Semidefinite Hierarchies in Quantum Information

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" summary

Semidefinite hierarchies convert hard nonnegativity and separability checks into tractable sum of squares and SDP feasibility problems by representing polynomials via a positive semidefinite moment matrix and enforcing increasingly strong multipliers (degree k). The method gives explicit certificates (positive semidefinite matrices) that detect entanglement, approximates quantum correlations and game values from below, and dualizes to symmetric extensions / PPT and k-extendible hierarchies. Convergence holds as k → ∞ but finite-dimensional attainment and rates depend on chosen norms and can be slow.

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" chapters (14)

1. 0:00Intro
2. 0:22Outline
3. 1:17What is semidefinite programming (SDP)?
4. 4:55SDP Hierarchies
5. 15:50Sum of Squares and Semidefinite Programming
6. 22:13Dual aside: moment consistency
7. 27:22From polynomials to entanglement
8. 30:43From entanglement to polynomials
9. 34:14Back to polynomials: hierarchies!
10. 36:24Hierarchy of entanglement tests
11. 41:25Dual interpretation: symmetric state extensions
12. 43:25State extensions and de Finetti
13. 44:46Noncommutative polynomials
14. 47:17Computing game values (DLTW)