Entangled qubit pairs

The aim of this dissertation is to study the kinematics of 2-qubit
pairs. A 2-qubit state is specified by the Bloch vectors for each
qubit and a 3 × 3 dyadic, representing the expection values of the
joint observables. In what follows the main results are listed. •
The behaviour of a 2-qubit state under a local transformation is
investi-gated, where we obtain a set of 9 quantities, constructed
from the Bloch vectors and the cross dyadic, which are invariant
under local transforma-tion. We establish a generic form for a
2-qubit state, which helps us decide if given two 2-qubit states
belong to the same family or not. Using this form, which depends on
the characteristic values of the cross dyadic, one can distinguish
between all two-qubit families, where we obtain six classes of
families of locally equivalent states. Those families classify all
2-qubit states, four of them consist of two subclasses each. • A
simple criterion is obtained to check if a 2-qubit state is
separable or not. This criterion does not depend on the eigenvalues
of the partial trans-position of the state in question but only on
three numbers, those are the coefficients of the eigenvalue
equation. These numbers are written as a function of the Bloch
vectors and the cross dyadic. Also using the charac-teristic values
of the cross dyadic, one can decide if the CHSH inequality is
obeyed or violated. • The properties of Lewenstein-Sanpera
decompositions are employed to split a given state into its optimal
separable and pure parts. The degree of separability is an
important part in this decomposition, where a state is more useful
for quantum information purposes, the smaller its degree of
separability. The optimal degree of separability is obtained
analytically for some important cases as a function of the Bloch
vectors and the cross dyadic of the state in question: For the
simple Werner state, where our result coincides with that obtained
numerically. Also for the self-transposeed states, the generalized
for Werner states and rank-2 states. • The technique of the rank-2
case is employed to study the case of rank-3 and the full rank,
where we obtain all possible decompositions. Among them, we choose
the one which has the largest splitting parameter. In this
situation a numerical procedure could help in obtaining the optimal
de-composition. Given a density matrix, one obtains its eigenvalues
and corre-sponding eigenvectors. Using the rank-2 method to obtain
all the possible decompositions, then choose the one corresponding
to the largest degree of separability. If this decomposition is the
optimal one, a technique is used to check the optimality. If it is
not optimal, an optimization procedure is performed. • There is an
important relation between the degree of separability S and what is
the so called concurrence C which appears in the definition of the
entanglement of formation. In general, the sum of the degree of
separabil-ity and the concurrence is less than one, but equal to
one for states with vanishing Bloch vectors. On the other hand both
of them run from 0 to 1. • A criterion is introduced to decide if a
given decomposition is the optimal one or not. Two inequalities are
obeyed if the decomposition in question is the optimal one and are
violated if it is not. • The effect of the unitary operators and
BCNOT operations are described on the dynamical variables. An
alternative presentation of the IBM and Oxford purification
protocols is obtained by using theses variables. One could
introduce the degree of separability as a purification parameter,
where the purified state has a smaller degree of separability than
the initial one. Employing the properties of the characteristic
values of the cross dyadic, makes the Oxford protocol much faster.