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: Information Theory:does not care about the physical realization of signals Quantum: Quantum: description of the carriers of information

Resources: Taksu Cheon Kochi University of Technology, Japan Private communication in Reinhard F. Werner Technical University of Braunschweig, Germany Course „Conceptual and mathematical foundations of quantum information “ given at Bressanone (Italy) in

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

Outline 1.Story on the quantum witch 2.Entangled states 3.Quantum teleportation 4.Quantum cryptography 5.Quantum computing 6.Quantum game theory QI contains more sexy topics than boring mathematical description…

Prerequisity Quantum mechanics, version 0.5 Starring Alice Bob

On the quantum witch Two ways of bark analysis: to dissolveto burn

On the quantum witch

100% 0%0% 70%30% 0%0% 100% 30%70%

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

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

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

100% 0%0% 70%30% 0%0% 100% 30%70% 30%30% 17%83%

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

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 d dd depends on the Alice’s choice of analysis!

On the quantum witch

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

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

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

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

On the quantum witch Alice can send a signals to Bob by encoding her message in her choice of the way of analysis. same colour 67% different colours 67% 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). CONTRADICTION with Einstein causality Transmission of information in infinite velocity!

On the quantum witch CONTRADICTION with Einstein causality Transmission of information in infinite velocity! 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 (produces particles) Measuring device (perfectly classical output, changes the state of particle) Object of QM: Object of QM: predict the probabilities of the outcomes Example: spin projection Preparing device q Measuring device 1 1,-1 11

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

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

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

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  Two pieces of “Magic bark” = = a couple of spins in entangled state Measuring device: projections to

Entangled states x

…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 Albert 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. Einstein – Podolsky – Rosen Paradox (EPR paradox) Modern experiments go against Albert!

Quantum teleportation Alice wants to teleport a “spin” to Bob. Two-level system (spin, qubit photon polariazation, …) = qubit q A ? E ? B Preparing device Measuring device q B 123 Teleporting one qubit requires one entangled pair of qubits and two bits of classical information.

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 Preparing device ↑ Preparing device → Measuring device → Measuring device ↑ Top secret Preparing device ↑ Preparing device → Measuring device ↑ Measuring device →

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

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

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