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Introduction HBB protocols Graph State Protocols Exploring Quantum Secret Sharing with the ZX Calculus Vladimir Nikolaev Zamdzhiev Oriel College, University of Oxford 29 October 2013 Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret

  1. Introduction Quantum Secret Sharing HBB protocols ZX Calculus Graph State Protocols Conditional Unitary Operations Q n CP(FHilb) Z s X s Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  2. Introduction Quantum Secret Sharing HBB protocols ZX Calculus Graph State Protocols Measuremens Q n CP(FHilb) Z-measure X-measure π π 2 2 Y-measure Bell basis measurement Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  3. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols HBB protcols overview In 1999, Hillery, Buzek and Berthiaume proposed the first quantum secret sharing protocols [4] Based on the GHZ state : ( | 000 � + | 111 � ) Two protocols – CQ(2,2) and QQ(2,2) Straightforward generalisation to CQ(n,n) based on generalised GHZ state ( | 0 n � + | 1 n � ) [7] Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  4. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols State and Secret Distribution In the distribution phase, the dealer prepares the (n+1)GHZ state and sends each player one qubit. The dealer also keeps one qubit for himself. Graphically, the quantum state and classical secret are given by : The classical secret is encrypted and shared in later steps, after a shared key has been established. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  5. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Protocol Description 1 The dealer and all players randomly choose a measurement direction - either X or Y . We can depict this by assigning a boolean variable x i to each player and the dealer. x i = 1 iff player i − 1 has chosen Y for his measurement dirrection ( x 1 is the direction of the dealer) 2 Each player and the dealer publicly announce their measurement directions 3 The players and the dealer restart the protocol if � x i is odd, i.e. there is an odd number of Y measurements. Otherwise, the protocol proceeds to the next step 4 The dealer measures his qubit in the selected direction 5 The dealer encrypts the classical bit S with the measurement outcome. This is achived by adding modulo 2 the two bits. 6 The dealer sends the encrypted message to all players (player 1 will decrypt it, so we depict only this scenario) Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  6. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Protocol Description 7 Every player measures his qubit in the selected direction 8 All players send their measurement outcomes to player 1. 9 Player 1 sums all measurement outcomes (including his) modulo 2. 10 Depending on the announced measurement directions, player 1 performs a negation on the result of the previous step. He performs a negation iff � x i is divisible by 2, but not by 4. 11 Now player 1 has obtained the shared key and he uses it to decrypt the bit he received from the dealer. This is done by adding modulo 2 the two bits. 12 Player 1 has the secret bit S Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  7. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Protocol Description in ZX Calculus Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  8. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Secret Reconstruction Now we have to show that the secret can be reconstructed by the players Proof is on the next few slides Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  9. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols � x i π 2 S 1 = x n + 1 π x n + 1 π x 1 π x 1 π x 2 π x 2 π 2 2 2 2 2 2 Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  10. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols � x i π 2 � x i π � x i π 2 2 Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  11. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols � x i π 2 � x i π 2 S 1 GB = = = � x i π � x i π 2 2 � x i π � x i π 2 2 Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  12. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols � x i π 2 � x i π � x i π B 1 N , S 1 2 = = = � x i π 2 � x i π 2 S 2 = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  13. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Secret inaccessibility Nothing to depict, because if one player is not collaborating, then he won’t announce a measurement direction and protocol stops. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  14. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols State and Secret Distribution Quantum secret and a GHZ state Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  15. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Protocol Description WLOG, we assume that the second player will receive the quantum secret. 1 The dealer measures his qubits in the Bell basis and sends two classical bits ( d 1 , d 2 ) to player 2 to inform him of the measurement outcome 2 Player 1 measures his qubit in the X basis and sends a classical bit ( p 1 ) to player 2 to inform him of the outcome 3 Player 2 performs the unitary correction Z p 1 ⊕ d 1 ⊗ X d 2 4 Player 2’s qubit is now in the state | S � Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  16. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Protocol Description in the ZX Calculus Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  17. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Secret Reconstruction Next, we have to show that the players can reconstruct the quantum secret, that is, the diagram is equal to the identity map. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  18. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols S 1 = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  19. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  20. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols T , N H = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  21. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  22. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols S 2 S 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  23. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  24. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols S 1 , S 2 , N H = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  25. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  26. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols S 2 , N H = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  27. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  28. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Secret inaccessibility We will show that player 2 is unable to reconstruct the state | S � , when player 1 performs an X measurement on his qubit and does not inform player 2 of the outcome. This means, that one player cannot independently obtain the secret and thus this is an example of a (2,2) QQ sharing scheme. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  29. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ S 1 = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  30. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  31. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ S 2 S 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  32. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  33. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ S 1 B 2 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  34. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  35. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ S 1 H = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  36. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Ψ Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  37. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Secret inaccessibility The last diagram reveals that, for any Ψ , the diagram cannot compute the identity function. This can be seen by plugging in states | + � and |−� , which result in the same behaviour. Therefore, the secret is denied to the player. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  38. Introduction CQ(n,n) HBB protocols QQ(2,2) Graph State Protocols Secret inaccessibility (not perfect) The sharing scheme is not perfect however. We can take Ψ to be the diagram representing the actions of player 2 where he measures in the computational basis and then compares his result with one of the bits which the dealer has send him. Then, he can prepare the correct quantum state to perfectly distinguish between | 0 � or | 1 � . Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  39. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  40. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Graph State protcols overview Introduced in 2008 by Markham and Sanders [6] Important class of QSS protocols Based on (extended) graph states Authors develop their own graphical formalism to reason about accessibility of information ZX Calculus is complete for stabilizer quantum mechanics and therefore graph states Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  41. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Graph States Definition Given an undirected graph G = ( V , E ) , with | V | = n , the graph state induced by G is the n -qubit state � ∧ Z e | + n � | G � := e ∈ E Therefore, any graph G = ( V , E ) gives rise to a graph state by : 1 Preparing the state | + n � , where n is the number of vertices 2 Applying a ∧ Z gate on qubits ( i , j ) iff ( v i , v j ) ∈ E Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  42. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Graph States in the ZX Calculus Graph states are represented in the ZX calculus by 1 Drawing two green dots for each vertex and connecting each green dot to an output box 2 For every edge ( v i , v j ) ∈ E connecting one of the green dots representing vertex v i to one of the green dots representing vertex v j by a wire and putting a Hadamard gate on the wire. Then do the same for the remaining pair of dots. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  43. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Example : C 4 graph state in ZX Example The graph state induced by the cycle graph C 4 is represented in the ZX calculus as : Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  44. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts CC(3,4) protocol Based on the cycle graph C 4 1 2 4 3 Figure: CC (3,4) graph Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  45. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts State and secret distribution The dealer prepares the graph state induced by the graph. The secret is distributed, by performing controlled unitary Z operations on each qubit of the graph state. Dealer doesn’t do anything else for the remainder of the protcol. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  46. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Protocol Description WLOG, we assume that the players who want to obtain the secret are players 1,2 and 3. 1 Players 1 and 3 measure in the computational basis 2 Player 2 measures his qubit in the X basis 3 Players 1 and 3 send their results to player 2 4 Player 2 sums all measurement results modulo 2 (including his own) 5 Player 2 now has the secret S Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  47. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Protocol Description in the ZX Calculus Figure: CC (3,4) protcol Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  48. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Secret Reconstruction We need to show that the previous diagram contains the identity function on the bit input Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  49. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts S 1 = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  50. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  51. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts C , S 1 H = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  52. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  53. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts C T = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  54. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  55. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts GB S 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  56. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts H S 2 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  57. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Secret inaccessibility The secret is denied to any set of two players. We use similar arguments as in the previous protocols - two players will measure their qubits without sharing the results with the others. We will see that no information is recovered by the players, which completes the proof of correctness. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  58. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts We consider only one case - the two collaborating players are neighbours. The other case is similar. Ψ Ψ S 1 = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  59. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Ψ Ψ H C , S 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  60. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Ψ Ψ H S 2 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  61. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Ψ Ψ S 1 , I H , S 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  62. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Ψ H = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  63. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts QQ(n,n) protocol Based on the following graph n 2 3 1 0 Figure: QQ (n,n) graph Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  64. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts State and Secret Distribution The dealer prepares the mentioned graph state. Then, the dealer does a Bell basis measurement on the input qubit | S � and the qubit 0. Depending on the measurement outcome, the dealer then applies unitary corrections to the remaining qubits. Sends each player one particle and doesn’t participate further. Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  65. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Protocol Description 1 Players 2, 3, ..., n measure in the computational basis 2 Players 2, 3, ..., n send their measurement results x i to player 1 � x i ◦ H 3 Player 1 performs the unitary correction Z 4 Player 1 now has the quantum secret | S � Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  66. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts S 1 , S 2 = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  67. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  68. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts H C = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  69. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  70. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts C , S 1 H , S 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  71. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  72. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts N , C U = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  73. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  74. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts S 1 S 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  75. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  76. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts GB B 1 = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  77. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

  78. Introduction CC(3,4) HBB protocols QQ(n,n) Graph State Protocols Other thoughts S 1 , N S 2 , I = = Vladimir Nikolaev Zamdzhiev Exploring Quantum Secret Sharing with the ZX Calculus

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