On mathematical information channels with a non-commutative intermediate system (Q1092870)

Recently the mathematical theory of non-commutative (quantum) information channels has been investigated by many authors. In the present paper we study mathematical information channels which have a non-commutative system between input and output spaces. For this, a concept of generalized channels is introduced. It is defined as a strongly measurable mapping from a measurable space \((A,{\mathcal A})\) to the space \(T_ s(H)\) of self-adjoint trace-class operators on a separable Hilbert space H. If a generalized channel \(\nu\) and an observable \(\alpha\) are given, we can construct a usual mathematical channel \(\alpha\cdot \nu:(A,{\mathcal A})\to M(R,{\mathcal R})\) by \(\alpha \cdot \nu (a;\cdot)=<\nu (a),\alpha (\cdot)>\), where \(M(R,{\mathcal R})\) is the space of bounded signed measures on \((R,{\mathcal R})\) and \(<\cdot,\cdot >\) is the dual form between \(T(H)\) and \(B(H)\). Such a channel is an object of our study. Then we can define the information quantity of the channel \(\alpha\cdot \nu\), if a probability measure on A is given. We write it by \(I(\nu;\alpha)\). Let \(\alpha\) and \(\beta\) be observables. If the range of \(\alpha\) is contained in the range of \(\beta\), it may be expected the inequality \(I(\nu;\alpha)\leq I(\nu;\beta)\). We shall show it and give a condition for it to be equal.