Study of continuous variable entanglement in multipartite harmonic oscillator systems

In this thesis we investigate the entanglement of Schrodinger cat states that derive from harmonic oscillator models. In order to extend the finite dimensional framework of entanglement to the infinite dimensional case we consider only initial conditions that have some type of symmetry. Systems with symmetry usually have fewer important parameters. In our case, symmetry allows us to discard the bulk of the Hilbert space as irrelevant to our particular entanglement problem. We are then left with an effectively finite dimensional Hilbert space, and the developed entanglement framework can therefore be followed. The dimension we derive for the reduced Hilbert space in each subsystem is equal to the number of coherent states in the Schrodinger cat superposition.;We investigate the entanglement vs. time of our Schrodinger cat state for closed and open systems. For closed systems, we place no limit on the number of coherently summed linearly independent coherent states. So the dimension of our effective Hilbert space can be quite high. We also place no restriction on the number of subsystems (or parties). Consequently, we use the entanglement measure developed by Barnum, Knill, Ortiz, and Viola (BKOV). This is the only measure to our knowledge that has no restriction on the dimension of the Hilbert space or the number of subsystems. We also place no constraint on the magnitude of our coherent states. The coherent value may be quite large, or quite small. We find that the entanglement of the Schrodinger cat state has nontrivial dependence on the above mentioned three variables. That is, the entanglement is a non-separable function of the values of the coherent states, the number of coherent states in the superposition, and the number of partitions of the Hilbert space.;For open systems, we model the reservoir as a harmonic oscillator zero temperature bath. Due to the interactions with the bath the Schrodinger cat state becomes a mixed density matrix. To investigate the time dependent entanglement of our density matrix, we apply the convex roof extension of the BKOV measure. This required development of an algorithm to search the space of decompositions of the density matrix. The time dependence depends on the symmetry of the system, naturally splitting the Hilbert space into a direct sum of two subspaces. One subspace interacts strongly with the bath resulting in rapid decoherence. The other complimentary subspace does not interact at all with the bath and is decoherence free. For initial states in the decohering subspace we find that for large values of the coherent state, the decay of entanglement corresponds to the decay rate of correlations in the bath. We are able to derive this result analytically. For small values of the coherent states, the loss of entanglement corresponds to the decay of the coherent state amplitude.;Finally, we consider initial states that live in the combined Hilbert space. Decoherence then provides a means of engineering a new state through decay of the component that resides in the decohering subspace of the complete Hilbert space. We develop a type of super-symmetry that gives us a new reduced Hilbert space that naturally encompasses the symmetry of the decohering subspace and the symmetry of the decoherence free subspace. In this new, "super" Hilbert space, we are able to characterize the entanglement of our density matrix vs time.