Distances of qubit density matrices on Bloch sphere

We recall the Einstein velocity addition on the open unit ball \B of $\R^{3}$ and its algebraic structure, called the Einstein gyrogroup. We establish an isomorphism between the Einstein gyrogroup on \B and the set of all qubit density matrices representing mixed states endowed with an appropriate addition. Our main result establishes a relation between the trace metric for the qubit density matrices and the rapidity metric on the Einstein gyrogroup \B.Comment: I thank to my supervisor, Jimmie Lawson. This was accepted in Journal of Mathematical Physics at September 26, 201

Weak log-majorization and inequalities of power means

As non-commutative versions of the quasi-arithmetic mean, we consider the Lim-P\'{a}lfia's power mean, R\'{e}nyi right mean and R\'{e}nyi power means. We prove that the Lim-P\'{a}lfia's power mean of order $t \in [-1,0)$ is weakly log-majorized by the log-Euclidean mean and fulfills the Ando-Hiai inequality. We establish the log-majorization relationship between the R\'{e}nyi relative entropy and the product of square roots of given variables. Furthermore, we show the norm inequalities among power means and provide the boundedness of R\'{e}nyi power mean in terms of the quasi-arithmetic mean.Comment: 18 page

Weak log-majorization between the geometric and Wasserstein means

There exist lots of distinct geometric means on the cone of positive definite Hermitian matrices such as the metric geometric mean, spectral geometric mean, log-Euclidean mean and Wasserstein mean. In this paper, we prove the log-majorization relation on the singular values of the product of given positive definite matrices and their (metric and spectral) geometric means. We also establish the weak log-majorization between the spectra of the two-variable Wasserstein mean and spectral geometric mean. In particular, for a specific range of the parameter, the two-variable Wasserstein mean converges decreasingly to the log-Euclidean mean with respect to the weak log-majorization.Comment: 16 page

Local midpoints on smooth manifolds

In this paper we consider three methods for obtaining midpoints, primarily midpoints of geodesics of sprays, but also midpoints of symmetry (in symmetric spaces), and metric midpoints (in Riemannian manifolds). We derive general conditions under which these approaches yield the same result. We also derive a version of the Lie-Trotter formula based on the midpoint operation and use it to show that continuous maps preserving (local) midpoints are smooth