monotonicity and wigneryanase-dyson metrics,” Infinite Dimensional Analysis, Quantum Probability and

We show that, for each value of α ∈ (−1, 1), the only Riemannian metrics on the space of positive definite matrices for which the ∇(α) and ∇(−α) connections are mutually dual are matrix multiples of the Wigner-Yanase-Dyson metric. If we further impose that the metric be monotone, then this set is reduced to scalar multiples of the Wigner-Yanase-Dyson metric.