On the space of positive definite matrices we consider distance functions of the form d(A, B) = [trA(A, B) − trG(A, B)]1/2 , where A(A, B) is the arithmetic mean and G(A, B) is one of the different versions of the geometric mean. When G(A, B) = A 1/2 B 1/2 this distance is A 1/2 − B 1/2 2 , and when G(A, B) = (A 1/2 BA 1/2 ) 1/2 it is the Bures-Wasserstein metric. We study two other cases: G(A, B) = A 1/2 (A −1/2 BA −1/2 ) 1/2 A 1/2 , the Pusz-Woronowicz geometric mean, and G(A, B) = exp log A+log B 2 , the log Euclidean mean. With these choices d(A, B) is no longer a metric, but it turns out that d 2 (A, B) is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of m positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy. √ 2 d(p, q) as the definition of the Hellinger distance. We have thenwhere A(p, q) is the arithmetic mean of the vectors p and q, G(p, q) is their geometric mean, and tr x stands for x i .A matrix/noncommutative/quantum version would seek to replace the probability vectors p and q by density matrices A and B ; i.e., positive semidefinite matrices A, B with tr A = tr B = 1. In the discussion that follows, the restriction on trace is not needed, and so we let A and B be any two positive semidefinite matrices. On the other hand, a part of our analysis 2010 Mathematics Subject Classification. 15B48, 49K35, 94A17, 81P45.