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, Debbie Leung, Matilde Baroni, Eduardo Martín-Martínez for their contributions. We are grateful for support by the NSF Challenge Institute for Quantum Computation.

The List (early draft)

1. The Spectral Theorem
2. Eigenstates in an infinite square well
3. The Quantum Harmonic Oscillator
4. Ehrenfest’s Theorem
5. Quantum tunneling through a finite potential barrier
6. Conservation of the probability (4-)current
7. Eigenstates of a δ-distribution well
8. Heisenberg Uncertainty Relation
9. Baker–Campbell–Hausdorff Formula
10. Trotter Product Formula (Lie–Trotter–Kato)
11. Simultaneous diagonalization of commuting observables
12. Noether’s Theorem
13. Wigner’s Theorem
14. Stone–von Neumann Theorem
15. Spectrum of the hydrogen atom (Coulomb potential)
16. Addition of angular momentum (e.g. Clebsch–Gordon)
17. Wigner–Eckart Theorem
18. Power-series spectrum of a perturbed time-independent Hamiltonian
19. Dyson series for time-dependent perturbation theory
20. The Adiabatic Theorem
21. Berry phase (Geometric phase)
22. Superdense coding
23. Quantum teleportation with shared entanglement
24. No-Cloning Theorem / No-Broadcast Theorem / No-Teleportation Theorem
25. No-Communication Theorem
26. No-Deleting Theorem
27. Schrödinger–HJW Theorem (purification of mixed states)
28. Schmidt decomposition for Hilbert spaces
29. Bell’s Theorem (QM violates CHSH inequality)
30. Monogamy of entanglement
31. Tsirelson bound
32. Correctness of the Deutsch–Jozsa algorithm
33. Correctness of Shor’s algorithm
34. Correctness of Grover’s algorithm
35. Quantum key distribution by BB84
36. Solovay–Kitaev Theorem
37. Universality of sets of two-qubit quantum gates
38. Universality of the Toffoli and the Hadamard
39. The Threshold Theorem (quantum fault-tolerance)
40. Petz recovery map
41. Choi’s Theorem on completely positive maps
42. Choi–Jamiołkowski isomorphism (channel-state duality)
43. Gelfand–Naimark–Segal construction
44. Gelfand–Naimark Theorem
45. Krein–Milman Theorem
46. Stinespring’s Factorization Theorem / Naimark’s Dilation Theorem
47. Continuous Functional Calculus
48. Gleason’s Theorem
49. Holevo’s Theorem
50. Pusey–Barrett–Rudolph Theorem
51. Kochen–Specker Theorem
52. The Spectral Theorem for PVMs
53. Strong subadditivity of quantum entropy
54. Entanglement-assisted classical capacity of quantum channels
55. Quantum state discrimination for two states (Ivanović–Dieks–Peres Limit)
56. The Lloyd–Shor–Devetak Theorem for quantum channel capacity
57. Eastin–Knill Theorem
58. Gottesman–Knill Theorem
59. Magic state distillation
60. MIP* = RE
61. BQP \(\subseteq\) PP
62. QIP = PSPACE
63. Quantum Stein Lemma
64. Quantum 3-SAT is QMA1-complete
65. Computing the Jones polynomial at roots of unity is BQP-hard
66. Onsager’s solution to the 2D Ising model
67. Uniqueness of 4-dimensional representations of the Clifford algebra (up to unitary equivalence)
68. Spin-statistics Theorem
69. Reeh–Schlieder Theorem
70. Wick’s Theorem
71. Elitzur’s Theorem
72. The CFT central charge is its entanglement entropy
73. Osterwalder–Schrader Reconstruction Theorem
74. Bisognano–Wichmann Theorem
75. Wood–Spekkens–Bell Theorem