Monografia: Fundamentos Matemáticos da Separabilidade - UFMG

Abstract
Entangled states are the key for the revolution that is happening in the foundations
of quantum mechanics: the discovery of nonlocality as the essential
trait of the quantum world, through the Bell inequalities, the advent of quantum
computation, with the famous Shor’s algorithm, and the quantum key
distribution, with its promise of a perfectly secure communication.
However, entanglement is a quite complicated characteristic, and we still
don’t know its complete mathematical characterisation. For this, we need
to develop criteria that can decided if a given quantum state is entangled or
not. This work makes a revision of the fundamental mathematical concepts
behind the famous positive partial transpose (PPT) criterion: the separating
hyperplane theorem, the Jamiołkowski isomorphism, and the decomposability
of the positive maps. The connection between this apparently disjoint concepts
is the Woronowicz theorem: every low-dimensional 2 positive map can be
written as the convex combination of completely positive and completely
copositive maps. The main focus of this work is its demonstration, presented
with a modern language and notation adequate for physics.
The work finishes with an geometrical exploration of the state space, by the
means of the Hilbert-Schmidt inner product. We analyse the basic symmetry
properties and the eigenvalue simplex representation, culminating with a
representation of the entangled states and the numerical calculation of the
relevant volumes in various dimensions.
2 Really low.
4