Qubism: self-similar visualization of a many-body wavefunction

Article, code and more: http://qubism.wikidot.com/

Transcript

1. self-similar visualization of many-body wavefunctions QUBISM: presented by: Piotr Migdał (ICFO, Barcelona)
2. Don’t take plots for granted!
5. bar chart - William Playfair (1786) scatter plot - Francis Galton (a century later)
6. Dmitri Mendeleev | Periodic Table of Elements (1869) periodic table - Dimitri Mendeleev (1869)
7. Back to the quantum world
8. ↵|"i + |#i
9. ↵|"i + |#i ⇠ = ↵| i + |•i
10. ↵|"i + |#i ⇠ = ↵| i + |•i ⇠ = ↵|0i + |1i
11. ↵|"i + |#i ⇠ = ↵| i + |•i ⇠ = ↵|0i + |1i ↵00 |00i + ↵01 |01i + ↵10 |10i + ↵11 |11i
12. ↵|"i + |#i ⇠ = ↵| i + |•i ⇠ = ↵|0i + |1i ↵00 |00i + ↵01 |01i + ↵10 |10i + ↵11 |11i ↵000 |000i + ↵001 |001i + ↵010 |010i + ↵011 |011i + ↵100 |100i + ↵101 |101i + ↵110 |110i + ↵111 |111i
13. ↵|"i + |#i ⇠ = ↵| i + |•i ⇠ = ↵|0i + |1i 2n complex parameters ↵00 |00i + ↵01 |01i + ↵10 |10i + ↵11 |11i ↵000 |000i + ↵001 |001i + ↵010 |010i + ↵011 |011i + ↵100 |100i + ↵101 |101i + ↵110 |110i + ↵111 |111i
16. 00 01 10 11
17. 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11
18. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11
19. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
20. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
21. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
22. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
23. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
24. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 FM: 000000... FM: 111111...
25. 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 FM: 000000... FM: 111111... AFM: 010101... AFM: 101010...
26. Examples
27. Dicke state |01i + |10i p 2
28. Dicke state |01i + |10i p 2 00 10 01 11
29. Dicke state (|0011i + |0101i +|0110i + |1001i +|1010i + |1100i) / p 6
30. Dicke state particles zeros ones 6 3 3
31. Dicke state particles zeros ones 8 4 4
32. Dicke state particles zeros ones 10 5 5
33. Dicke state particles zeros ones 12 6 6
34. Dicke state particles zeros ones 14 7 7
35. Product state (↵|0i + |1i)n
36. Heisenberg AFM X ~ Si · ~ Si+1 (periodic boundary cond.)
37. Heisenberg AFM (1,2) (3,4) (5,6) (7,8) ... X ~ Si · ~ Si+1 (periodic boundary cond.)
38. Heisenberg AFM (1,2) (3,4) (5,6) (7,8) ... (n,1) (2,3) (4,5) (6,7) ... X ~ Si · ~ Si+1 (periodic boundary cond.)
39. Heisenberg AFM (1,2) (3,4) (5,6) (7,8) ... (n,1) (2,3) (4,5) (6,7) ... X ~ Si · ~ Si+1 (open boundary cond.)
40. It works for any qudit 1D spin chains
41. -- -0 -+ 0- 00 0+ +- +0 ++ + qutrits (spin-1) 0 -
42. AKLT state Affleck, Lieb, Kennedy and Tasaki (| +i + |00i + | + i)/ p 3 + 1 3 ⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1
43. AKLT state particles 4 Affleck, Lieb, Kennedy and Tasaki + 1 3 ⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1
44. AKLT state Affleck, Lieb, Kennedy and Tasaki + 1 3 ⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1 particles 6
45. AKLT state Affleck, Lieb, Kennedy and Tasaki + 1 3 ⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1 particles 8
46. AKLT state Affleck, Lieb, Kennedy and Tasaki + 1 3 ⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1 particles 10
47. Alternative qubistic schemes
48. 00 01 11 10 anti-ferromagnetic ferromagnetic
49. Heisenberg AFM X ~ Si · ~ Si+1
50. X z i z i+1 X x i Ising transverse field
51. X z i z i+1 X x i Ising transverse field = 1
52. X z i z i+1 X x i Ising transverse field
53. X z i z i+1 X x i Ising transverse field = 1
55. Product state
56. Product state Dicke half-filled
57. Product state Dicke half-filled Ising transverse field (ground state)
58. Product state Dicke half-filled Ising transverse field (ground state) Heisenberg (ground state)
59. You can see entanglement
60. entanglement: (1,2) vs (3,4,5,6,7,8,9,...)
61. entanglement: (1,2) vs (3,4,5,6,7,8,9,...) Schmidt rank: A A A A 1 (not entangled)
62. entanglement: (1,2) vs (3,4,5,6,7,8,9,...) Schmidt rank: A A A A 1 (not entangled) A B B C 3 (entangled!)
63. entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank:
64. entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A A A A A A A A A A A A A 1 (not entangled)
65. entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A A A A A A A A A A A A A 1 (not entangled) A
66. A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A A A A A A A A A A A A A 1 (not entangled) A
67. A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A A A A A A A A A A A A A 1 (not entangled) A B B C B C C B C C C A
68. A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A A A A A A A A A A A A A 1 (not entangled) A B B C B C C D B C C D C D D A B B C B C C B C C C A
69. A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A A A A A A A A A A A A A 1 (not entangled) A B B C B C C D B C C D C D D A B B C B C C B C C C A 5 (entangled!) A B B C B C C D B C C D C D D E
70. {|0i, |1i}⌦4 {|+i, | i}⌦4 ⌦4 x ⌦4 z Schmidt number: 1 2 2 3 4 |0000i |GHZi |Wi Dicke half-filling
71. Renyi fractal dimension (and box counting)
72. AKLT ground state also works for qutrits (e.g. spin-1) log(4) log(3) ⇡ 1 . 26 and its fractal dimension
73. 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 dq arctan(K) q=0 q=0.5 q=1 q=2 q =104 X ( i ) z ( i +1) z ( i ) x Ising transverse field surface-like line-like point-like
74. 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 dq arctan(K) q=0 q=0.5 q=1 q=2 q =104 X ( i ) z ( i +1) z ( i ) x Ising transverse field = 1 surface-like line-like point-like
75. And how about going the other way?
76. Jose I. Latorre, arXiv:quant-ph/0510031 (2005) QPEG! matrix product states for image compression JPEG?
77. Javier Rodriguez-Laguna Piotr Migdał Miguel Ibanez Berganza Maciej Lewenstein German Sierra
78. http://qubism.wikidot.com/ Thanks! paper, code, etc: J.Rodriguez-Laguna, P. Migdał, M. Ibánez Berganza, M. Lewenstein and G. Sierra. Qubism: self-similar visualization of many-body wavefunctions. New J. Phys. 14, 053028 (2012), arXiv:1112.3560.