Simultaneous decompositions of a pair of states into pure ones are examined. There are privileged decompositions which are distinguished from all the other ones. Presently we witness that quantum information theory is becoming an interdisciplinary, quickly growing field of research. In its history Roman S. Ingarden is playing a significant role, both by his own research and by posing stimulating questions and problems . It is about 40 years ago that I met Roman the first time, and he was already thinking about the role of information in quantum physics and, in particular, whether one can found the concept of probability onto that of information . I feel honored by the possibility to dedicate to him the following paper. 1 Decomposing one density operator A density operator, representing a state, is a positive operator with trace one. However, it is convenient for the following considerations not to insis
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