Top 100 Quantum Theorems

M. David, J. Palsberg

This list collects 100 important theorems in quantum physics, quantum information and quantum computing. Inspired by the list of the top 100 mathematical theorems of which Freek Wiedijk tracks the formalization progress, this is a similarly arbitrary and necessarily incomplete list with no particular order.

We welcome submissions from your interactive theorem proving community, and will link formal statements and proofs below.

Acknowledgements. Thank you to Rodolfo R. Soldati, Alex Meiburg, Serge Massar, Tein van der Lugt for their contributions. We are grateful for support by the NSF Challenge Institute for Quantum Computation.

The List (early draft of the first half)

1. The Spectral Theorem
2. Heisenberg Uncertainty Relation
3. Baker–Campbell–Hausdorff Formula
4. Superdense Coding
5. Quantum Teleportation with Shared Entanglement
6. No-Cloning Theorem / No-Broadcast Theorem / No-Teleportation Theorem
7. No-Communication Theorem
8. No-Deleting Theorem
9. Purification Theorem for Mixed States
10. Schmidt Decomposition for Hilbert Spaces
11. Bell’s Theorem (QM violates CHSH Inequality)
12. Monogamy of Entanglement
13. Tsirelson Bound
14. Correctness of the Deutsch–Jozsa Algorithm
15. Correctness of Shor’s Algorithm
16. Correctness of Grover’s Algorithm
17. Quantum Key Distribution by BB84
18. Solovay–Kitaev Theorem
19. Universality of Sets of Two-Qubit Quantum Gates
20. Universality of the Toffoli and the Hadamard
21. Petz Recovery Map
22. Choi’s Theorem on Completely Positive Maps
23. Choi–Jamiołkowski Isomorphism (channel-state Duality)
24. Gelfand–Naimark Theorem
25. Stinespring’s Factorization Theorem / Naimark’s Dilation Theorem
26. Continuous Functional Calculus
27. Gleason’s Theorem
28. Holevo’s Theorem
29. Pusey–Barrett–Rudolph Theorem
30. Kochen–Specker Theorem
31. The Spectral Theorem for PVMs
32. Strong Subadditivity of Quantum Entropy
33. Entanglement-assisted Classical Capacity of Quantum Channels
34. Quantum State Discrimination for Two States (Ivanović–Dieks–Peres Limit)
35. The Lloyd–Shor–Devetak Theorem for Quantum Channel Capacity
36. Eastin–Knill Theorem
37. Gottesman–Knill Theorem
38. Magic State Distillation
39. Spin-statistics Theorem
40. Reeh–Schlieder Theorem
41. MIP* = RE
42. BQP \(\subseteq\) PP
43. QIP = PSPACE
44. Quantum Stein Lemma
45. Quantum 3-SAT is QMA1-complete
46. Computing the Jones Polynomial at Roots of Unity is BQP-hard
47. Onsager’s solution to the 2D Ising Model
48. Wick’s Theorem
49. Osterwalder–Schrader Reconstruction Theorem
50. Wood–Spekkens–Bell Theorem