Quantum Information, Communication and Computing

1. Quantum Information, Communication and Computing Jan Kříž Department of physics, University of Hradec Králové Doppler Institute for mathematical physics and applied mathematics

2. Quantum Information, Communication and Computing Information Theory: does not care about the physical realization of signals Quantum: description of the carriers of information

3. Taksu Cheon Kochi University of Technology, Japan Private communication in 2004 Resources: http://www.mech.kochi-tech.ac.jp/cheon/q-inf/q-inf00_e.html Reinhard F. Werner Technical University of Braunschweig, Germany Course „Conceptual and mathematical foundations of quantum information“ given at Bressanone (Italy) in 2007 http://www.imaph.tu-bs.de/qi/qi.html

4. optimists: within next 30 years When will we have a quantum computer? pessimists: NEVER! IBM (in 1998): Probably in the next millenium R.F.Werner: “Even if the Quantum Computer proper were never to be built, the effort of building one, or at least deciding the feasibility of this project, will turn up many new results, likely to have applications of their own.”

5. Hilbert Space: we associate a Hilbert space  to each quantum system Sorry!!! Preliminaries •  is a vector space over  •  has a sesquilinear scalar product , for z, satisfying the positivity condition •  is complete, i.e.

6. QI contains more sexy topics than boring mathematical description… Outline • Story on the quantum witch • Entangled states • Quantum teleportation • Quantum cryptography • Quantum computing • Quantum game theory

7. Quantum mechanics, version 0.5 Prerequisity Starring Alice Bob

8. On the quantum witch Two ways of bark analysis: to dissolve to burn

9. On the quantum witch

10. On the quantum witch Oughhh!

11. 0% 100% On the quantum witch 70% 30% 100% 0% 30% 70%

12. 30% 70% On the quantum witch 17% 83% 70% 30% 83% 17%

13. 0% 100% On the quantum witch 70% 30% 100% 0% 30% 70%

14. On the quantum witch ?????? 1.There is a “symmetry” in reddish and greenish property !

15. 0% 100% On the quantum witch 70% 30% 100% 0% 30% 70% 30% 70% 17% 83%

16. 100% 0% On the quantum witch 30% 70% 70% 30% 83% 17%

17. On the quantum witch ?????? 1.There is a “symmetry” in reddish and greenish property ! 2.There is no “symmetry” in ways of analysis, i.e. Bob’s result depends on the Alice’s choice of analysis!

18. On the quantum witch IMPOSSIBLE MACHINES, Corp.

19. On the quantum witch

20. 70% On the quantum witch 30% 0% 0%

21. 0% On the quantum witch 0% 30% 70%

22. 11% On the quantum witch 59% 5% 25%

23. 25% On the quantum witch 5% 59% 11%

24. same colour 70% On the quantum witch different colours 30% same colour 36% different colours 64%

25. Alice can send a signals to Bob by encoding her message in her choice of the way of analysis. On the quantum witch same colour 67% different colours 67% Bob’s guesses are better than chance! We have proper transmission of information (although in a “noisy channel”)

26. However, Alice (in Amsterdam) and Bob (in Boston) can carry out their experiments at the same time (or even Bob can do his measurements sooner than Alice). On the quantum witch Transmission of information in infinite velocity! CONTRADICTION with Einstein causality

27. Transmission of information in infinite velocity! On the quantum witch CONTRADICTION with Einstein causality This may happen in the story, where the crucial roles are played by … By the way, nobody can be forced to accept Einstien causality as a fundamental principle

28. Experiment in quantum mechanics: Entangled states Preparing device Measuring device (produces particles) (perfectly classical output, changes the state of particle) Object of QM: predict the probabilities of the outcomes Example: spin projection Preparing device Measuring device 1 1 -1 1 -1 q 1,-1

29. Entangled states q (Arbitrary) state q can be thus interpreted as some mixture of states ↑ and ↓ Such mixture in QM - SUPERPOSITION On the other hand: any (normalised) superposition of quantum states is again a legitimate quantum state

30. Assume now the system of two particles, we have four possible combinations of basis states: Entangled states Any superposition of these states is again a quantum state, which can be prepared in suitable preparing device, e.g.

31. Spins in entangled state can be send to different places on the Earth, they still remain entangled… Entangled states ? ? What does the measurement bring? Measuring device: ↑or↓

32. Thus, we can “translate” the story on the quantum witch to QM… Entangled states Quantum witch = a person (traditionally called Eve) who possesses a preparing device for the entangled state |W Measuring device: projections to Two pieces of “Magic bark” = = a couple of spins in entangled state Measuring device: projections to

33. Entangled states x

34. Entangled states …really impossible machine However, the impossibility to construct it is not a consequence of Einstein causality breakdown. It follows from QM itself! (known as No Cloning Theorem)

35. Since this "instanteneous comunication" between faraway Alice and Bob is a direct result of the fundamental principle of quantum mechanics, and also this is against the local causality, it could only be that either quantum physics or the interpretation of the standard quantum state must be wrong. Albert Entangled states Einstein – Podolsky – Rosen Paradox (EPR paradox) Modern experiments go against Albert!

36. Alice wants to teleport a “spin” to Bob. Teleporting one qubit requires one entangled pair of qubits and two bits of classical information. Quantum teleportation Two-level system (spin, photon polariazation, …) = qubit q ? ? A E B Measuring device 1 2 3 q Preparing device B

37. Alice wants to send a secret message to Bob… Quantum cryptography Eve is now a rival of Alice… Observes the signals of Alice and tries to send the identical signals to Bob. Has all quantum devices as Alice and Bob.

38. Quantum cryptography Top secret Measuring device↑ Measuring device→ Preparing device↑ Preparing device→ Measuring device↑ Measuring device→ Preparing device↑ Preparing device→

39. Top secret Quantum cryptography ↑ → ↑ → ↑→→ ↑ ↑→ 1 0 1 1 0 1 0 0 0 1 ↑ ↑ → ↑ →→ → ↑ → ↑ 1 1 1 1 0 1 0 0 0 0 If these bits match 100%, OK. In such a way Alice and Bob can obtain shared (random) secret sequence of numbers. They can use it to code messages classically. If not… BB84 protocol according to inventors Bennet, Brassard.

40. How does the quantum computer look like? Quantum computing Why? We have perfectly good classical computers.

41. Why? We have perfectly good classical computers. Quantum computing P. Shor converted a classical hard task into a tracktable one…