123 research outputs found
Distances of qubit density matrices on Bloch sphere
We recall the Einstein velocity addition on the open unit ball \B of
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 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
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