Sum Uncertainty Relations: Uncertainty Regions for Qubits and Qutrits

Abstract

We investigate the notion of uncertainty region using the variance based sum uncertainty relation for qubits and qutrits.We compare uncertainty region of the qubit (a 2-level system) with that of the qutrit (3-level system) by considering sum uncertainty relation for two non-commuting Pauli-like observables, acting on the two dimensional qubit Hilbert space. We identify that physically valid uncertainty region of a qubit is smaller than that of a qutrit. This implies that an enhanced precision can be achieved in the measurement of incompatible Pauli-like observables acting on the 2-dimensional subspace of a qutrit Hilbert space. We discuss the implication of the reduced uncertainties in the steady states of Lambda, V and Cascade types of 3-level atomic systems. Furthermore, we construct a two-qubit permutation symmetric state, corresponding to a 3-level system and show that the reduction in the sum uncertainty value -- or equivalently, increased uncertainty region of a qutrit system { is a consequence of quantum entanglement in the two-qubit system. Our results suggest that uncertainty region can be used as a dimensional witness.