Norm Minimization, Invariant Theory, And Quantum Tomography
summary
Shows the exponential rate that the probability of measuring total angular momentum equals zero equals a concrete norm minimization over SL2, and more generally links that rate to the moment map (geodesic gradient) of the log-norm. Uses averages over the unitary group and a stationary-phase style estimate to get one-over-poly K lower bounds, gives deterministic algorithms because the optimization is geodesically convex for fixed groups, explains spectrum estimation and tomography converging to the moment map and spectrum, recovers matrix-scaling capacity/permanent limits, and points to the Mathieu → Jacobian conjecture as the remaining obstacle.
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chapters (13)
- 0:00Introduction
- 1:50Workshop Overview
- 4:00Presentation
- 5:30Angular Momentum
- 9:20Optimization Problem
- 10:00Norm Minimization Problem
- 12:20Invariant Theory
- 15:00Quantitative Invariant Theory
- 18:45Norm Minimization
- 26:45Representation Theory
- 39:05Moment Map
- 42:05Results
- 57:30Connection
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