Let H-1, H-2 be complex Hilbert spaces, H be their Hilbert tensor product and let tr(2) be the operator of taking partial trace, with respect to the space H-2, of trace class operators in H. The operation tr(2) maps states in H (=positive trace class operators in H with trace equal to 1) into states in H-1. In this paper we give the full description of mappings that are linear right inverse to tr(2). More precisely, we prove that any affine mapping F(W) of the convex set of states in H-1 into the states in H that is right inverse to tr(2) is given by W bar right arrow WxD for some state D in H-2. In addition we investigate a representation of the quantum mechanical state space by probability measures on the set of pure states and a representation-used in the theory of stochastic Schrodinger equations-by probability measures on the Hilbert space. We prove that there are no affine mappings from the state space of quantum mechanics into these spaces of probability measures. (C) 2001 American Institute of Physics. [References: 14]
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