Positive Mappings in Quantum Physics

Abstract

This chapter offers some results which will help to understand some foundational aspects of quantum mechanics. It relies on some results presented in the last chapter. The first section discusses the general form of σ-additive probability measures on the complete lattice of orthogonal projections on a Hilbert space (Gleason’s theorem) and its variations. In quantum mechanics and in quantum information theory quantum channels or quantum operations are defined mathematically as completely positive maps between density operators which do not increase the trace (see for instance the book “Quantum Computation and Quantum Information” by M.A. Nielsen and I. L. Chuang, Cambridge University Press 2000). Thus in the next section we determine the general form of quantum operations on a separable Hilbert space, i.e., we prove Kraus’ first representation theorem for operations. Usually quantum information theory studies systems of some finite dimension n and then density operators are just positive $n \times n$ matrices with complex coefficients which have trace 1. In this context the relevant C$^*$-algebra is just the space $M_n(\mathbb C)$ of all $n \times n$ matrices with complex entries, for some $n \in \mathbb{N}$. Therefore in the last section we determine the general form of completely positive maps for these algebras (Choi’s results). Of course, this is a special case of Stinespring factorization theorem, but some important aspects are added.