1. 1 Transactional Nature of Quantum Information Subhash Kak Computer Science, Oklahoma State Univ © Subhash Kak, June 2009

2. 2 Quantum information and processing Fourier transform in (log n) 2 rather than n log n Database search in n 1/2 rather than n [Grover] Factorization in log n rather than n 1/2 [Shor]

3. 3 Qubits versus bits |Ψ> = a |0> + b |1> where | a| 2 + |b| 2 = 1 |Ψ>|Ψ>

4. 4 Qubit dynamics A quantum computer (U) rotates a state |Ψ>|Ψ>

5. 5 Quantum register Each cell has a qubit. Number of states is 2 n 1 2 3 4 5..... n

6. 6 Information and Entropy Classical information I(x) = - log p(x) Information is additive Classical Entropy H(X) = - ∑ i p(x i ) log p(x i ) Von Neumann entropy = - ∑ i λ i log λ i

7. 7 Von Neumann entropy Entropy of the mixed state is equal to

8. 8 Mixed states Its von Neumann entropy equals 0.81 bits. This mixed state can be viewed to be generated from a variety of ensembles of states.

9. 9 An Entangled State

10. 10 The von Neumann entropy is zero. But the two objects individually have entropy of 1 bit. Von Neumann entropy is not additive. An entangled state is pure and its entropy is zero, but the state of its components is mixed and their entropy is non=zero!

11. 11 Multiple copies of a state Consider multiple copies of the qubit where tr(Xρ) etc are average values

12. 12 A more constrained example Preparer of states A trying to communicate the ratio a/b = m to B, where a and b are real and the state is either pure or a mixture or it is the mixture of the states: It is the pure state:

13. 13 Informational entropy A proposed measure for informational entropy: It depends only on the diagonal terms and therefore it reflects the mutual relationship between the sender and the recipient

14. 14 0 ≤ S inf (ρ) ≤ n S inf (ρ) ≥

15. 15 Example Von Neumann entropy is less than Inf. Entropy by 0.07 bits The eigenvalues of ρ are 0.242 and 0.758 and, therefore, the von Neumann entropy is: =-0.242 log2.242 – 0.758 log2.758 =.242  2.047 +.758 .400=0.798 bits S inf (ρ) = -0.71 log2.71 – 0.29 log2.29 = 0.868 bits

16. 16 Example (contd.)-- Partial Information case 1 We know that the ensemble consists of a pure and mixed component as follows: Entropy is 0.3 + 0.7  0.722= 0.805 bits

17. 17 Example (contd.)-- Partial Information case 2 We know that the ensemble consists of a pure and mixed component as follows: Entropy is 0.316  0.918+0.684  0.84 =0.290+0.575=0.865 bits

18. 18 Quantum cryptography protocol

19. 19 Concluding Remarks If information cannot be defined independent of the experimenter, quantum computing may be harder to implement Amongst other things, it implies greater attention to errors Is transactional nature of quantum information of relevance to other fields of physics?

20. 20 References S. Kak, Prospects for quantum computing. arXiv:0902.4884v1 arXiv:0902.4884v1 S. Kak, Quantum information and entropy, International Journal of Theoretical Physics 46, pp. 860-876, 2007. S. Kak, Information complexity of quantum gates, International Journal of Theoretical Physics, vol. 45, pp. 933-941, 2006. S. Kak, Three-stage quantum cryptography protocol, Foundations of Physics Letters, vol. 19, pp. 293-296, 2006. S. Kak, General qubit errors cannot be corrected, Information Sciences, vol. 152, pp. 195-202, 2003