4 Qubit Quantum State Tomography

Abstract

Much like its classical counterpart, which aims at reconstructing three-dimensional images via a series of two-dimensional projections along vari-ous 'cuts', quantum tomography characterizes the complete quantum state of a particle or particles through a series of measurements in different bases. While the characterization of a classical object can involve a series of measure-ments on the same subject, measuring a single quantum particle perturbs its state, often making its further investigation uninformative. For this reason, quantum tomography must be carried out in stages on a number of identical copies of the same state, and can never be successfully applied to a single unknown particle. The etymology of tomography is therefore descriptive (the Greek tomos means section) – a series of measurements on identical particle ensembles each allow a glimpse into a distinct aspect of a quantum state's reality. Each new type of measurement illuminates a new dimension of an unknown state; subjecting more identical copies of that state to a single type of measurement brings that particular observable into sharper relief. This chapter will present a review of the representation of quantum states, a procedure for and an explanation of the reconstruction of an unknown state from a series of ideal measurements on an ensemble of identical particles, and the adaptation of that method to real systems, i.e., non-ideal measurements. Each section will be presented first in general, without respect to a particular physical qubit implementation, followed by the application of that theory to the specific case of qubits encoded into the polarization states of photons. We chose this system for convenience and availability of clean results. However, the techniques presented here can be applied to determine the quantum state of any system of one or more qubits (2-level systems). This includes photons [1-10], spin-1 2 particles (as, e.g., are used in NMR quantum computing [11, 12, 13, 14]), and (effectively) 2-level atoms [15, 16]. In order to facilitate the use of these techniques by groups and individuals working in any field, a website is available which provides both further details about these techniques and working, documented code for implementing them. 3