A General Introduction to Quantum Tomography -- and a Specialized Report on Property Testing...

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Quick, concrete guide: the first half surveys four concrete tomography strategies and their proven scalings — local asymptotic normality maps N qubits to a Gaussian oscillator and is optimal for fixed dimension; Schur–Weyl/Keyl–Werner uses the Schur transform to estimate spectrum with sample cost ~R·D/ε^2; compress sensing recovers rank-R states from O(RD) Pauli measurements via an SDP; projected least squares is a simple, fast estimator giving sample bounds ≈R^2·D^1.6 for product-basis setups. The second half proves the Clifford-tensor commutant is generated by discrete orthogonal maps and CSS projections and yields a 6-copy stabilizer test (dimension-independent), plus bounds on adding few non-Cliffords for designs.

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" chapters (17)

1. 0:00Intro
2. 1:09What's the point?
3. 5:22The design problem
4. 7:10The estimation problem
5. 13:001/4 Local asymptotic normality
6. 23:483/ 4 Compressed Sensing
7. 28:564/4 Projected Least Squares
8. 31:25Summary
9. 36:02Testing under symmetry
10. 39:06Schur-Weyl duality 2: Transversality
11. 41:20Clifford group, prior results
12. 43:22Statement of main result
13. 43:58Stochastic orthogonal transformations
14. 49:27Stabilizer testing
15. 51:26exponential de Finetti
16. 51:42Quantum Homeopathy
17. 59:18Outlook