1. Q UANTUM C OMMUNICATION Aditi Sen(De) Harish-Chandra Research Institute, India

2. O UTLINE Communication Secure Communication Quantum Cryptography Communication

3. O UTLINE Communication Secure Communication Quantum Cryptography Communication Without security Classical info transmission Quantum state transmission

4. O UTLINE Communication Secure Communication Quantum Cryptography Communication Without security Classical info transmission Quantum state transmission

5. O UTLINE Communication Secure Communication Quantum Cryptography Communication Without security Classical info transmission Classical info transmission Quantum state transmission

6. C OMMUNICATION

8. W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

9. W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

10. W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

11. W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

12. W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob a process by which information is sent by a sender to a receiver via some medium.

13. W HAT IS C OMMUNICATION ? Alice (Encoder) Sends encodes Bob (Decoder) receives & decodes

14. W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/Decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

15. W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/Decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

16. W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

17. W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

18. W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer Do quantum states advantageous?

19. Classical Information Transmission via Quantum States Part 1

20. Quantum Dense Coding Bennett & Wiesner, PRL 1992

21. C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

22. C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

23. C LASSICAL P ROTOCOL Sunny Windy

24. C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

25. C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

26. C LASSICAL P ROTOCOL Sunny Snowing Windy Raining 2 bits

27. C LASSICAL P ROTOCOL Sunny Snowing Windy Raining 2 bits Classical computer unit: Bit = one of {0, 1}

28. C LASSICAL P ROTOCOL Message Sunny Snowing Windy Raining EncodingDecoding Distinguishable by color Alice Bob Sending 00 01 10 11

29. C LASSICAL P ROTOCOL Message Sunny Snowing Windy Raining EncodingDecoding Distinguishable by color Alice Bob 2 bits 4 dimension

30. What abt Quantum?

31. Q UANTUM P ROTOCOL Message Sunny Snowing Windy Raining Alice Bob B A Singlet state

32. Message Sunny Snowing Windy Raining Alice Bob B A I U Alice performs unitary on her particle

33. Message Sunny Snowing Windy Raining Alice Bob B A I U Creates 4 orthogonal states Singlet, Triplets Alice performs unitary on her particle

34. Message Sunny Snowing Windy Raining Alice Bob B A I U Alice sends her particle to Bob

35. Message Sunny Snowing Windy Raining Alice Bob I A B Bob has 2 particles: one of the triplets or singlet

36. Message Sunny Snowing Windy Raining Alice Bob I A B Decoding 4 orthogonal states Possible to distinguish 4 orthogonal states Possible to distinguish

37. Message Sunny Snowing Windy Raining Alice Bob I A B Decoding 4 orthogonal states Possible to distinguish 4 orthogonal states Possible to distinguish Decodes message

38. Message Sunny Snowing Windy Raining Alice Bob I A B Decoding 4 orthogonal states Possible to distinguish 4 orthogonal states Possible to distinguish 2 bits 2 dimension

39. M ORAL Classical Quantum Vs. Task: sending 2 bits Encoding: 4 Dimensions Encoding: 2 Dimensions

40. M ORAL Classical Quantum Vs. Task: sending 2 bits Encoding: 4 Dimensions Encoding: 2 Dimensions Bennett & Weisner, PRL 69, 2881 (’92).

41. D ENSE C ODING FOR ARBITRARY STATE Hiroshima, J. Phys. A ’01; Ziman & Buzek, PRA ’03, Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 Hiroshima, J. Phys. A ’01; Ziman & Buzek, PRA ’03, Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04

42. B A Alice & Bob share a state

43. B A Alice’s aim: to send classical info i Alice’s aim: to send classical info i Encoding

44. B A Alice’s aim: to send classical info i which occurs with probability p i Alice’s aim: to send classical info i which occurs with probability p i Encoding

45. UiUi UiUi B A Alice performs p i, U i Encoding

46. UiUi UiUi B A Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice performs p i, U i she produces the ensemble E = {p i,  i } Encoding

47. UiUi UiUi B A Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice performs p i, U i she produces the ensemble E = {p i,  i } Encoding

48. UiUi UiUi B A Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice sends her particle to Bob Sending

49. AB Alice Bob Decoding

50. AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding

51. AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Bob measures and obtains outcome j with prob q j

52. AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Post measurement ensemble: E |j= {p i|j,  i|j }

53. AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Post measurement ensemble: E |j= {p i|j,  i|j } Mutual information: i

54. AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Mutual information: i I acc = max I(i:M)

55. AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i = Maximal classical information from E= {p i,  i }. = Maximal classical information from E= {p i,  i }. Decoding I acc = max I (i:M)

56. H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i }

57. H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i }

58. H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i } d: dimension of  i

59. H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i } Bit per qubit

60. AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Accessible information = Maximal classical information from E = {p i,  i }. Accessible information = Maximal classical information from E = {p i,  i }. Decoding

61. DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc

62. DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max Holevo quantity obtained by Bob

63. DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max Holevo quantity obtained by Bob Holevo can be achieved asymptotically Schumacher, Westmoreland, PRA 56, 131 (’97)

64. DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max

65. DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max

66. C = Max DC C APACITY

67. C = Max DC C APACITY

68. C = Max DC C APACITY

69. C = log 2 d A + S(ρ B ) - S(ρ AB )

70. DC C APACITY C = log 2 d A + S(ρ B ) - S(ρ AB ) I B = S(ρ B ) - S(ρ AB ) > 0 A state is dense codeable

71. C LASSIFICATION OF STATES Entangled S DC In 2  2, 2  3

72. DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Solved

73. D ENSE C ODING N ETWORK

74. W HY QUANTUM DENSE CODING NETWORK ? Point to point communication has limited commercial use

75. W HY QUANTUM DENSE CODING NETWORK ? To build a quantum computer, or communication network To build a quantum computer, or communication network

76. W HY QUANTUM DENSE CODING NETWORK ? To build a quantum computer, or communication network, classical info transmission To build a quantum computer, or communication network, classical info transmission

77. W HY QUANTUM DENSE CODING NETWORK ? To build a quantum computer, or communication network, classical info transmission via quantum state in network To build a quantum computer, or communication network, classical info transmission via quantum state in network

78. Dense Coding Network 1

79. D ENSE C ODING N ETWORK Bob Debu Charu Nitu........ Alice Receivers Sender

80. D ENSE C ODING N ETWORK Bob Debu Charu Nitu........ Alice Receivers Sender Task: Alice individually sends classical info to all the receivers Task: Alice individually sends classical info to all the receivers

81. D ENSE C ODING N ETWORK Bob Debu Charu Nitu........ Alice Receivers R. Prabhu, A. K. Pati, ASD, U. Sen, PRA ’ 2013 R. Prabhu, ASD, U. Sen, PRA’ 2013 R. Nepal, R. Prabhu, ASD, U. Sen, PRA’ 2013 Sender

82. D ENSE C ODING N ETWORK Bob Debu Charu Nitu........ Alice Receivers R. Prabhu, A. K. Pati, ASD, U. Sen, PRA ’ 2013 R. Prabhu, ASD, U. Sen, PRA’ 2013 R. Nepal, R. Prabhu, ASD, U. Sen, PRA’ 2013 Sender Ujjwal’s Talk Prabhu’s Talk Ujjwal’s Talk Prabhu’s Talk

83. Dense Coding Network 2

84. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver

85. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver Several senders & a single receiver

86. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver Several senders & a single receiver

87. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver

88. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver senders perform U i k, k=1,2,..N on her parts

89. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver Senders create ensemble

90. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver Senders create ensemble

91. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver Senders send ensemble to Bob

92. D ENSE C ODING N ETWORK Alice Debu Charu Nitu........ Bob Senders Receiver Bob’s task: gather info abt

93. DC C APACITY NETWORK DC capacity network maximization over all encodings i.e. over all { p { i}, U { i } } C = Max I acc = Max Holevo quantity obtained by Bob

94. DC C APACITY N ETWORK C =C =C =C = Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 Bruss, Lewenstein, ASD, Sen, D’Ariano, Macchiavello, Int. J. Quant. Info. ’05

95. DC C APACITY N ETWORK C =C =C =C = Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 Bruss, Lewenstein, ASD, Sen, D’Ariano, Macchiavello, Int. J. Quant. Info. ’05 Tamoghna’s Poster

96. DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved

97. Dense Coding Network 3

98. D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 )

99. D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 ) LOCC i1i1 i2i2

100. D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 )

101. D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 ) Alices send her particles to Bobs

102. D ISTRIBUTED DC: T WO RECEIVERS Bob (B 1 ) Bob (B 2 ) Bobs task: gather info abt i k by LOCC

103. D ISTRIBUTED DC: T WO RECEIVERS Bob (B 1 ) Bob (B 2 ) Bobs task: gather info abt i k by LOCC LOCC

104. C = Max D ISTRIBUTED DC: T WO RECEIVERS

105. C = Max Max LOCC Holevo bound Maximization over all encodings i.e. over all {p i, U i } D ISTRIBUTED DC: T WO RECEIVERS

106. C = Max Max LOCC Holevo bound Maximization over all encodings i.e. over all {p i, U i } Badziag, Horodecki, ASD, Sen, PRL’03 D ISTRIBUTED DC: T WO RECEIVERS

107. C = Max Max LOCC Holevo bound Maximization over all encodings i.e. over all {p i, U i } Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 D ISTRIBUTED DC: T WO RECEIVERS

108. DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved

109. DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved Many Senders – Two Receivers

110. DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved Many Senders – Two Receivers Partially Solved

111. DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved Many Senders – Two Receivers Partially Solved Many Senders – Many Receivers Not Solved 