On Pins and Needles: The Geometry of Quantum Measurements

Stepping up to big quantum systems with the intuitions learned from qubits is dangerous business, but studying the geometric properties of these higher-dimensional quantum state spaces can keep us from stumbling. I will discuss properties of tangent cones relevant to the complexity of compressed sensing and convex optimization in quantum state tomography, revealing the state space to be quite sharply pointed. I will also describe the measurement-based metrics in which these properties must be quantified, highlighting their incompatibility with the most widely used metrics on the state space.