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In this paper, we study the convex quadratic optimization problem withindicator variables. For the bivariate case, we describe the convex hull of theepigraph in the original space of variables, and also give a conic quadraticextended formulation. Then, using the convex hull description for the bivariatecase as a building block, we derive an extended SDP relaxation for the generalcase. This new formulation is stronger than other SDP relaxations proposed inthe literature for the problem, including Shor's SDP relaxation, the optimalperspective relaxation as well as the optimal rank-one relaxation.Computational experiments indicate that the proposed formulations are quiteeffective in reducing the integrality gap of the optimization problems.

2x2 convexifications for convex quadratic optimization with indicator variables