Some Studies On The Differential Entropy

In this paper,we take several special distributions as examples to study some special forms of differential entropy.First of all,we take the joint distribution of the eigenvalues of the Wishart matrix,the joint distribution of the random quantum state eigenvalues,and the diagonal distribution of the random density matrix as examples to define and obtain the formula of the Tsallis differential entropy under these three distributions.When the parameter approaches 1,we find that the Tsallis differential entropy under these three distributions includes the Shannon differential entropy under the corresponding distributions.Secondly,we take the joint distribution of eigenvalues of Wishart matrix,the joint distribution of random quantum state eigenvalues,and the diagonal distribution of random density matrices as examples to define and obtain the formula of the Rényi differential entropy under these three distributions.Similarly,we find that the Rényi differential entropy under these three distributions includes the Shannon differential entropy under the corresponding distributions.Once again,we take the joint distribution of random density matrix eigenvalues induced by partial traces on the pure state of the two-body Haar distribution and the joint distribution of its diagonal elements as examples to define and obtain the formula of the Tsallis relative differential entropy under two types.Finally,we take the joint distribution of the random density matrix eigenvalues induced by partial traces on the pure state of the two-body Haar distribution and the joint distribution of its diagonal elements as examples to define and derive the formula of the Rényi relative differential entropy under two types.When the parameter approaches 1,the Tsallis relative differential entropy and Rényi relative differential entropy of the first type include Shannon's relative differential entropy under the corresponding distribution.The second type of Tsallis relative differential entropy and Rényi relative differential entropy limit values are equal.