藤井 淳一

Corach-Porta-Rechtが採用した正定値行列のファイバー束を使った幾何学的考察,いわゆる「CPR幾何学」は,その手法が特殊なため,あまり正当な評価を受けているとは言いがたかった。最近Hiai-Petzが導入した測地線に対して,筆者がCPR型の解釈を試みたことで,元のCPR幾何学がなぜそのような描像を選んだのかが明らかになり,簡潔にして要を得た手法だったことが判明した。ここではその利点を余すところなく解説し,ファイバー束の幾何学の直観的解釈に基づいて,Hiai-Petzの測地線を導く中で,教育的な視点も交えながら,CPR幾何学の本質を論じてみたい。Since it is hard to explain why Corach-Porta-Recht should use fiber bundles and their actions in their geometry of the positive-definite matrices, almost all researchers have kept away from their method in this subject. Recently Hiai-Petz introduced interest parametrized Riemannian geometries whose geodesics are paths of operator means. So the author discussed their geometry from the viewpoint of the Corach-Porta-Recht. Consequently we now learned that they used a very natural and brief explanation for their geometry. Based on a intuitive interpretation of basic concepts of their geometry, we show these facts in the discussion of the Hiai-Petzgeodesics.

This is an extension of results represented in ISIT2003. Concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is proven for 0 /spl les/ s /spl les/ 1.