Quantum Information A Glimpse at the Strange and Intriguing Future of Information

1. Quantum Information A Glimpse at the Strange and Intriguing Future of Information Dan C. Marinescu School of Computer Science University of Central Florida Orlando, Florida 32816, USA dcm@cs.ucf.edu

2. Boole Lecture - February 15, 2006

3. Boole Lecture - February 15, 2006

4. Acknowledgments The material presented is based on the books Approaching Quantum Computing ISBN 013145224X, Prentice Hall, March 2004 Approaching Quantum Information Theory (in preparation) by Dan C. Marinescu and Gabriela M. Marinescu Work supported by National Science Foundation grants MCB9527131, DBI0296107,ACI0296035, and EIA0296179. Boole Lecture - February 15, 2006

5. Information • 2,450,000,000 Google hits for the word “information”. • The earliest historical meaning of the word information in English was related to the act of informing, or giving form or shape to the mind, as in education, instruction, or training. A quote from 1387: "Five books come down from heaven for information of mankind." (Oxford English Dictionary)…..Amazon.com was established later…. • 2667?( Japanese imperial year based on the mythical founding of Japan by Emperor Jimmu in 660 BC) Boole Lecture - February 15, 2006

6. Information (cont’d) • Information is a primitive concept (like matter or energy). • Information abstracts properties of and allows us to distinguish objects/entities/phenomena. • There is a common expression of information, strings of bits, regardless of the object/entity/process it describes. Bits are independent of their physical embodiment. • Information is transformed using logic operations. Gates implement logic operations and allow for automatic processing of information. The usefulness of information increases if the physical embodiments of bits and gates become smaller and we need less energy to process, store, and transmit information. Boole Lecture - February 15, 2006

7. Classical Information • Can be copied without altering it. • Deterministic; the result of measuring/observing it is deterministic (unless affected by noise). • Cannot travel faster than light or backward in time. • It is processed by conventional computers using irreversible gates. During processing we experience an irretrievable loss of information. • Information Theory was developed by Shannon for macroscopic bodies at a time when microscopic systems carrying information were not known. Boole Lecture - February 15, 2006

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10. Moore’s Law Boole Lecture - February 15, 2006

11. Limits of Solid-State Technology • To increase the clock rate we have to pack transistors as densely as possible because the speed of light is finite. • The power dissipation increases with the cube of the clock rate. When we double the speed of a device its power dissipation increases 8 (eight) fold. • The computer technology vintage year 2000 requires some 3 x 10-18 Joules/elementary operation. Boole Lecture - February 15, 2006

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13. Hitting a Wall… • An exponential growth cannot be sustained indefinitely; sooner or later one will hit a wall. • Revolutionary rather than evolutionary approach to information processing and to communication: • Quantum computing and communication  quantum information. • DNA Computing  biological information. Boole Lecture - February 15, 2006

14. Quantum Information Processing A happy marriage between two of the greatest scientific achievements of the 20th century: quantum mechanics stored program computers. In 1985 Richard Feynman wrote: “..it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms and quantum behavior holds sway.” Quantum information Information encoded as the state of atomic or sub-atomic particles. Boole Lecture - February 15, 2006

15. Richard Feynman I think I can fairly say that nobody understandsQuantumMechanics Boole Lecture - February 15, 2006

16. Light • Light  electromagnetic radiation. • The electric and magnetic field • oscillate in a plane perpendicular to the direction of propagation and • are perpendicular to each other. • The dual, wave and corpuscular, nature of light: • Diffraction phenomena  wave-like behavior • Photoelectric effect corpuscular/granular The light consists of quantum “particles” called photons. Boole Lecture - February 15, 2006

17. Polarization of Light Polarization is given by the electric field vector • Linearly polarized (vertical/horizontal)  the tip of the electric field vector oscillates along any straight line in a plane perpendicular to the direction of propagation. • Circularly polarized (right- /left-hand)  the tip of the electric field vector moves along a circle in a plane perpendicular to the direction of propagation: • Elliptically polarized light  the tip of the electric field vector moves along an ellipse in a plane perpendicular to the direction of propagation. • In a beam of linearly polarized light each photon has a random orientation of the polarization vector. Boole Lecture - February 15, 2006

18. The Spin of Atoms and Sub-atomic Particles • Spin  the intrinsic angular momentum; it takes discrete values (the spin quantum number s.) • Two classes of quantum particles: • fermions - spin one-half (e.g., electrons). • s=+1/2 and • s=-1/2 • bosons - spin one particles (e.g., photons). • s=+1, • s=0, and • s=-1 Boole Lecture - February 15, 2006

19. The spin of the electron: + ½ spin up,- ½ spin down. Boole Lecture - February 15, 2006

20. Quantum Mechanics Quantum mechanics mathematical model of the physical world. Quantum concepts such as: • Uncertainty, • Superposition, • Entanglement, • No-cloning, do not have a correspondent in classical physics. Boole Lecture - February 15, 2006

21. Heisenberg's Uncertainty Principle The position and the momentum of a quantum particle cannot be determined with arbitrary precision. • h=6.6262 x 10-34 Joule x second  Planck’s constant • Non-determinism  basic tenet of quantum mechanics. Boole Lecture - February 15, 2006

22. Max Born’s Nobel Prize Lecture, Dec. 11, 1954 “... Quantum Mechanics shows that not only the determinism of classical physics must be abandoned, but also the naive concept of reality which looked upon atomic particles as if they were very small grains of sand. At every instant a grain of sand has a definite position and velocity. This is not the case with an electron. If the position is determined with increasing accuracy, the possibility of ascertaining its velocity becomes less and vice versa.” Boole Lecture - February 15, 2006

23. “Liebe Gott würfelt nicht” (Dear God does not play dice) - Albert Einstein Boole Lecture - February 15, 2006

24. Superposition Principle States of a quantum system: - Orthogonal. - Non-orthogonal. - Superposition – a weighted sum (some elements appear with a – sign because the phases are negative). Schrödinger’s cat. In a quantum system, in addition to reliably distinguishable states there are states that cannot be reliably distinguishable. Incomplete distinguishability is one of the tenets of quantum mechanics. Boole Lecture - February 15, 2006

25. From Lewis Carroll to…. Incomplete Distinguishability in Quantum Physics “I have a very long and sad tale’’ said the Mouse. “I see that your tail is long, but why do you say it is sad?’’ asked Alice. Boole Lecture - February 15, 2006

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27. Entanglement (Vërschränkung) • Discovered by Schrödinger. • An entangled pair is a single quantum system in a superposition of equally possible states. • The entangled state contains no information about the individual particles, only that they are in opposite states. • Einstein called entanglement “Spooky action at a distance.” Boole Lecture - February 15, 2006

28. EPR (Einstein, Podolski,Rosen) Effect • A source generates two entangled particles e.g., two photons entangled in polarization. • A measurement of one of them, say using a V-H basis, produces a random result (V) or (H) and at the same time forces the other particle to enter the same state. Boole Lecture - February 15, 2006

29. No Cloning Principle Monogamy of Entanglement Quantum states cannot be cloned. Cloning - would increase distinguishability of states, - it is a non-linear transformation. Boole Lecture - February 15, 2006

30. Charles Bennett and Peter Shor: “classical information can be copied freely, but can only be transmitted forward in time to a receiver in the sender's forward light cone. Entanglement, by contrast cannot be copied, but can connect any two points in space-time. Conventional data-processing operations destroy entanglement, but quantum operations can create it, preserve it and use it for various purposes, notably speeding up certain computations and assisting in the transmission of classical data or intact quantum states (teleportation) from a sender to a receiver.” Boole Lecture - February 15, 2006

31. Quantum Information • Embodied by the state of atomic or sub-atomic particles. • Superposition - we cannot reliably recognize differences between the states of a quantum system except under special conditions. • The state of a quantum system cannot be measured or copied without disturbing it. • Quantum state can be entangled. Two or more systems have a definite state though neither has an identifiable state of its own. • Qubits – elementary units of quantum information. Boole Lecture - February 15, 2006

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34. Classical versus Quantum Information Classical information is information written in stone… Quantum information is more like the information in a dream. Recalling a dream inevitably changes your memory of it. Eventually you remember only your own description, not the original dream. Charles Bennett at QIPP workshop, 2002 Boole Lecture - February 15, 2006

35. Entropy • Thermodynamic:  the number of microstates • Informational, Shannon’s: X is a random variable pXi -probability of outcome Xi • Quantum, von Neumann’s: is the density matrix Boole Lecture - February 15, 2006

36. A Bit Versus a Qubit Boole Lecture - February 15, 2006

37. Qubit Measurement Boole Lecture - February 15, 2006

38. Quantum Gates • One-qubit gates  X - transposes the components of a qubit; Z - flips the sign of a qubit; Hadamard - creates a superposition state. • Two-qubit gates  CNOT • Three-qubit gates  Toffoli • Quantum gates are reversible  in principle no power dissipation. Boole Lecture - February 15, 2006

39. Universal Quantum Gates • Any Boolean expression can be written as a sum (logical OR) of products (logical AND) of Boolean variables and/or negation of Boolean variables. Thus, any classical logic circuit can be implemented using only AND, OR, and NOT gates. • NAND and NOR are classical universal gates. • Similarly, we can simulate any complex n-qubit quantum circuit using a small set of one-qubit and CNOT gates. Boole Lecture - February 15, 2006

40. Decoherence Decoherence randomization of the internal state of a quantum computer due to interactions with the environment. Conceptually decoherence can be prevented using: - Quantum fault-tolerant circuits. - Quantum Error Correcting Codes. - Entanglement Purification and Distillation extract a subset of states of high entanglement and high purity from a large set of less entangled states. Boole Lecture - February 15, 2006

41. Di Vicenzo’s Criteria for Physical Implementation of a Quantum Computer • Scalable physical system with well characterized qubits. • Initialize the qubits state as |000…00>. • Long decoherence times. • Universal set of quantum gates (operations). • Qubit specific measurements Boole Lecture - February 15, 2006

42. Entering the Quantum Wonderland …. • We now have: • quantum gates and quantum circuits • quantum communication channels. • What should we be excited about? • Quantum parallelism • Quantum teleportation • Communication with entangled particles • Quantum key distribution Boole Lecture - February 15, 2006

43. Quantum Parallelism • In quantum systems the amount of parallelism increases exponentially with the size of the system, thus with the number of qubits. For example, a 21-qubit quantum computer is twice as powerful as as a 20-qubit one. • An exponential increase in the power of a quantum computer requires linear increase in the amount of matter and space needed to build the larger quantum computing engine. • A quantum computer will enable us to solve problems with a very large state space. Boole Lecture - February 15, 2006

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45. Boole Lecture - February 15, 2006

46. Quantum Teleportation • The process of transferring the state of a quantum particle to possibly distant one. • Based upon the entanglement. • No cloning - the original state is destroyed in the quantum teleportation process. Boole Lecture - February 15, 2006

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48. A Teleportation Experiment • Francesco De Martini, University of Rome, 1997. • Based upon an idea of Sandu Popescu. • A UV laser beam interacts with a non-linear medium, a crystal of dihidrogen phosphate to generate two photons for an incoming one – parametric downconversion. • The polarization entanglement of the two photons is converted into a path entanglement. Boole Lecture - February 15, 2006

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50. Communication with Entangled Particles • Even when separated, two entangled particles continue to interact with one another. • Basic idea. Consider three particles • Two particles (particle 1 and particle 2)  in an anti-correlated state (spin up and spin down). • We measure particle 1 and particle 3 and set them in an anti-correlated state. • Then particle 2 ends up in the same state particle 3 was initially in. Boole Lecture - February 15, 2006