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Berkeley COMPSCI 294 - Lecture 1: Axioms of QM + Bell Inequalities

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Young's double-slit experimentBasic Quantum MechanicsThe superposition principleThe Geometry of Hilbert SpaceMeasurement PrincipleQubitsExamples of QubitsMeasurement example I: phase estimationBra-ket notation.Measurement example II.Unitary OperatorsTwo qubits:Two Qubit GatesEPR Paradox:Bell's Thought ExperimentLecture 1: Axioms of QM + Bell Inequalities0.1 Young’s double-slit experimentIs light transmitted by particles or waves? The basic dilemna here (which dates as far back as Newton) is toreconcile the evidence that light is transmitted by particles (called photons), with experiments demonstratingthe wave nature of light. To be concrete, let us recall Young’s double-slit experiment from high schoolphysics, which was used to demonstrate the wave nature of light. The apparatus consists of a source of light,an intermediate screen with two very thin identical slits, and a viewing screen (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 pathand falls off in either direction. However, if both slits are open, then the intensity oscillates according to thefamiliar interference pattern predicted by wave theory. These facts can be qualitatively and quantitativelyexplained by positing that light travels in waves (as you did in high school physics).Let us now introduce the particle nature of light into this experiment. To do so, we turn down the intensityof the light source, until a photodetector clicks only occassionally to record the emission of a photon. As weturn down the intensity of the source, the magnitude of each click remains constant, but the time betweensuccessive clicks increases. This is consistent with light being emitted as discrete particles (photons) — theintensity of light is proportional to the rate at which photons are emitted by the source. So now with the lightsource emitting a single photon every so often, we can ask where this single emitted photon hits the viewingscreen. The answer is no longer deterministic, but probabilistic. We can only speak about the probabilitythat a photodetector placed at point x detects the photon. If only a single slit is open, then plotting thisprobability of detection as a function of x gives the same curve as the intensity as a function of x in theclassical Young experiment. What happens when both slits are open? Our intuition would strongly suggestthat the probability we detect the photon at x should simply be the sum of the probability of detecting it atx if only slit 1 were open and the probability if only slit 2 were open. In other words the outcome shouldno longer be consistent with the interference pattern. In the actual experiment, the probability of detectiondoes still follow the interference pattern. Reconciling this outcome with the particle nature of light appearsimpossible, and that is the dilemna we face.Let us spell out in more detail why this contradicts our intuition: for the photon to be detected at x, eitherit went through slit 1 and ended up at x or it went through slit 2 and ended up at x. Now the probability ofCS 294, Spring 2009, 1seeing the photon at x should be the sum of the probabilities in the two cases. To make the contradictionseem even more stark, notice that there are points x where the detection probability is zero (or small) if bothslits are open, even though it is non-zero (large) if either slit is open. How can the existence of more waysfor an event to happen actually decrease its probability?Let us now turn to quantum mechanics to see how it explains this phenomenon.Quantum mechanics introduces the notion of the complex amplitudeψ1(x) ∈ C with which the photon goesthrough slit 1 and hits point x on the viewing screen. The probability that the photon is actually detectedat x is the square of the magnitude of this complex number: P1(x) = |ψ1(x)|2. Similarly, letψ2(x) be theamplitude if only slit 2 is open. P2(x) = |ψ2(x)|2.Now when both slits are open, the amplitude with which the photon hits point x on the screen is just the sumof the amplitudes over the two ways of getting there:ψ12(x) =ψ1(x) +ψ2(x). As before the probabilitythat the photon is detected at x is the squared magnitude of this amplitude: P12(x) = |ψ1(x) +ψ2(x)|2. Thetwo complex numbersψ1(x) andψ2(x) can cancel each other out to produce destructive interference, orreinforce each other to produce constructive interference or anything in between.Some of you might find this ”explanation” quite dissatisfying. You might say it is not an explanation atall. Well, if you wish to understand how Nature behaves you have to reconcile yourselves to this type ofexplanation — this wierd way of thinking has been successful at describing (and understanding) a vast rangeof physical phenomena. But you might persist and (quite reasonably) ask “but how does a particle that wentthrough the first slit know that the other slit is open”? In quantum mechanics, this question is not well-posed.Particles do not have trajectories, but rather take all paths simultaneously (in superposition). As we shall see,this is one of the key features of quantum mechanics that gives rise to its paradoxical properties as well asprovides the basis for the power of quantum computation. To quote Feynman, 1985, ”The more you see howstrangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomenaactually work. So theoretical physics has given up on that.”0.2 Basic Quantum MechanicsThe basic formalism of quantum mechanics is very simple, though understanding and interpreting (andaccepting) the results is much more challenging. There are three basic principles, enshrined in the four basicpostulates of quantum mechanics:• The superpostion principle: this axiom tells us what are the allowable (possible) states of a givenquantum system.• The measurement principle: this axiom governs how much information about the state we can access.• Unitary evolution: this axoim governs how the state of the quantum system evolves in time.0.3 The superposition principleConsider a system with k distinguishable (classical) states. For example, the electron in a hydrogen atom isonly allowed to be in one of a discrete set of energy levels, starting with the ground state, the first excitedstate, the second excited state, and so on. If we assume a suitable upper bound on the total energy, then theelectron is restricted to being in one of k different


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Berkeley COMPSCI 294 - Lecture 1: Axioms of QM + Bell Inequalities

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