norm and the vector norm,and the relationship between the mutually unbiased measurements and the density matrix,we present separability criteria for quantum states.Moreover,we illustrate that our approach can detect more entangled states by some detailed examples.