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On quantum information

We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator ρ, tr ρ=1. Let I(P)\in R be defined for any partition P = (P_1,...,P_m), P_1+ ... +P_m=1_H, P_i \in proj H$ and let I(P_i Qj, i \leq m, j \leq n) = I(P) + I(Q) for Q =(Q_1,..., Q_n), \sum Q_j = 1_H and P_iQ_j = Q_j P_i, tr ρ P_iQ_j = tr ρ P_i tr ρ Q_j (P, Q are physically independent). Assuming some continuity properties we give a general form of generalised information I.