States of quantum systems and their liftings

TL;DR

This paper characterizes all affine right-inverse mappings to the partial trace operation in quantum systems and explores the limitations of representing quantum states via probability measures.

Contribution

It provides a complete description of affine right-inverse mappings to the partial trace and analyzes the non-existence of affine mappings into certain probability measure spaces.

Findings

Any affine right-inverse to the partial trace is a tensor product with a fixed state.

No affine mappings exist from quantum state space into probability measures on pure states.

No affine mappings exist from quantum state space into probability measures on the Hilbert space.

Abstract

Let H(1), H(2) be complex Hilbert spaces, H be their Hilbert tensor product and let tr2 be the operator of taking the partial trace of trace class operators in H with respect to the space H(2). The operation tr2 maps states in H (i.e. positive trace class operators in H with trace equal to one) into states in H(1). In this paper we give the full description of mappings that are linear right inverse to tr2. 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 tr2 is given by the tensor product of W with 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 Schroedinger 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.