| This dissertation explores several topics pertaining to quantum computation and information theory. First, we discuss the distinction between entangled and separable states from a geometric point of view. In particular, we construct the ellipsoid of smallest volume that bounds the set of separable states for systems of n qubits, although the results generalize easily to larger spaces. This ellipsoid serves as an approximation of the boundary between separable and entangled states. Notably, we show that when restricted to pure states all separable states lie on the ellipsoid boundary, and all entangled states lie outside. We demonstrate that this distinguishing power motivates an entanglement measure on pure states. For 2 qubits, this measure can be written in a particularly convenient form, while for 3 or more qubits the ellipsoid structure provides a natural weighting of entanglement shared between subsystems of varying size.;We then address classical models of quantum noise. Though the classical noise models are not fully general, it is known that certain classes of quantum noise can be realized classically. In particular, dephasing noise can always be simulated classically. For a single qubit, we explicitly construct classical models to simulate arbitrary dephasing noise. For two qubits, we construct classical models that reproduce a subset of the dephasing noise; these models can be combined to create more complicated dephasing behavior. Additionally, we show that depolarizing noise is classical for quantum systems of arbitrary dimension.;Lastly we discuss error correction. Motivated by experimental capabilities and limitations of neutral atom qubits, we explore the practical possibility of measurement-free error correction. For three well known error correction codes---the bit-flip, Bacon-Shor, and Steane codes---we adapt standard measurement-based procedures to measurement-free circuits on neutral atom systems. In particular, we present a novel syndrome extraction technique to achieve fault-tolerance. Using numerical simulation we estimate first-level depolarizing thresholds for these circuits. We find that simulating realistic conditions for the bit-flip, Bacon-Shor, and Steane codes produced error thresholds of pth ≈ 10-2, 10 -3, and 10-4, respectively. Encouragingly, these results are within the range of expected neutral atom capabilities and compare well to measurement-based threshold values. |
