1. 1 Combinatorics and Quantum Information Simone Severini Department of Combinatorics & Optimization, and Institute for Quantum Computing, University of Waterloo, Canada www.iqc.ca/~sseverin Isaac Newton Institute, Cambridge, March 2008

2. 2 Quantum Information Quantum information may be defined as the study of the achievable limits to information processing possible within quantum mechanics. Goals of quantum information: Determine limits on the class of information processing tasks which are possible in quantum mechanics (e.g., limitations on the class of measurements that may be performed on a quantum system). Provide constructive means for achieving information processing tasks (e.g., development of unbreakable schemes for doing cryptography based upon the principles of quantum mechanics, study of the power of quantum computational devices). (M. Nielsen)‏

3. 3 Quantum teleportation (applications: quantum memories, quantum repeaters). Superdense coding (primitive in quantum protocols). Secure key distribution in cryptography. Quantum computing (e.g., discrete Log, search in databases). Simulation of quantum mechanical evolutions (e.g., many-body systems, applications in condensed matter physics). Secret sharing (e.g., secure distributed computation). Quantum Games (e.g., lower bounds in algorithms, study of phase transitions). Quantum Lithography and Imaging (e.g., increased signal intensities, extensions of optical microscopy). Quantum Sensors (improve the sensitivity of measurements across the electromagnetic spectrum, measuring microwaves). Phenomena and Applications

4. 4 Entanglement Theory What is entanglement? Entanglement is a property of certain states of quantum systems. Entanglement is a physical resource (like energy or time). It is one of the key ingredients for a quantum computer, allowing to solve certain tasks more quickly. It is an NP hard problem to decide whether a given state is entangled or not. Goals of entanglement theory: Find easily implementable criteria for detecting and classifying entanglement. Isolate the properties that characterize entanglement. Determine how useful entanglement is. I would not call entanglement one but rather the characteristic trait of quantum mechanics (E. Schrödinger)‏

5. 5 Reconstruction of quantum states: Young diagrams, Marginal inequalities (Christandl et al. 2005, Klyachko, 2004)‏ Quantum state transformations: Schubert calculus and varieties (Hayden et al. 2004, Knutson 2004)‏ Classification of entangled states: Hyperdeterminants (Miyake et al., 2003), Combinatorics of permutations (Clarisse et al., 2005), Knots invariants (Kauffman et al., 2004)‏ Toy states and entanglement detection: graph laplacians (Braunstein et al., 2005)‏ One-way quantum computation: local operations on graphs (Briegel et al., 2002)‏ Quantum-error correction: codes over Z ₄ (Calderbank et al., 2003)‏ Bell's inequalities: cut polytopes (Avis et al., 2005)‏ Non-locality, CHSH games: extremal set theory (Cleve et al., 2005)‏ Optimal measurements: finite geometry, projective designs (Wootters et al., 2005)‏ Randomization of quantum states: quantum expanders (Ta-Shma et al., 2006)‏ Information transfer and spin network control: spectral graph theory (Cambridge people, Bose >2000), power domination in graphs (Aazam et al., 2008)‏ Topological quantum computation: Hopf algebras (Friedman, Kitaev, 1998)‏ Complexity parameters and simulation: (Valiant 2000, Shi, Josza 2006)‏ Combinatorics

6. 6 Classification of entangled states: combinatorics of permutations Toy states and entanglement detection: graph laplacians Quantum-inspired algorithm for graph matching Information transfer and spin network control: spectral graph theory, power domination in graphs Non-locality, CHSH games: extremal combinatorics Plan of this talks

7. 7 Clarisse, Ghosh, Sudbery, S, Phys. Rev. A 72, 012314 (2005)‏ quant-ph/0502040 Clarisse, Ghosh, Sudbery, S, Phys. Lett. A, 365 (2007)‏ quant-ph/0611075 Classification of entangled states and combinatorics of permutations

8. 8 What is a quantum state?

9. 9 What is entangled?

10. 10 How to measure entanglement?

11. 11 How to entangle?

12. 12 What is required to be a good entangler ?

13. 13 What are the best entanglers among all the elements of the unitary group?

14. 14 Latin square 123 231 312

15. 15 Mutually Orthogonal Latin Squares (MOLS)‏ 123 231 312 132 213 321 1,12,33,2 2,23,11,3 3,31,22,1

16. 16 MOLS permutation 1,12,33,2 2,23,11,3 3,31,22,1

17. 17 What are the best entanglers ?

18. 18 Open problems Combinatorial problems: What about dimension 6? Classify permutations according to the their entangling power. What about multipartite systems (i.e., many subsystems)? Notice: The max. # of MOLS of side l is For a larger # of subsystems can we do it with other unitaries? If we can not? 1. e(U) is not plausible; 2. entanglement for a few subsystems is special (or MOLS are objects from a larger class?).

19. 19 (Pictures - Courtesy of filmforum.org) 1/4

20. 20 Braunstein, Ghosh, S, Ann. Comb. 10 (2006), 291-327 quant-ph/0406165 Braunstein, Ghosh, Mansour, S, Wilson, Phys. Rev. A 73, 012320 (2006) quant-ph/0508020 Hildebrand, Mancini, S, Math. Structures Comput. Sci. 18:1 (2008), 205-219 cs.CC/0607036 Toy states and entanglement detection

21. 21 Associating graphs to states (Spin networks, bosonic graphs, graph states, etc.)‏

22. 22 3/ How hard is to detect entanglement?

23. 23 Partial transpose

24. 24 Positivity of Partial Transpose (PPT) Criterion

25. 25 Associating a lattice to the graph

26. 26 Partial transpose on the lattice

27. 27 Degree Criterion

28. 28 Degree Criterion and PPT

29. 29 Degree Criterion and entanglement

30. 30 The concurrence of entangled laplacian states in dimension four is exactly fractional: 1/(# edges). Characterize graphs whose laplacian state is entangled independently of the labelling. Consider multipartite systems: Wang, Wang, Elec. J. Comb. 14 (2007)‏ Hassan, Joag, J. Phys. A: Math. Theor. 40 (2007); J. Math. Phys. 49 (2008)‏ Open problems

31. 31 2/4

32. 32 Emms, Hancock, S, Wilson, Elec. J. Comb. 13 (2006), #R34. quant-ph/0505026 See this morning talk by E. Hancock, for applications in pattern recognition: Emms, Hancock, Wilson, ICPR, IEEE Computer Society (2006). For a different approach see Audenaert, Godsil, Royle, Rudolph, J. Comb. Theory, Ser. B 97(1): 74- 90 (2007). math.CO/0507251 A quantum-inspired graph matching algorithm

33. 33 Graph matching Problem. How “similar” graphs G and H are? (Pictures - Courtesy of M. Schultz)

34. 34 Graph isomorphism Problem. Are graphs G and H actually the “same graph”? (Pictures - Courtesy of D. Bacon)

35. 35 Representing a graph with an orthogonal matrix

36. 36 Graph Isomorphism Problem

37. 37 Strongly regular graphs

38. 38 Result

39. 39 3/4

40. 40 Aazami, S, A covering problem with a propagation rule: formulations and algorithms, submitted to SWAT08. Important reference: Burgardt, Giovannetti, Phys. Rev. Lett. 99 (2007). A game for controlling spin systems Perfect state transfer Saxena, S, Shparlinski, Int. J. Quantum Inf. (2007), 417-430. quant-ph/0703236 Important reference: Christandl, Datta, Ekert, Landahl, Phys. Rev. Lett. 92 (2004).

41. 41 Nondiscriminatory propagation

42. 42 Nondiscriminatory propagation

43. 43 Nondiscriminatory propagation

44. 44 Nondiscriminatory propagation

45. 45 Nondiscriminatory propagation

46. 46 Nondiscriminatory propagation

47. 47 Nondiscriminatory propagation

48. 48 Nondiscriminatory propagation

49. 49 Nondiscriminatory propagation

50. 50 Nondiscriminatory propagation

51. 51 Nondiscriminatory propagation

52. 52 Nondiscriminatory propagation problem

53. 53 Example

54. 54 Example

55. 55 Example

56. 56 Example

57. 57 Result

58. 58 For the purpose of studying state transfer (in the XY model), we can introduce the following: Perfect state transfer

59. 59

60. 60 Result: circulant graphs

61. 61 4/4

62. 62 Parallel repetitions of Clauser-Horne-Shimoni-Holt (CHSH) Games R. Cleve, W. Slofstra, F. Unger, S. Upadhyay quant-ph/0608146

63. 63 The CHSH game Bob Alice Referee No communication

64. 64 The CHSH game Bob Alice Referee No communication

65. 65 Best classical strategy 01 000 101 Input Output

66. 66 Best classical strategy 01 000 101 Input Output 00 000 000

67. 67 Best classical strategy 01 000 101 00 000 000 Input Output

68. 68 Best classical strategy

69. 69 Best quantum strategy

70. 70 Best quantum strategy Optimal by the Tsirelson’s Inequality

71. 71 There is a quantum strategy which is better than any classical strategy

72. 72 The repeated CHSH game Bob Alice Referee No communication

73. 73 The repeated CHSH game Bob Alice Referee No communication

74. 74 Input for n = 2 00011011 00 0100010001 1000 10 1100011011

75. 75 Best quantum strategy for playing the game n times

76. 76 A classical strategy

77. 77 Is this the best strategy? If not, what is the best strategy?

78. 78 For n = 2 there is better 00011011 00 0100010001 1000 10 1100011011 Input Output

79. 79 For n = 2 there is better 00011011 00 0100010001 1000 10 1100011011 00 10 00 010001 00 10 0100011011 Input Output

80. 80 For n = 2 there is better 00011011 00 0100010001 1000 10 1100011011 00 10 00 10 00 10 00 10 01 11 Input Output

81. 81 Best classical strategy for playing the game 2 times

82. 82 Best classical strategy for playing the game 3 times

83. 83 M. Bordewich, S, Ann. of Benians Court, Vol.1:1 (2008), p. 1.

84. 84 Best classical strategy for playing the game 4 times

85. 85 Best classical strategy for playing the game for an arbitrary number of times

86. 86 Upper bound Semidefinite programming (Feige-Lovasz relaxation)‏ Density of squares in (0,1)-matrices (Peleg)‏

87. 87 Natural upper bound

88. 88 Open problem