The uncertainty relation places lower bound on the product in the variances of observables, a part of which is attributable to their commutator. Since non-commutivity of observables is a quantum theoretic characteristic, the said part in the uncertainty relation places a fundamental quantum limit on the error in any measurement process. Yet there are situations, like gravitational wave detection and spectroscopy, which require measurements with precision better than what is permitted by the said fundamental quantum limit. Is it possible to make measurements with precision better than what is permitted by the fundamental quantum limit? That question shall be addressed in the talk for measurements involving light and two-level atoms. It will be shown that the use of so called non-classical states of light and two-level atoms can circumvent the fundamental quantum limit in the respective processes of measurement. To that end, the concept of non-classical states shall be introduced. It is based on the question of whether the statistical nature of quantum theoretic predictions is attributable to some hidden influence acting randomly on physical systems. The theory which assumes that to be the case is the so called Local Hidden Variable (LHV) theory in which the hypothetical hidden influence is incorporated as hidden variables acting locally on physical systems. The LHV theory leads also to the concept of quantumness of information which characterizes quantum information processing like quantum computing. The concept and measure of quantumness of information shall be introduced.