Quantum state visualization, verification and validation via phase space methods

Since its introduction in the 1930s by Wigner, and its generalisations by Moyal and Weyl, the ability to associate an operator on Hilbert space by a quasi-probability distribution function on phase space has found extensive use in the physics of con- tinuous variable systems. Lacking, however, is finite system applications; to date, such functions have taken a back seat to state vector, path integration, and Heisen- berg representations.

In recent work, this lack of application has been addressed by giving a general framework to generate phase-space distribution functions for any system. Where the Wigner function for any system can be expressed in displaced parity form. This construction of a general framework for treating quantum mechanics in phase space will be presented in full in this thesis. Demonstrating a general approach to quantum mechanics as a statistical theory.

Using this work, it will be shown how varied quantum systems can be easily represented in phase space as well as visualise certain quantum properties, such as entanglement, within these systems. In particular, formalism is applied to directly measure phase space coordinates of multiple qubit states, including a five-qubit GHZ state, on IBM’s Quantum Experience. Further, how these methods can be extended for use in general composite quantum systems, such as hybrid atom-cavity systems, will be presented, demonstrating how these phase-space methods are an optimal method for quantum state analysis, entanglement testing, and state characterisations.