Theoretical investigations of separability and entanglement of bipartite quantum systems

A key distinguishing feature of quantum theory is the possibility of entanglement between subsystems. The significance of this phenomenon is now unquestioned, as it lies at the core of several of the most important achievements of quantum information science. Although the nonclassical nature of entanglement has been recognized for many years, only recently has a considerable effort been put into trying to understand and characterize its properties precisely.; There is a general method for quantifying the degree of entanglement of a pair of qubits, and a criterion, the partial transposition condition, which determines whether a general state of two qubits is entangled and whether a general state of a qubit and a qutrit is entangled. This partial-transposition condition fails, however, to provide a criterion for entanglement in other cases, where the constituents have higher Hilbert-space dimensions or where there are more than two constituents. At present there is no general criterion for determining whether the joint state of N qudits is entangled, nor is there any general way to quantify the degree of entanglement if such a state is known to be entangled. We consider states of two qudits that are a mixture of the maximally mixed state and a maximally entangled state. We show that such states are separable if and only if the probability for the maximally entangled state in the mixture does not exceed 1/(1 + D). We also consider the separability of mixed states of N two-dimensional systems near the maximally mixed state. We find both lower and upper bounds on the size of the neighborhood of separable states around the maximally mixed state.; To quantify the amount of entanglement in a given entangled state, several measures of entanglement have been introduced and investigated, an example being the entanglement of formation.; The universal inverter turns out to be the ideal inverter of pure states, since it takes a pure state to the maximally mixed state in the subspace orthogonal to the pure state.; We define the I-concurrence of mixed states of D 1 x D2 quantum systems as the minimum average I-concurrence of ensemble decompositions of the joint density operator. We also investigate a quantity which looks similar to I-concurrence called the tangle. (Abstract shortened by UMI.)...