Flat connections and Wigner-Yanase-Dyson metrics

TL;DR

This paper characterizes the unique conditions under which flat dual affine connections exist on the manifold of positive definite matrices, linking them specifically to Wigner-Yanase-Dyson metrics.

Contribution

It proves that such flat dual connections exist only when the metric is the Wigner-Yanase-Dyson skew information, providing a new geometric characterization.

Findings

Existence of flat dual connections is exclusive to Wigner-Yanase-Dyson metrics.

Dual connections can be constructed via $oldsymbol{ extit{ extalpha}}$-embeddings or contrast functionals.

The result links geometric structures to specific quantum information metrics.

Abstract

On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the $\alpha$-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that in both cases, the existence of such a pair of connections is possible if and only if the metric is given by the Wigner-Yanase-Dyson skew information.