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Quantum Information, Communication and Computing Jan Kříž Department of physics, University of Hradec Králové Doppler Institute for mathematical physics and applied mathematics Quantum Information, Communication and Computing Information Theory: does not care about the physical realization of signals Quantum: description of the carriers of information Resources: Taksu Cheon Kochi University of Technology, Japan Private communication in 2004 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 When will we have a quantum computer? pessimists: NEVER! optimists: within next 30 years 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.” Preliminaries Hilbert Space: we associate a Hilbert space to each quantum system is a vector space over has a sesquilinear scalar product , z z z for z, the positivity 0 for 0. satisfying condition 2 m 0 isn complete, i.e. , such that n 0. Outline QI contains more sexy topics than boring mathematical description… 1. Story on the quantum witch 2. Entangled states 3. Quantum teleportation 4. Quantum cryptography 5. Quantum computing 6. Quantum game theory Prerequisity Quantum mechanics, version 0.5 Starring Alice Bob On the quantum witch Two ways of bark analysis: to dissolve to burn On the quantum witch On the quantum witch On the quantum witch 100% 0% 70% 30% 0% 100% 30% 70% On the quantum witch 70% 17% 30% 83% 30% 83% 70% 17% On the quantum witch 100% 0% 70% 30% 0% 100% 30% 70% On the quantum witch 1.There is a “symmetry” in reddish and greenish property ! On the quantum witch 100% 0% 70% 30% 0% 70% 100% 30% 30% 17% 70% 83% On the quantum witch 0% 100% 30% 70% 30% 83% 70% 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! On the quantum witch On the quantum witch On the quantum witch 70% 30% 0% 0% On the quantum witch 0% 0% 30% 70% On the quantum witch 11% 59% 5% 25% On the quantum witch 25% 5% 59% 11% On the quantum witch same colour 70% different colours 30% same colour 36% different colours 64% On the quantum witch Alice can send a signals to Bob by encoding her message in her choice of the way of analysis. 67% same colour 67% different colours Bob’s guesses are better than chance! We have proper transmission of information (although in a “noisy channel”) On the quantum witch 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). Transmission of information in infinite velocity! CONTRADICTION with Einstein causality On the quantum witch Transmission of information in infinite velocity! 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 Entangled states Experiment in quantum mechanics: 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 p(1) cos Preparing device q 2 2 , p(1) sin 2 Measuring 2 device 1 1 -1 1 -1 1,-1 Entangled states q p cos 2 p sin 2 2 2 (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 cos 2 sin 2 , Entangled states Assume now the system of two particles, we have four possible combinations of basis states: , 1 2 , 1 2 , 1 2 1 2 Any superposition of these states is again a quantum state, which can be prepared in suitable preparing device, e.g. 1 1 S , 1 2 2 1 2 2 1 1 W 1 2 1 2 2 2 Entangled states Spins in entangled state can be send to different places on the Earth, they still remain entangled… ? ? What does the measurement bring? 1 1 Measuring W 1 2 1 device: ↑or↓ 2 2 2 Entangled states Thus, we can “translate” the story on the quantum witch to QM… Quantum witch = a person (traditionally called Eve) who possesses a preparing device for the entangled state |W Measuring projections Two piecesdevice: of “Magic bark” = to = a couple of spins in entangled state Measuring device: projections to Entangled states x 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) Entangled states Since this "instanteneous Albert 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 Einstein – standard quantum state must Podolsky – Rosen be wrong. Paradox (EPR paradox) Modern experiments go against Albert! Quantum teleportation Alice wants to teleport a “spin” to Bob. Teleporting one qubit requires one Two-level (spin, entangled pairsystem of qubits and two photon …) = qubit bits ofpolariazation, classical information. q ? ? A E B 1 Measuring S 2 device E 1 B 2 E ~ X B ~ X B X AE X AE Y AE Y AE q Preparing device B ~ Y B 1 ~ Y B 2 B 3 Quantum cryptography Alice wants to send a secret message to Bob… 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. Quantum cryptography Top secret Preparing Preparing device ↑ device → Measuring device ↑ Measuring device → Measuring device ↑ Measuring device → Preparing device ↑ Preparing device → Quantum cryptography Top secret 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) If not… secret sequence of numbers. They can use it to code messages classically. BB84 protocol according to inventors Bennet, Brassard. Quantum computing How does the quantum computer look like? Why? We have perfectly good classical computers. Quantum computing Why? We have perfectly good classical computers. P. Shor converted a classical hard task into a tracktable one…