C CS Phys C191 Fall 2009 Introduction Quantum States Superposition 8 27 09 Lecture 1 1 Readings Benenti Casati Strini Introduction Kaye Laflamme Mosca Chs 1 6 1 7 2 1 2 2 2 Course Philosophy Outline Over the last decade there has been great foundational progress at the interface of quantum physics and computer science The remarkable power of computing devices based on quantum mechanics is the subject of the emerging area of quantum computation which incidentally provides an alternative to the exponential Moore s law dash towards fundamental limits in classical computation This course provides an introduction to this area the basic ideas of quantum mechanics the formal model of quantum computers basic quantum algorithms and more concrete proposals for experimental realization of quantum computers Qubits are the building blocks of quantum computation quantum information and cryptography They also provide a particularly simple setting in which to introduce the basic concepts of quantum mechanics such as the superposition principle tensor products measurements and the enigmatic Bell s inequalities and Heisenberg uncertainty principle The first part of this course provides a simple introduction to quantum mechanics for non physics majors while providing physics majors an opportunity to deepen their understanding of this important topic This course will focus on the enormous computational power latent in quantum mechanics and how it can be used to design quantum computers and quantum algorithms that provide algorithmic speedup relative to classical algorithms We will also discuss schemes for quantum error correction and for implementing unconditionally secure cryptography based on the principles of quantum mechanics Finally we will turn our attention to physical realization we will discuss in detail the spin properties of elementary particles the vehicle of choice for carrying a qubit The course will conclude with a survey of schemes for implementing quantum computers in the laboratory There are four main properties of quantum systems that are useful in quantum computation cryptography and Information Interference Superposition Entanglement Measurement In particular the detailed study of entanglement is the most important point of departure from more traditional approaches to the subject For example quantum computation derives its power from the fact that the C CS Phys C191 Fall 2009 Lecture 1 1 description of the state of an n particle quantum system grows exponentially in n This enormous information capacity is not easy to access since any measurement of the system only yields n pieces of classical information Thus the main challenge in the field of quantum algorithms is to manipulate the exponential amount of information in the quantum state of the system and then extract some crucial pieces via a final measurement Quantum cryptography relies on a fundamental property of quantum measurements that they inevitably disturb the state of the measured system Thus if Alice and Bob wish to communicate secretly they can detect the presence of an eavesdropper Eve by using cleverly chosen quantum states and testing them to check whether they were disturbed during transmission 3 Quantum duality Young s double slit experiment Recall Young s double slit experiment from high school physics which was used to demonstrate the wave nature of light The apparatus consists of a source of light a screen with two very thin identical slits and a screen to view the interference pattern created by transmitted light see picture on next page If only one slit is open then intensity of light on the viewing screen is maximum on the straight line path and falls off in either direction However if both slits are open then the intensity oscillates according to the interference pattern predicted by wave theory In the quantum version of this experiment the light source is replaced by a source of single photons Instead of the intensity of light falling on a point x on the viewing screen we can only speak about the probability that a detector at point x detects the photon If only a single slit is open then plotting this probability of detection as a function of x gives the same curve as the intensity as a function of x in the classical Young experiment What happens when both slits are open Could the probabability plot duplicate the interference pattern Classical intuition strongly suggests that this is impossible After all for the photon to be detected at x either it went through slit 1 and ended up at x or it went through slit 2 and ended up at x The probability that it is detected at x is just the sum of the probabilities of these two events However there are points x where the detection probability is large if only one slit is open although it is zero or small in the interference pattern If the photon actually goes through slit 1 why should it matter whether slit 2 is open or shut How could the probability that the photon goes through slit 1 and ends up at x be affected by whether or not slit 2 is open Nonetheless the probability of detection when both slits are open does duplicate the interference pattern How does quantum mechanics explain this Quantum mechanics explains this as follows although this might not be very satisfactory as an explanation it does provide a good formal was of thinking about the phenomenon Quantum mechanics introduces the notion of the complex amplitude 1 x C with which the photon goes through slit 1 and hits point x on the viewing screen The probability that the photon is actually detected at x is the square of the magnitude of this complex number P1 x 1 x 2 Similarly let 2 x be the amplitude if only slit 2 is open P2 x 2 x 2 Now when both slits are open the amplitude with which the photon hits point x on the screen is just the sum of the amplitudes over the two ways of getting there 12 x 12 1 x 12 2 x As before the probability that the photon is detected at x is the squared magnitude of this amplitude P12 x 1 x 2 x 2 The two complex numbers 1 x and 2 x can cancel each other out to produce destructive interference or reinforce each other to produce constructive interference or anything in between But in this quantum mechanical explanation how does a particle that went through the first slit know that the other slit is open In quantum mechanics this question is not well posed Particles do not have trajectories C CS Phys C191 Fall 2009 Lecture 1 2 but rather take all paths simultaneously This