Tensors and Complexity in Quantum Information Science

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Claims that classification of pure multipartite entanglement reduces to algebraic geometry, using SL(2,C) local actions to label SLOCC orbit closures and singularities of the dual of the Segre variety. Demonstrates the 2x2x2 case is governed by the degree‑4 hyperdeterminant, separates GHZ vs W classes that do not convert under SLOCC, and shows many‑party systems produce infinitely many orbits. Connects tensor network and graph‑state constructions (control‑Z gates, cluster states) to complexity via the entanglement width criterion for universality.

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" chapters (21)

1. 0:00Intro
2. 0:16Outline Subject: math of tensors - physics of entanglement Aim: Talk by a quantum information theorist about elementary concepts/results and many open questions
3. 2:53Stochastic LOCC Entanglement: quantum correlation, never increasing on average
4. 5:24Bipartite entanglement under SLOCC
5. 14:26Encounter with tensors What is the SLOCC classification of 2x2x2 (3-qubit) case?
6. 16:20Dual variety
7. 18:13Hyperdeterminat
8. 19:273-qubit (2x2x2) SLOCC classification
9. 24:482x2x4 SLOCC classification
10. 26:37Entanglement measures in 2x2x4 case
11. 28:38SLOCC conversions among different classes
12. 30:374-qubit hyperdeterminant
13. 31:46Asymptotic bipartite case Classification of entanglement when there are infinitely many identical
14. 37:54quantum circuit as tensor composition
15. 39:282-qubit Controlled-Z gate
16. 41:52Multipartite entanglement as graphs
17. 45:37Entanglement width
18. 47:53Tensor network as quantum computation
19. 48:25Examples of universal resources
20. 49:51Complexity of entanglement Universal resources are necessarily macroscopic entanglement!
21. 53:35Tensor network states