Quantum measurements with post-selection

Abstract

A quantum measurement with post-selection defines measurement outcomes consistent with given initial and final states of the system being measured, through a change recorded on a measuring device or pointer. As the initial and final states of the system, known as pre-selection and post-selection respectively, need not be eigenstates of the measurement operator, post-selected measurement can consistently give rise to non-eigenvalue measurement outcomes, particularly if the initial pointer state has a large associated uncertainty. In the limit of a very weak interaction between the system and the device, the measurement outcome is the weak value, which can be larger than any known eigenvalue and can even be complex. Even though their physical meaning is debated, weak values show patterns consistent with classical logic and have been used to address conceptual problems in quantum mechanics. Weak measurements have found immense use in experimental techniques to amplify and detect small sig- nals, and for precision measurements. In this thesis, we review the theory of post-selected measurements and consequently that of weak measurements. We demonstrate the possibility of achieving higher signal-to-noise ratio of amplification using simplified Hermite Gauss and Laguerre Gauss modes instead of Gaussian wavefunction as pointer. We also explore the upper limit of amplification by calculating exact expressions of pointer shifts. Finally, we propose a method of reconstructing the state of a spin- 1/2 particle using post-selected quantum measurements.