Chapter 6

Measurements and two-levelquantum systems

In the previous chapter we saw how the quantum state, represented by thewave function of a quantum system, evolves in space and time as governedby the Schrodinger equation. We discussed the interpretation of the wavefunction as the probability amplitude, in which the modulus squared is equalto the probability density of finding the particle in a small region about apoint in space at a given time. Furthermore, we discussed the idea of expec-tation values, that is, if we measured say the position of the particle at time tprepared in some initial state, the specific measurement outcome is random,and we can only talk about the most probable outcome or average outcomethat we expect if we repeat the same measurement on multiple copies ofidentically prepared systems. This inherent uncertainty in the outcome of ameasurement is a crucial difference between quantum and classical physics.In the previous discussion we did not mention the implications of this ran-dom outcome nor what happens to a quantum system after a measurementis made though, and these will be topics of the current chapter. In additionwe will set out the mathematics of two level quantum systems, which arethe most basic (simplest) quantum systems upon which modern quantumthinking is based.

6.1 Observables, operators, and expectation val-ues

6.1.1 Classical measurement

As mentioned in the introductory chapter, the concept of measurement inquantum mechanics differs significantly from its classical counterpart. Clas-sically, measurements reveal intrinsic properties or observables of the system,for example the position, momentum, angular momentum, or energy of a

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particle are classical observables

Classical observables: x, p, L, and E.

Classical measurement postulates: Two key postulates for classicalmeasurements that fail for quantum measurements involve the implicit as-sumptions that

1. Independent reality: Measurements reveal “elements of physical re-ality” that exist independent of the observation. In other words, mea-surements reveal information about properties possessed by a particle,which existed prior to the measurement, and we are simply ignorantof the value of the observable before we make the measurement. Forexample, classically we would say that measurement of the positionof an electron bound to a hydrogen nucleus just tells us where theelectron was before the measurement.

2. No disturbance: Measurements can be performed that do not dis-turb the system. In classical physics, all that is needed to do this isto make our measurement interaction sufficiently weak so that thereis no disturbance of the system.

In classical physics, and most experiences we have in the macroscopic world,we do not see any difficulties with these postulates. We have already dis-cussed the concept of measurement disturbance in quantum mechanics interms of the uncertainty principle and Heisenberg microscope. The ideathat by trying to determine the position of a particle with ever increasingprecision (smaller uncertainty ∆x), at the cost of gaining momentum uncer-tainty of a high energy (and thus high momentum) photon seems plausibleto our classical ways of thinking. However, the idea that the position of themoon does not exist unless we look seems absurd to our classical intuition.As we will see, this is precisely what quantum mechanics prescribes.

6.1.2 Quantum measurement

In quantum mechanics measurements are represented by operators that acton the wave function Ψ(x, t). For example the observables of position, mo-mentum, angular momentum, and energy correspond to the following oper-ators

XΨ(x, t) = xΨ(x, t), (6.1)

pΨ(x, t) = −i~∇Ψ(x, t), (6.2)

LΨ(x, t) = −i~x×∇Ψ(x, t), (6.3)

HΨ(x, t) = i~∂tΨ(x, t). (6.4)

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The position operator acting on a wave function results in multiplication bythe coordinate x inside the argument of the wave function. The momentumoperator acting on a wave function takes a spatial gradient (multiplied by−i~). The orbital angular momentum operator follows from these and thecorresponding classical definition. The Hamiltonian operator, or energy op-erator, acting on a wave function takes a time derivative (multiplied by i~).

As mentioned in the previous chapter, the outcome of a particular mea-surement cannot in general be predicted. One can only talk about theprobability to obtain a particular measurement outcome, which is governedby the wave function. This is a fundamental feature of quantum mechanics.Let us take the example of the particle confined to an infinite potential well,as described in Sec. 5.6 of the previous chapter, and consider the situationin which the particle is prepared in the symmetric superposition state

Ψ(+)(x, t) =1√2

(φ1(x)e−iω1t + φ2(x)e−iω2t

), (6.5)

whose wave function and probability density are shown in Fig. 6.1 at timet = 0.

If we make a measurement of the position, any value in the range −L/2 <x < L/2 will be obtained for a single measurement, with the exception ofL/6, where the wave function is zero. The most probable position to findthe particle can be calculated as the expectation value

〈x(t)〉 =1

2

∫ ∞−∞

x[|φ1(x)|2 + |φ2(x)|2 + 2<

{φ∗2(x)φ1(x)ei∆ωt

}]dx

= −16L

9π2cos(∆ωt), (6.6)

where ~∆ω = ~ω2 − ~ω1 is the difference between the energies of the twoenergy eigenstates in the superposition.

Suppose we choose to measure the particle energy instead. The expec-

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-0.4 -0.2 0.2 0.4x

ΨHxL

-0.4-0.2 0.2 0.4x

ÈΨHxL 2

Figure 6.1: Probability amplitude and probability distribution for the sym-metric superposition state of the ground and first-excited states of the infi-nite potential well at time t = 0. The horizontal axes is in units of the boxwidth L. Note that the probability to find the particle at L/6 is zero.

tation value of the energy is

〈H〉 =

∫ ∞−∞

Ψ∗(x, t)HΨ(x, t)dx

=

∫ ∞−∞

Ψ(+)∗(x, t)i~∂tΨ(+)(x, t)dx

=1

2

∫ ∞−∞

(φ1(x)e−iω1t + φ2(x)e−iω2t

)i~∂t

(φ1(x)e−iω1t + φ2(x)e−iω2t

)dx

=1

2

∫ ∞−∞

(φ1(x)e−iω1t + φ2(x)e−iω2t

)∗ (~ω1φ1(x)e−iω1t + ~ω2φ2(x)e−iω2t)

dx

=1

2

∫ ∞−∞

[~ω1|φ1(x)|2 + ~ω2|φ2(x)|2 + (~ω1φ1(x)φ∗2(x)ei∆ωt + ~ω2φ

∗1(x)φ2(x)e−i∆ωt)

]dx

=1

2(~ω1 + ~ω2), (6.7)

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where in going to the last line we used the orthogonality of the energy eigen-states. Note that the expected value is simply the average energy of the twostates.

Now, what happens on a single measurement when we choose to measurethe energy? Since the allowed energy values of the particle confined to thebox only take on discrete, quantized values En, a measurement of the energycan only yield one of these allowed values and nothing in between. Note thatthe energy expectation value does not equal one of the energy eigenvalues,and thus cannot be observed in a single measurement. However, if we repeatthe measurement many times on identically prepared systems, we would findthat the average value is given by the expectation value in Eq. (6.7).

Starting with the symmetric superposition state, we have equal prob-ability of 1/2 to obtain energy value E1 or E2 on any given energy mea-surement. Suppose on a given measurement we find energy E1. Initially,just before the measurement the particle was in the symmetric superposi-tion state Ψ(+)(x, t) in Eq. (6.5). However, our knowledge of the energyE1 implies that the particle most certainly must occupy the correspondingenergy eigenstate φ1(x) just after the measurement. Because of the time-evolution behavior of energy eigenstates as governed by the time-dependentSchrodinger equation, i.e. they evolve in time by gathering an unobservableglobal phase e−iωnt, any subsequent measurement of the particle energy willgive the same value of energy! Thus, we can predict with certainty the out-come of any subsequent measurement of the energy, in this case we wouldfind E1. This should seem strange to you, in the sense that the measurementcauses the wave function of the particle to “collapse” into the eigenstate ofthe observable being measured. By observing the particle, we can causeirreversible evolution of the state. The collapse of the wave function is oneof the key non-classical features of quantum mechanics.

The measurement collapse hypothesis can be generalized to observablesother than energy as well. For example, measurement of the particle posi-tion causes the wave function to collapse into an eigenstate of the positionoperator. To determine the subsequent evolution of the quantum state af-ter such a measurement requires development of some further mathematics,for example one must define the position eigenstates (these turn out to bewhat are called Dirac delta functions) and how these evolve in time. Thisis beyond the scope of the current course, and you will spend a significantamount of time on this in the future. For now, we will summarize the pos-tulates associated with quantum measurements as taught in most schoolsof thought. This approach to measurement was put forth by Niels Bohr,Werner Heisenberg and Wolfgang Pauli, who were all working in Copen-hagen at the time it was developed and is thus known as the Copenhagen

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interpretation, and formalized by John von Neumann who emphasized themeasurement collapse hypothesis. We will then examine the consequencesof these postulates on the interpretation of measurement outcomes.

(a) Niels Bohr, Werner Heisenberg, and Wolfgang Pauli (b) John von Neumann

Quantum Measurement Postulates

1. Single-value measurement outcome: When a measurement cor-responding to an operator A is made, the result is one of the operatoreigenvalues, an.

2. Wave function collapse: As a result of a measurement yieldingeigenvalue an, the wave function ‘collapses’ into the correspondingeigenstate of the measurement operator

ψ(x, t)→ φn(x). (6.8)

3. Outcome probability: The probability of a particular measurementoutcome equals the squared modulus of the overlap between the wavefunctions before and after the measurement. For example, the proba-bility to obtain measurement outcome an, corresponding to the eigen-state φn(x) is given by

Pn(t) = |〈φn|ψ(t)〉|2 =

∣∣∣∣∫ ∞−∞

φ∗n(x)ψ(x, t)dx

∣∣∣∣2 . (6.9)

A useful corollary to Postulate 3, gives us a recipe to determine the expan-sion coefficients of an arbitrary state. For example, let us consider a stateexpanded in the energy eigenstates {φn(x)}, given by

ψ(x, t) =

∞∑n=1

cnφn(x)e−iωnt, (6.10)

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with expansion coefficients {cn}. Here I have include the time dependenceof the energy eigenstates given by the phase factors e−iωnt explicitly. (Recallthat ωn = En/~ is the angular frequency associated with the energy eigen-value En.) By taking the overlap of the state in Eq. (6.10) with one of theenergy eigenstates, say φm(x), we find

〈φm|ψ(t = 0)〉 =

∫ ∞−∞

φ∗m(x)ψ(x, t = 0)dx,

=∞∑n=1

cn

∫ ∞−∞

φ∗m(x)φn(x)dx,

=

∞∑n=1

cnδm,n,

= cm, (6.11)

where in going from the second to third lines, we used the orthonormalityof the energy eigenstates. So we see that the expansion coefficients {cn}associated with the energy eigenstates of an arbitrary state are given by theoverlap of the state at t = 0 with the corresponding eigenstates {φn(x)}.

6.2 Interpretation of quantum measurement

A major challenge associated with the measurement collapse hypothesis isthat it is a fundamentally stochastic process, that is the actual measure-ment outcome is completely random and unpredictable. We can only saywith what probability we expect to obtain a particular measurement out-come. Another difficulty lies with the instantaneous collapse of the wavefunction itself. Once the wave function is known, the Schrodinger equationis completely deterministic in describing its evolution. There is no ran-domness associated with the state evolution between measurements. Thus,there appears to be two different types of time evolution in quantum mechan-ics: “unitary evolution” under the action of the time-dependent Schrodingerequation and “collapse” associated with measurement. We now want to ex-plore some of the consequences of the measurement collapse hypothesis andhow this affects our interpretation of quantum physics.

6.2.1 Schrodinger’s cat

An often cited example of the implications and challenges to our classi-cal way of thinking associated with quantum measurement is Schrodinger’scat. This “gedanken experiment” (thought experiment), was proposed bySchrodinger in 1935 to highlight the apparent conflicts associated with thetheory of quantum superpositions and quantum measurement when applied

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to the macroscopic level of everyday experience.

Figure 6.2: Schematic of the Schrodinger cat thought experiment.

The basic idea for the experiment, which to my knowledge has thank-fully not been carried out, consists of placing a live cat inside a sealed steelchamber, which also has a vial of poison that can be smashed by a hammerconnected to a Geiger counter, as depicted in Fig. 6.2. There is a smallamount of radioactive material inside the chamber as well, which can decayand cause the Geiger counter to trigger the hammer. From the amount ofradioactive material used and its known half-life, we expect that within onehour there is a 50:50 chance that one atom has decayed. If this occurs, theGeiger counter will trigger the hammer, which smashed the vial containingthe poison, and subsequently killing the cat.

Prior to measuring the decay, the state of the radioactive atoms must bedescribed as a superposition of decayed and not decayed states

ψatoms =1√2

(φu + φd) , (6.12)

where u and d correspond to undecayed atoms and decayed atoms respec-tively. The ‘state’ of the cat being alive or dead is exactly correlated withthe state of the radioactive material, so that their joint state is written

Ψatoms+cat =1√2

(φu × χalive + φd × χdead) . (6.13)

Clearly, when we open the box and look inside we will find the cat eitherdead or alive, depending on whether or not one or more of the atoms havedecayed. However, prior to opening the box and looking inside, i.e. mak-ing a measurement, the state of the cat and atoms must be given by thesuperposition state in Eq. (6.2). Prior to measurement the state of thecat is therefore ‘blurred’ – it is neither alive nor dead, but in some peculiar

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combination of both states as depicted in the cartoon in Fig. ??. We canperform a measurement on the state of the cat by opening the box to seeif it has survived. The major dilemma is concerned with the time at whichthe measurement, that is the collapse of the state of the cat, occurs. Do wesuppose that the measurement occurs at precisely the time that we catchthe first glimpse of the cat (either lying there motionless or springing outto greet us), and record the observation dead or alive as appropriate? Ordoes the cat somehow observe itself? One could easily replace the cat witha friend (as Wigner suggested), in which he suggested “consciousness” playsan important role in observation.

The example of Schrodinger’s cat is meant to illuminate the apparent con-tradiction between our classical notions of reality and measurement andthose of quantum physics. Classically, we imagine that measurement simplyreveals properties of physical systems, i.e. a physical system has an innate‘real’ existence independent of and prior to measurement that is illuminatedby the act of measurement. Our classical way of thinking tells us that thecat must surely be either alive or dead before looking in the box. However,this is in contradiction to quantum theory and in some cases does not yieldthe correct predictions. Furthermore, the notion that there are two differenttypes of time evolution for the quantum state is unsatisfactory. One wouldlike to have a theory in which the state evolution and measurement processare given by the same description.

There have been various approaches to resolve these dilemmas, for ex-ample, some have argued that quantum mechanics is incomplete and doesnot fully describe the natural world. Such approaches are collectively knownas “hidden-variable” theories, in which there are certain yet-to-be observedproperties of quantum systems hidden from current experiments that deter-mine the properties of physical systems and measurement outcomes. Suchhidden-variable theories should reproduce the results of quantum mechanicsin every case where these have been confirmed. However, it was shown byJohn Bell in 1964 that under certain circumstances quantum theory andall ‘local’ hidden-variable theories predict different results. (Note: Localhidden-variable theories are hidden-variable theories in which the hiddenproperties of the physical systems are localized to each individual systemand do not depend on the properties at another point in space.) This re-sult, known as Bell’s Theorem, allows one to experimentally test whetherquantum mechanics or local hidden-variable theories accurately predict ex-perimental outcomes. Thus far, no experiments have shown deviations fromquantum predictions, thus ruling out all local hidden-variable theories. Aswe will see in the next chapter, Bell’s theorem forces us to make a choicebetween quantum theory or non-local hidden-variable theories.

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6.2.2 Many worlds

Figure 6.3: Hugh Everett III

One approach to alleviate the difficulties associated with the measure-ment collapse hypothesis is to reinterpret what we mean by measurement. In1957 a young American physicist, Hugh Everett, introduced what he calledthe “relative state” formulation of quantum measurement, in which he as-sumed that no wave-function collapse actually occurs when we perform a“measurement”. Instead Everett proposed that the universe splits into mul-tiple universes, or ‘branches’, one for each possible measurement outcome.The “multiverse,” which consists of multiple universes (or branches), eachlabeled by n, can be described by a single quantum state

|Ψ〉 =

∞∑n=1

cn|ψn〉, (6.14)

in which each universe is represented by a state |ψn〉. This universal wavefunction |Ψ〉 evolves according to the Schrodinger equation, with no discon-tinuous collapse.

In the case of Schrodinger’s cat, there are two possible outcomes (deador alive), and the universe splits into two “worlds” one in which the cat isalive and another in which it is dead, as depicted in the cartoon in Fig. 6.4.Once the splitting has occurred, not only can the initial state not be recon-structed, but there is no way in which the different branches can interferewith one another. Each branch evolves independently of the others and,as far as it is concerned, its future behavior is the same as if collapse had

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occurred and the other branches disappeared.

Figure 6.4: Many-worlds cartoon depiction of Schrodinger’s cat thoughtexperiment. In the many-worlds interpretation, both outcomes actually oc-cur. The point in time at which the measurement occurs the universe splitsinto two branches corresponding to the two possible measurement outcomes.These worlds have no knowledge of the other world, and cannot interact withone another.

This reinterpretation of quantum measurement eliminates the problemof wave-function collapse and reinstates realism to quantum physics, thatis, a particle in a superposition of two possible states does actually endsup in both state! These ideas were not well received at the time Everettproposed his relative state framework, leading to his departure from physicsresearch for a career in the defense industry. It was not until the 1960s and1970s that physicists began to take a serious look at Everett’s proposal. Itwas also during this time that the term “many-worlds interpretation” wascoined by Bryce DeWitt to describe Everett’s theory.

The many-worlds interpretation of quantum physics may resolve themeasurement problem, but it introduces its own interpretation problems.If we accept the premises of the many-worlds interpretation, then one majordifficulty that arises is the idea that there are an infinite-number of universeswith which we can never interact or gain knowledge about, but neverthelessexists independent of ourselves. This is an extremely uneconomical predic-tion, which seems to defy the idea of “Occam’s razor”, that is given a choicebetween theories that predict the same outcomes of events, one should choosethe theory with the fewest number of postulates or assumptions. There isa second, more difficult problem with the many-worlds interpretation apartfrom the multiverse extravagance. The problem resides in the question ofhow to talk about the probabilities of events when all possible events actuallyoccur. If a particle’s spin is either up or down, it makes sense to attribute

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probability to each outcome (up or down), which can be verified by makingmeasurements on a large number of systems and associating the fraction ofup and down outcomes with the corresponding probabilities. However, ifthe spin is in a superposition of up and down, what does probability mean?The question of probabilities is particularly difficult when we realize thatthe probabilities postulated in quantum mechanics are not related to thenumber of branches associated with each measurement outcome. The diffi-culties of probabilities and the multiverse in the many-worlds interpretationare still open problems. Nevertheless, many well-respected physicists chooseto side with the Everett interpretation.

6.2.3 Copenhagen interpretation

The standard interpretation of quantum measurements developed by NielsBohr, Werner Heisenberg, and Wolfgang Pauli in Copenhagen during the1930s, is known as the Copenhagen interpretation. The basic idea is thatquantum theory is not a theory of reality, but rather a set of mathematicalrules that allow us to predict the probabilities of measurement outcomes.The interpretation assumes that the wave function has no counterpart ‘in re-ality’. In other words, the wave function does not represent a physical entity,but is only a mathematical object that enables us to calculate statistical pre-dictions about experiments. The concept of “complementarity” introducedby Bohr to describe wave-particle duality, plays a central role in the Copen-hagen interpretation. Recall that complementarity implies that one cannotobserve “complementary aspects” or properties of a quantum system in thesame experiment. For example, in the double slit experiment, if we try todetermine the path of the particle through the double slit, we cannot sub-sequently observe the interference pattern due to the wave properties of thequantum system. Here the wave and particle nature of quantum systemsare complementary to one another. Similarly, the position and momentumare complementary observables in that measurement of one destroys anypossible knowledge about the other. This led Bohr to the conclusion thatnot only does it not make sense to discuss measurement of both the waveand particle properties of a system simultaneously, but that these propertiesdo not actually exist independent of the measurement. Unfortunately theCopenhagen interpretation does not really address the measurement prob-lem at all, but circumvents it by an operational approach, that is to saythat quantum mechanics does not tell us about what is (no reality associ-ated with the wave function), but only what can be (probabilities for certainmeasurement outcomes).

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6.3 Two-level quantum systems

There are many situations in which a quantum system can only take on oneof two possible values when measured. Examples of such two-level systemsinclude photon polarization, electron spin, and the path of a photon througha beam splitter. Such two-level quantum systems allow simple demonstra-tions of nonclassical predictions of quantum theory and form the founda-tions of quantum information. In the latter, a two-level quantum systemis known as a quantum bit or qubit. Here we look at three examples oftwo-level quantum systems and how to calculate the state evolution andmeasurement outcomes for different experimental scenarios.

6.3.1 Dirac notation

Figure 6.5: Paul Adrian Maurice Dirac

Before moving to the examples of two-level quantum systems, we shouldintroduce some notation that will make our calculations easier. To simplifymany calculations in quantum mechanics, Dirac introduced a shorthandnotation that now bears his name. It puts emphasis on the overlap integralof two wave functions ψ(x) and φ(x)

〈ψ|φ〉 =

∫ ∞−∞

ψ∗(x)φ(x)dx, (6.15)

which is the probability amplitude to observe the particle in state ψ(x) whenit is initially prepared in state φ(x), as discussed in Eq. (6.9) above. In Diracnotation, the integral on the right is written in the form shown on the left.

More generally, the integral operation 〈ψ|φ〉 denotes

1. Take the complex conjugate of the object in the first position (ψ →ψ∗).

2. Integrate the product ψ∗φ.

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This operation has the following simple properties. If a is any complexnumber and the functions ψ, φ satisfy∫ ∞

−∞ψ∗(x)φ(x)dx <∞, (6.16)

then the following relations hold

〈ψ|aφ〉 = a〈ψ|φ〉 (6.17)

〈aψ|φ〉 = a∗〈ψ|φ〉 (6.18)

〈ψ|φ〉∗ = 〈φ|ψ〉 (6.19)

〈ψ + φ| = 〈ψ|+ 〈φ| (6.20)

and∫ ∞−∞

(ψ1 + ψ2)∗(φ1 + φ2)dx = 〈ψ1 + ψ2|φ1 + φ2〉 (6.21)

= (〈ψ1|+ 〈ψ2|) (|φ1〉+ |φ2〉)= 〈ψ1|φ1〉+ 〈ψ1|φ2〉+ 〈ψ2|φ1〉+ 〈ψ2|φ2〉

The object 〈ψ| (called a “bra”) joins in to make the inner product withthe object |φ〉 (called a “ket”) to form a “bracket,” 〈ψ|φ〉, which is a com-plex number representing the complex overlap of the two wave functions (orstates).

Note that an operator acting on a Dirac bra or ket follows from its actionon the wave function. For example, the position operator acts by multiplyingthe wave function by x, as in Eq. (6.2). This implies that the expectationvalue of an operator A when the system is known to be in state ψ is

〈A〉 = 〈ψ|A|ψ〉 =

∫ ∞−∞

ψ∗(x)Aψ(x)dx. (6.22)

The operator acts from left to right on the ket, or similarly on the wavefunction ψ(x) and not its complex conjugate.

For our purposes here, we will only begin to use Dirac notation to sim-plify the discussion of two-level systems, in which the two possible states,generically denoted a and b are orthogonal. This implies that the state ketfor an arbitrary state, can be written in terms of |a〉 and |b〉, for example

|ψ〉 = α|a〉+ β|b〉, (6.23)

where the expansion coefficients must satisfy |α|2+|β|2 = 1 for normalizationof the wave function,

〈ψ|ψ〉 = |α|2〈a|a〉+ |β|2〈b|b〉+ α∗β〈a|b〉+ β∗α〈b|a〉= |α|2 + |β|2 = 1. (6.24)

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Here we use the orthonormality of the wave functions, 〈i|j〉 = δi,j , wherei and j label the different states a and b, and δm,n is the Kronecker deltafunction, which is zero for m 6= n, and one for m = n.

The formalism of two-level systems can also be expressed in matrix no-tation, in which the state is represented by a vector. The states a and brepresent basis vectors

|a〉 =

(10

), |b〉 =

(01

), (6.25)

along with their corresponding conjugates

〈a| =(1∗ 0

), 〈b| =

(0 1∗

). (6.26)

6.3.2 Photon beam path

Consider a photon confined to occupy a beam path. Such a system is readilycreated in the laboratory using nonlinear optics. The photon is incident ona beam splitter, which transmits a photon with probability T = t2 andreflects a photon with probability R = r2. Note that since the photonmust go somewhere, either transmit or reflect, we have R + T = 1, whichis a statement of conservation of probability, photon number, or energy (allequivalent). If we label the input modes a and b, with the output modes cand d, as show in Fig. 6.6. Denoting the photon occupying a given modej = a, b, c, d by |j〉, then the state of the photon before the beam splitter isgiven by

|ψinitial〉 = |a〉. (6.27)

The shorthand notation |j〉 implies that the photon is localized in modej = a, b, c, d, which could also be represented with an appropriately definedspatial wave function. The state at the output of the beam splitter is givenby a superposition of the photon having transmitted or reflected with ap-propriate weighting

|ψout〉 = t|c〉+ ir|d〉. (6.28)

The factor of i arise from the π/2 phase shift between transmission andreflection from a surface. (For more in depth derivation of the beam splitterinput-output relations, please see the excerpt (Beam splitter relations) fromR. Loudon’s book ”The Quantum Theory of Light” on the course website.)We can thus use the matrix representation of the two-level system to writethe input-output relations for the beam splitter as(

|c〉|d〉

)=

(t irir t

)(|a〉|b〉

). (6.29)

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Figure 6.6: Beam splitter input and output mode labels. Two inputs, a andb, are transformed into two output modes, c and d, by the beam splitter. Thetransmission (t) and reflection (r) coefficients correspond to the amplitudes(not intensities) and generally depend on many factors such as frequency,incidence angle, and polarization. However, for many cases beam splittersare often designed to take on specific values transmission and reflectioncoefficients. For example, a 50:50 beam splitter transmits and reflects equalamounts leading to t = r = 1/

√2.

The probability to find the photon in output mode c is thus given bythe modulus squared of the overlap between the state representing c and theoutput state

Pc = |〈c|ψout〉|2 = |〈c| (t|c〉+ ir|d〉)|2 ,= |t〈c|c〉+ ir〈c|d〉|2 ,= t2 = T, (6.30)

where we use the orthonormality of the states |c〉 and |d〉 to simplify thesecond line. Note that the probability to find the particle in mode c is givenby the expectation value of the operator

Πc = |c〉〈c|. (6.31)

This operator projects the state of the photon into mode c

Πc|ψout〉 = |c〉〈c|ψout〉 = |c〉〈c|(t|c〉+ ir|d〉) = t|c〉, (6.32)

and is thus known as a projection operator. It corresponds to a measure-ment of the photon in mode c. A similar projector exists for mode d. Theprobability to find the photon in a particular mode is thus given by theexpectation value of a projector onto that mode, for example

Pc = 〈ψ|Πc|ψ〉 = 〈ψ|c〉〈c|ψ〉 = 〈c|ψ〉∗〈c|ψ〉 = |〈c|ψ〉|2. (6.33)

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From Eq. (6.28) we see that the beam splitter creates a superpositionstate of the photon being in two modes (or states), which is a wave phe-nomenon (superposition of amplitudes). We can observe this wave behaviorby interfering the two output paths of the beam splitter, creating a Mach-Zehnder interferometer as in Fig. 6.7. If we vary the path length of thelower interferometer path by a distance L (note that this is not the amountthe mirror moves, but is related to it through the angle of reflection), weintroduce a phase difference Φ = kL between the upper and lower paths.This gives the following input on the second beam splitter

|ψin〉 = t1|c〉+ eiΦir1|d〉, (6.34)

where t1(r1) is the transmission (reflection) coefficient of the input beamsplitter. The output state of the interferometer is given by

|ψout〉 = (t1t2 − r1r2eiΦ)|e〉+ i(r1t2 + t1r2e

iΦ)|f〉, (6.35)

where we note that the transmission and reflection coefficients for both beamsplitters could differ. For 50:50 beam splitters, in which t = r = 1/

√2, this

Figure 6.7: Mach-Zehnder interferometer. Two input modes, a and b, areinterfered on the first beam splitter, which has transmission and reflectioncoefficients t1 and r1, respectively. Two mirrors direct the output modes ofthe first beam splitter, c and d, to the inputs of the second beam splitter,which has transmission and reflection coefficients t2 and r2, respectively. Thepath lengths between the two interior ‘arms’ (or paths) of the interferometerare initially balanced (equal), but the lower path (c) can be adjusted tointroduce an additional path length difference, L, and thus a phase differenceΦ = kL. The outputs of the second beam splitter, e and f , will generallybe superpositions of the input modes, dependent upon the phase differenceΦ between the two interior interferometer arms.

17

simplifies to

|ψout〉 =1

2(1− eiΦ)|e〉+

i

2(1 + eiΦ)|f〉,

= −ieiΦ/2(

sin

(Φ

2

)|e〉 − cos

(Φ

2

)|f〉). (6.36)

Now, suppose instead of letting the amplitudes recombine on the secondbeam splitter, we decide to find out which path the photon takes through theinterferometer by looking just before the input of the second beam splitter.What happens to the output state? Well, if the photon is observed in mode c,the state in Eq. (6.34) collapses into just |c〉 and similarly for d. The outputstate will then be just |c〉 → t2|e〉+ir2|f〉 or |d〉 → ir2|e〉+t2|f〉 respectively.The interference fringes that depend upon the phase Φ disappear and thewave aspect is washed out. This demonstrates the concept of wave-particleduality and complementarity in a new situation.

Note that for a 50:50 beam splitter the following two states are invariantunder the beam splitter transformation (i.e. eigenstates of the beam splittertransformation)

|+〉 =1√2

(|a〉+ |b〉), (6.37)

and

|−〉 =1√2

(|a〉 − |b〉). (6.38)

You should verify that if one of these states is put into a beam splitter, thenit emerges from the beam splitter in the same state unchanged. These statesare also orthonormal, so that 〈+|−〉 = 〈−|+〉 = 0 and 〈+|+〉 = 〈−|−〉 = 1.

6.3.3 Spin-1/2 system

The two paths of a beam splitter or interferometer is a perfectly acceptabletwo-level system, but really only represents a subset of all possible spatialstates that a photon can occupy. For example, there are an infinite num-ber of input modes for a beam splitter that differ by their transverse modeshape. A common example of a naturally occurring two-level system is thatof a spin-1/2 system, for example an electron or certain atoms have totalspin-1/2. The concept of particle spin was first introduced by WolfgangPauli in 1924, as a “two-valued quantum degree of freedom” associated withthe outer electrons of an atom. This allowed him to formulate the exclusionprinciple, i.e. that two electrons cannot occupy the same quantum state,and describe the structure of the periodic table. (NOTE: When we speakof a ‘quantum state’ of an electron in an atom, we need only specify a fulllist of ‘quantum numbers’ describing the energy level, orbital angular mo-mentum value, orbital angular momentum projection onto the z-axis, spin,and spin projection along the z-axis. These quantum numbers specify the

18

particular wave function of the electrons. This is similar to the way thequantum number n specified the wave function ψn(x, t) for the particle in abox from Chapter 5.)

The nature of this additional degree of freedom was not initially identi-fied. In 1925, Ralph Kronig, and George Uhlenbeck and Samuel Goudsmitsuggested that Pauli’s additional degree of freedom is associated with theself-rotation of the electron, and thus an intrinsic angular momentum. Al-though strictly speaking this concept is incorrect since the speed at whichthe particle would have to rotate is much faster than relativity allows, itdoes give the correct line of thinking. Spin, just as electric charge, is anintrinsic property of elementary (electrons, quarks, and photons for exam-ple) and composite particles (protons and neutrons for example), as well asatoms, and is associated with intrinsic angular momentum. For elementaryparticles, with no known substructure, spin cannot be explained by postu-lating that such particles are composed of smaller particles rotating about acommon center of mass. The spin of elementary particles is a truly intrinsicphysical property.

From experimental observation we find in nature that elementary par-ticles only have integer (s = 0, 1, 2, 3, ..., known as bosons) or half-integer(s = 1/2, 3/2, 5/2, ..., known as fermions) ‘spin’. The expectation value ofthe total spin vector squared is given by

〈S2〉 = ~2s(s+ 1), (6.39)

where s is often just called the particle spin. In quantum mechanics, theprojection of spin angular momentum measured along any cartesian coordi-nate axis, x, y, or z, can only take on quantized multiplies of ~. For example,we commonly choose to talk about the spin projection along the z-axis as amatter of convention. The possible values for this spin projector are givenby

〈Sz〉 = ~ms, (6.40)

where ms = −s,−(s − 1), ..., s − 1, s, which for a spin-1/2 system givesms = −1/2, 1/2. Quantization of spin angular momentum is a naturalextension of the concept of orbital angular momentum quantization as pro-posed by Bohr in his model of the hydrogen atom. However, notice that theorbital angular momentum may only take on integer multiples of ~, whereasspin can also have half-integer multiples in the case of fermions.

Particles with charge may also possess an intrinsic magnetic dipole mo-ment associated with the spin. The naive idea is that a spinning chargedistribution has rotating current, which leads to a magnetic dipole. Justas the projection of spin angular momentum along a measurement axis is

19

quantized, so too is the associated magnetic dipole moment. The first directexperimental evidence for the quantization of electron spin was the Stern-Gerlach experiment, which set out to demonstrate the quantization of orbitalangular momentum associated with the motion of an electron orbiting anatom. However, as we will see below, their anticipated interpretation wasincorrect due to the fact that the atomic species they chose, silver, has zeroorbital angular momentum.

Stern-Gerlach Experiment

In 1922, two German physicists in Frankfurt, Otto Stern and Walter Gerlach,were attempting to demonstrate the prediction made by Arnold Sommerfeldand Paul Ehrenfest in 1913 that projection of orbital angular momentumalong a particular measurement axis should be quantized. However, it wasnot immediately clear that their results actually showed the existence ofelectron spin and its quantization.

Figure 6.8: Schematic of Stern-Gerlach experiment. An oven emits silveratoms with a range of velocities that are subsequently filtered using a pairof slits to give a fairly uniform atomic beam with velocity vx. The atomicbeam is passed through a non-uniform magnetic field derived from a pairof magnets. The magnetic moment of the outer shell electron undergoes aforce due to the magnetic field gradient. The direction and magnitude ofthe force depends on the projection of the magnetic moment onto the z-axis.Classically, one expects a continuous range of magnetic moment orientationsalong the z axis. However, quantum theory predicts quantized values of themagnetic moment projection, which is experimentally observed.

The Stern-Gerlach experiment, depicted in Fig. 6.8 consists of a beam ofneutral silver atoms traveling in the x-direction with velocity vx and directedthrough an inhomogeneous magnetic field Bz(z) oriented in the z-direction.Stern and Gerlach anticipated that the magnetic moment of the outer-shell

20

electron orbiting the silver atoms should experience a force due to the in-homogeneous magnetic field, and due to the quantization of orbital angularmomentum along the z-axis one should observe three deflected positions forthe beam.

Prior to discussing the results of the Stern-Gerlach experiment, let usfirst go through the classical and quantum predictions. Assuming that theelectron orbiting the nucleus occupies a circular orbit of radius r, the mag-netic moment associated with this motion is given by

µ = IA

= −n( ev

2πr

) (πr2)

= −n e

2mmvr

= −n e

2mL

= −n e

2m~n

= −nµBn, (6.41)

where I is the current associated with the electron orbiting the nucleus, givenby the charge −e, multiplied by the speed v divided by the circumferenceof the orbit 2πr. A is the vector associated with the area of the electronorbit, with unit direction vector n, normal to the surface area as depictedin Fig. 6.9. In going from line 3 to 4 we use the relationship betweenangular momentum for a circular orbit of radius r, and momentum, i.e.L = mvr = rp. In the second-to-last line we have used the quantization ofangular momentum, L = n~, where n = 1, 2, 3, ..., and in the last line weintroduce the Bohr magneton defined as

µB =e~2m

, (6.42)

which represents the fundamental unit of magnetic moment.A magnetic dipole moment µ placed in a magnetic field B, will experience

a torqueτ = µ×B. (6.43)

Depending on the orientation of the magnetic moment with respect to themagnetic field, the system will have different energy associated with thisinteraction as depicted in Fig. 6.10.

We can define a potential energy associated with this interaction

U = −µ ·B. (6.44)

Here we see that a magnetic moment aligned along the magnetic field hasthe lowest possible energy, while a magnetic moment aligned anti-parallel

21

Figure 6.9: Pictoral representation of the magnetic moment associated withthe orbital motion of an electron (e− in a circular orbit of radius r around thenucleus (+). The electron has velocity v, sweeps out an area A = πr2, andthe unit vector n corresponds to the direction of orbital angular momentumL = r×p. The magnetic moment, µ, points in the opposite direction owingto the negative sign of the electronic charge.

with the magnetic has the largest energy. Now, if the magnetic field is nothomogeneous, not only will there be a difference in energy associated withdifferent orientations, but also a net force on the magnetic dipole propor-tional to the gradient of the magnetic field. This can be viewed in terms ofa force due to a potential energy F = −∇U , leading to the following forceon the magnetic dipole in the Stern-Gerlach experiment

F = −∇U= ∇(µ ·B)

= (µ · ∇)B

= µz(∂zBz)z (6.45)

where we have made use of the fact that the magnetic moment is assumedto be constant, and the following vector calculus identity

∇(A ·B) = (A · ∇)B + (B · ∇)A + A× (∇×B) + B× (∇×A). (6.46)

We have also used the fact that the curl of the magnetic field is zero, sincethere is no free current or time-varying electric field, and the magnetic fieldis oriented and varies only along the z-direction.

The force acts on the beam only during the time that the atoms passthrough the magnetic field gradient, which is equal to ∆t = L/vx, where L isthe magnet length, and vx is the velocity in the x-direction of the particlesin the beam. Due to the magnetic force on the particle in Eq. (6.45),the beam will gain some transverse velocity in the z-direction and will bedeflected at an angle θ on the output. The transverse velocity will be given

22

Figure 6.10: Energy dependence of magnetic dipole moment orientation ina uniform external magnetic field (B). The lowest energy is associated witha magnetic dipole moment oriented parallel (with) the magnetic field (onthe left), whereas the highest energy is associated with a magnetic dipolemoment oriented anti-parallel (against) the magnetic field (on the right).

by the kinematic relations

vz = az∆t

=Fzm

∆t

=µz(∂zB)L

vxm. (6.47)

This leads to a deflection angle

θ =vzvx

=µz(∂zB)L

v2xm

, (6.48)

as sketched in Fig. 6.11.

Figure 6.11: The deflection angle θ from the Stern-Gerlach apparatus isrelated to the initial velocity in the x-direction and the acquired velocitycomponent in the z-direction, vz due to the interaction of the electron mag-netic moment and the inhomogeneous magnetic field.

Classical prediction

23

Equation (6.48) implies that the beam is deflected through a range ofangles proportional to the projection of the atomic magnetic moment ontothe z-axis. Silver has only one valence electron that can contribute to theorbital angular momentum (the filled subshells are completely symmetricand therefore do not contribute to the angular momentum). If we assumea classical model of the atom, in which the electron orbits in a circular or-bit with arbitrary orientation of the orbit (the projection onto the z-axis isgiven by a sinusoidal distribution), we see that the deflection angle shouldvary smoothly over a range of values, as depicted in Fig. 6.12. Classically,the projection of magnetic momentum onto the z-axis is not quantized.

Figure 6.12: Predicted angular deflection probability distributions for classi-cal (top), quantum spin-1 (middle), and quantum spin-1/2 (bottom) models.The classical distribution has a continuous distribution across the deflectionangles θ. The spin-1 model predicts three discrete peaks, while the spin-1/2 model predicts only two peaks. Stern and Gerlach observed two peaksindicating that the electron does indeed have quantized spin.

Stern-Gerlach prediction

The predicted results that Stern and Gerlach were hoping to obtain werebased on the idea that the orbital angular momentum of the electron in thesilver atoms was quantized. Having only one electron in the valence shell,

24

they anticipated that there would be total orbital angular momentum valueof ~. This would then imply the magnetic moment is also quantized, as inEq. (6.41), and would have three projections onto the z-axis, leading tothree deflection angles (one for each of the three projections of the orbitalangular momentum onto the z-axis n = +1, 0,−1)

θn =µBn(∂zB)L

v2xm

. (6.49)

However, the theory at the time (due to Sommerfeld) predicted two lines,which corresponds to the experimental results which only showed two de-flected paths (“up” and “down”), with no straight through path. At thetime, they were satisfied that they had indeed showed the quantization ofangular momentum. However, the interpretation that the magnetic momentcausing the deflection was due to the orbital angular momentum of the elec-tron was incorrect. Recall that I mentioned briefly that the Bohr modelpredicts the wrong value of orbital angular momentum for and electron inthe ground state – it predicts L = ~ for the ground state, but it is actually0! The same follows for the ground state of silver atoms. It was only yearslater when the concept of particle spin was introduced, did a satisfactoryexplanation of the Stern-Gerlach experiment arise.

Stern-Gerlach spin-1/2 description

The total magnetic moment of the silver atoms in the Stern-Gerlachexperiment is given by a vector sum of the contributions due to the nuclearspin, electron orbital angular momentum, and electron spins. This turnsout to give a spin-1/2 system. Thus the correct interpretation of the Stern-Gerlach experiment is not in terms of the quantization of the orbital angularmomentum of the valence electron, but the quantization of all contributionsto the angular momentum and thus magnetic moment. For a spin 1/2 systemthere are only two possible projections onto the z-axis (m = ±1/2), leadingto two deflection angles

θ = ±µB(∂zB)L

2v2xm

. (6.50)

This explanation matches the observed behavior as depicted in Fig. 6.12and was the first direct experimental observation of electron spin, althoughnot known at the time. When a particle is detected in the z+ deflecteddirection, it is said to have “spin up”, while a particle in the z− deflecteddirection is said to have “spin down”. The Stern-Gerlach apparatus mea-sures the projection of magnetic moment (and thus spin) along the magneticfield gradient.

Subsequent measurements

25

We now want to consider sequential Stern-Gerlach measurements, inwhich the atomic beam goes through two or more SG magnets. We beginby considering a beam of unpolarized spin-1/2 atoms emitted from an oventhat pass through a SG apparatus with the field gradient aligned along thez-axis, which we denote SGz as depicted in Fig. 6.13. At the output ofthe first SG apparatus, the atoms will have split into two beams with equalnumbers of atoms in well-defined spin projection onto the z-axis, denotedby z+ and z−. We denote the spin states associated with these by |Sz;±〉,and the measurement performed by the SGz apparatus by Sz, which gives

Sz|Sz;±〉 = ±~2|Sz;±〉. (6.51)

Note that these states are orthonormal, that is

〈Sz; i|Sz; j〉 = δi,j (6.52)

where the Kronecker delta symbol is 0 for non-identical indices (+ or −),and 1 for identical indices. If we block the z− output, and send the z+beam into a second SG apparatus with the same magnetic-field gradientorientation (i.e. we use another SGz setup), then we will only see one beamemerge from the second SG in the z+ port with nothing coming out of z−.This is not too surprising if we think of the atom spins are all aligned in the“up” state before entering the second SGz setup.

What happens if we rotate the SG apparatus by 180◦ around the beamaxis, so that the z+ and z− outputs are flipped? It turns out that we willagain see only one beam emerge, but this time all the atoms will come outfrom the z− port at the top of the apparatus. (To understand why thisis the case requires further mathematics to describe how spin-1/2 systemstransform under spatial rotations. This is quite different from the rotationproperties of vectors with which most students are familiar. I will not gointo detail about this, but the interested student can see for example J. J.Sakurai’s book, Modern Quantum Mechanics, for a nice description of therotation properties of spin-1/2 systems.)

Another interesting situation, depicted in Fig. 6.13, consists of an initialSG apparatus again with its magnetic field gradient aligned along z, but thesecond SG apparatus is aligned along the x-direction, denoted SGx. Thez+ polarized beam that enters the second SG setup is split into two beamsat the output with well-defined spin projections onto the x-axis, denoted byx+ and x−. It is tempting to assume that the beam output from SGz hasatoms with well-defined projections of spin onto both the z-axis and the x-axis simultaneously. However, this is not correct as we will see, measurementof the spin component along z is incompatible with measurement along x.This is analogous to trying to measure both the position and momentum of a

26

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(*% +,%%|Sz;−�

!"#$% &'(%()%

(*%&'-%

-)%

-*% |Sx;−�

|Sx; +�

|Sz; +�&'(%

()%

(*% |Sz;−�

./0%,1%2$345%

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(*%&'-%

-)%

-*%

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Figure 6.13: Sequential Stern-Gerlach (SG) measurements for different SGapparatus orientations. An oven emits unpolarized (randomly oriented spin)atoms into a SG apparatus with its field gradient oriented along the z-axis(denoted SGz). Blocking the z− port, we create an atomic beam withwell-defined spin projection along the z-axis in the state |Sz; +〉. If wepass this through a second SGz setup, we only find |Sz; +〉 again (top).However, passing the z+ polarized atoms through a SG apparatus withits field gradient along the x-axis, we find both x+ and x− at the output(middle). If we further take only the |Sx; +〉 output from the SGx setup,and pass this through an SGz, we find both z+ and z− polarized atoms atthe output.

particle simultaneously. We can only measure the spin component along oneaxis, and subsequent measurement of the spin projection along another axiswill give a random value. This means that we cannot obtain simultaneousknowledge of the spin projection along both the x- and z-axes.

To understand the incompatibility of measuring the spin projection alongthe z-axis or x-axis, we need to determine the spin state associated withthe SGx setup in terms of the eigenstates of the z-axis spin measurementSz, i.e. |Sz;±〉. Again, this requires an understanding of how spin-1/2systems transform under rotations, and I will only state the result. Thespin eigenstates for a SG apparatus rotated by an angle θ with respect tothe z-axis, which we will denote |Sθ;±〉, are given by

|Sθ; +〉 = cos

(θ

2

)|Sz; +〉+ sin

(θ

2

)|Sz;−〉, (6.53)

and

|Sθ;−〉 = sin

(θ

2

)|Sz; +〉 − cos

(θ

2

)|Sz;−〉. (6.54)

From these we can see that the spin eigenstates associated with projection

27

onto the x-axis are those with θ = π/2, in which case we have

|Sx; +〉 =1√2

(|Sz; +〉+ |Sz;−〉) , (6.55)

and

|Sx; 〉 = − 1√2

(|Sz; +〉 − |Sz;−〉) . (6.56)

You should verify that the pairs of states {|Sθ; +〉, |Sθ;−〉}, and {|Sx; +〉, |Sx;−〉}are orthonormal.

Now, at the output of the first SG apparatus, the spin state of the atomsis |Sz; +〉. In terms of the x-measurement eigenstates, we can rewrite thisas

|Sz; +〉 =1√2

(|Sx; +〉+ |Sx;−〉) . (6.57)

In other words, the well defined spin projection along the z-axis is a super-position of spin-projection states along the x axis. We see that when we usethe second SG apparatus oriented along the x-axis, SGx, and look to seewhere the atoms emerge (either x+ or x−), we collapse the superposition.For example, the probability to measure x+ is given by the modulus squaredof the overlap between 〈Sx; +| and |Sz; +〉

P (Sx,+) = |〈Sx; +|Sz; +〉|2 =

∣∣∣∣ 1√2

∣∣∣∣2 =1

2. (6.58)

Similarly, we can ask what is the probability to obtain the + measurementoutcome when our SG apparatus is rotated by an angle θ, when the inputspin state is |Sz; +〉? We simply take the modulus squared overlap betweenthe rotated + state with the initial state

P (Sθ,+) = |〈Sθ; +|Sz; +〉|2 = cos2

(θ

2

). (6.59)

Similarly, the probability to obtain the − measurement outcome for therotated case is

P (Sθ,−) = |〈Sθ;−|Sz; +〉|2 = sin2

(θ

2

), (6.60)

which could also be determined by considering that there are only two pos-sible outcomes for the measurement so the probabilities for both outcomesmust sum to unity, P (Sθ,+) + P (Sθ,−) = 1.

Now suppose we put another SGz after the SGx apparatus on the x+output, as shown in Fig. 6.13. How do we determine the probability thatan atom will be observed in the z+ output of the last SG setup? Well, the

28

probability to emerge from the first SGz after the oven in the z+ state is1/2, which arises from the fact that the spin state emerging from the ovenis completely random, and thus equally probable to have spin up or down(note that this is true for any orientation of the first SG apparatus). Theprobability that an atom, now in the z+ state, incident on the second SGapparatus, SGz, will emerge in the x+ spin state is also 1/2. This howeverarises from a different reason, i.e. the modulus squared overlap betweeninput and output states. Finally, the probability for an atom in the x+state sent into the final SG setup aligned along z to emerge in the z+ stateis given by the overlap

P (Sz,+) = |〈Sz; +|Sx; +〉|2 = |〈Sx; +|Sz; +〉|2 =1

2. (6.61)

Thus the total fraction of atoms that emerge from the third SG apparatus(SGz) in the z+ port is given by P1 ·P2 ·P3 = 1/2 · 1/2 · 1/2 = 1/8 = 12.5%,where Pj is the probability that an atom will transmit through the jth SGapparatus in the appropriate port.

6.3.4 Photon polarization

Photon polarization is analogous to electron spin. In fact, photon polariza-tion is actually equivalent to the photon spin. However, photons differ inmany ways from electrons – they are massless, zero-charge, spin-1 “parti-cles”. As we discussed above, spin-1 particles should have three spin projec-tions along the z-axis giving Sz = ~, 0,−~. However, owing to the zero massof the photon, the 0 spin projection does not occur. The reasons for this arewell beyond the scope of the discussion here, but I thought I would bringthis up to wet your appetite. The full story can be found when consideringrelativistic quantum theory.

For our purposes, we will use the photon polarization as a two-levelsystem. Assuming a beam of photons, we typically talk about horizontal(|H〉) and vertical (|V〉), diagonal (|+〉) and anti-diagonal (|−〉), and right(|R〉) and left (|L〉) polarization states. These states can all be writtenas superpositions of one another, similar to the relationship between thex and z Stern-Gerlach spin states in the previous section. Typically, weuse polarizing beam splitters to analyze and measure the polarization statesof photons, which pass horizontal photons and reflect vertical photons, asdepicted in Fig. 6.14.

Thus, we often choose to use the horizontal and vertical states as ourbasis states in which to represent other states. The diagonal / anti-diagonalstates are then given by

|±〉 =1√2

(|H〉 ± |V〉) , (6.62)

29

Figure 6.14: A polarizing beam splitter (PBS) transmits horizontal (H)polarization and reflects vertical (V) polarization, as depicted above (top).This is analogous to a Stern-Gerlach apparatus for photons. Polarizingbeam splitters are often drawn as a square with a diagonal for the reflectivesurface, while horizontal polarization is represented by a double arrow line,and vertical is represented with a dotted circle (bottom).

while the right and left circular polarization states are

|R〉 =1√2

(|H〉+ i|V〉) , (6.63)

and

|L〉 =1√2

(|H〉 − i|V〉) . (6.64)

The polarizing beam splitter (PBS) acts like a Stern-Gerlach apparatus inthat it splits orthogonal polarization amplitudes into two separate spatialpaths. If we rotate our polarizing beam splitter by an angle Φ, the trans-mitted and reflected eigenstates become

|ΦH〉 = cos (Φ) |H〉+ sin (Φ) |V〉, (6.65)

and|ΦV〉 = sin (Φ) |H〉 − cos (Φ) |V〉. (6.66)

Again, we can consider what happens when we perform sequential mea-surements on the polarization state of photons, as we did for the spin-1/2systems considered in the previous section. However, it is perhaps more

30

enlightening to consider the setup in Fig. 6.15, in which a photon initiallyprepared in the |+〉 state is incident on a polarizing beam splitter (PBS).The two outputs of the PBS are recombined on a second PBS, in which thephase of the horizontal path differs from the vertical path by Φ. The pho-ton will clearly emerge from only one beam splitter port of the second PBS.However its polarization state will depend on the value of Φ. The outputstate of the photon will be

|ψ(Φ)〉 =1√2

(eiΦ|H〉+ |V〉

), (6.67)

where I have included the additional phase on the horizontal component ofthe state.

Figure 6.15: Polarization interferometer. A photon with diagonal polariza-tion is incident on a polarizing beam splitter (PBS). The vertical amplitudeis reflected on the lower path while the horizontal amplitude is transmit-ted along the upper path. The phase of the upper path can be varied in amanner similar to the Mach-Zehnder interferometer in Fig. 6.7. The twopolarization amplitudes are recombined at the second PBS, resulting in amodified output polarization state |ψ(Φ)〉.

We now consider what happens when this photon is incident on a po-larizing beam splitter oriented at +45◦. What is the probability that thephoton will be transmitted? Well this is just given by the modulus squared

31

of the appropriate state overlap

P (+,Φ) = |〈+|ψ(Φ)〉|2

=

∣∣∣∣12 (〈H|+ 〈V|)(eiΦ|H〉+ |V〉

)∣∣∣∣2=

1

4

∣∣1 + eiΦ∣∣2

=1 + cos(Φ)

2

= cos2

(Φ

2

), (6.68)

where we used the fact that the transmitted state of the final beam splitterat +45◦ is just the diagonal state, |+〉. So we see that as Φ varies, we changethe output polarization state of the second PBS.

Measurement operators

As we discussed at the beginning of the chapter, in quantum mechan-ics, observable properties of physical systems become operators in quantummechanics. These operators can be expressed in the Dirac notation and ma-trices we introduced earlier in the chapter, see Eq. (6.31). For example, ifwe consider measurement of the polarization of a photon using a polarizingbeam splitter in which horizontal (H) transmits and vertical (V) reflects, andwe associate H with the value +1 and V with −1, then the correspondingmeasurement operator is

Sz = |H〉〈H| − |V〉〈V|,

=

(1 00 −1

). (6.69)

The combination of a ket followed by a bra, |ket〉〈bra| is known as an ‘outerproduct’. If we take the expectation value of this operator for a photon inH polarization, we have

〈H|Sz|H〉 = 〈H| (|H〉〈H| − |V〉〈V|) |H〉= 1, (6.70)

where in the last line we used the orthonormality of the H and V states.Similarly, if we take the expectation value of Sz when the photon is knownto be in the V state, we obtain 〈Sz〉 = −1. The z subscript is associatedwith the concept of spin projections along different spatial axes. There arealso projectors onto the “x” and “y” axes, which for photon polarizationcorrespond to projection onto diagonal polarizations at ±45◦ and right- and

32

left-circular polarizations:

Sx = |+〉〈+| − |−〉〈−|,= |H〉〈V|+ |V〉〈H|,

=

(0 11 0

), (6.71)

and

Sy = |R〉〈R| − |L〉〈L|,= −i (|H〉〈V| − |V〉〈H|) ,

=

(0 −ii 0

). (6.72)

Note that the operator Pψ = |ψ〉〈ψ| is called a projector, since it projects astate onto the state |ψ〉.

We will use these properties in the next chapter to discuss and analyzevarious experimental situations.

33

Chapter 7

Two-particle systems andentanglement

7.1 Introduction

In the previous chapter we discussed the standard formalism associated withquantum measurement, and the measurement-induced collapse of the wavefunction. The difficulties that arise from the measurement-collapse hypoth-esis, i.e. the non-deterministic (probabilistic) nature of the outcome and theill-defined time at which the collapse process occurs, led some of the foundersof quantum theory (particularly Einstein and Schrodinger) to challenge thisstandard (Copenhagen) interpretation of quantum mechanics.

The discourse on the interpretation of quantum theory is often said tohave begun at the fifth Solvay Conference on ‘Electrons and Photons’ inBrussels during the end of October 1927. There the famous Bohr-Einsteindebate began after Bohr described his Copenhagen-interpretation of quan-tum mechanics, which includes measurement collapse. Einstein thought ofquantum theory as still incomplete and posed to Bohr a series of gedankenexperiments (thought experiments) that he believed pointed out the incom-pleteness of quantum mechanics. He began by describing the collapse of thewave function of a particle on a screen after a single slit. He argued thatprior to collapse the wave function is spread across the entire screen, andupon detection the particle is registered at a point A. He argued that inmaking this observation, not only do we learn that the particle arrived atA, but also that it did not arrive at B. Moreover, we learn that the parti-cle’s non-arrival at B instantaneously with its observation at A. However,prior to observation the probability amplitude is smeared out over the wholescreen. Einstein believed that the collapse of the wave function implies apeculiar ‘action at a distance’. The particle, which is somehow spread outover a large region of space, becomes localized instantly, the act of mea-

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Figure 7.1: Photograph from the 1927 Solvay Conference on Electrons andPhotons.

surement appearing to change the physical state of the system far from thepoint where the measurement is actually recorded. Einstein thought thatthis kind of action at a distance violated the postulates of special relativity.However, according to the Copenhagen interpretation, the wave function isnot associated with any physical reality of the system, but simply a math-ematical object used to calculate probabilities of measurement outcomes.Thus the collapse of the wave function according to Bohr, is not a physicalcollapse, but rather a mathematical procedure.

The debate between Bohr and Einstein continued after the conferencethrough correspondence, as well as at the sixth Solvay conference in 1930.Still dissatisfied with what he saw as an incomplete theory, Einstein con-tinued to work on the interpretation of quantum mechanics which led himto one of the truly unique implications of quantum theory. After movingto the United States in 1933, working at the Institute for Advance Studyat Princeton, Einstein, and two young colleagues – Russian, Boris Podolskyand American, Nathan Rosen – published a paper in the journal PhysicalReview, entitled “Can quantum-mechanical description of physical reality beconsidered complete?” that would become one of the quintessential paperson the foundations of quantum physics.

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Figure 7.2: Niels Bohr and Albert Einstein in conversation.

In the paper, Einstein, Podolsky and Rosen (EPR) consider measure-ments not on a single quantum system, but on two quantum systems. Theyshowed that if one considers systems that are correlated with one anotherand we make a measurement on one of the particles, we immediately changethe quantum state of its twin. For example if the particles are correlatedin their momenta and the momentum of one particle is p1, its twin musthave momentum p2 = −p1 due to their correlations. Measurement of themomentum of particle 1 immediately changes the physical state of particle2. EPR believed that this instantaneous affect violates special relativity and

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therefore quantum theory had to be incomplete.

This is the crux of the EPR paradox – according to quantum theorymeasurement on one particle instantly allows one to affect the state of itstwin no matter how far away it is from the first. The initial EPR paperwas framed in terms of measurements on two particles’ positions and mo-menta, but a simplification of the problem was developed by David Bohmin 1951. The EPR paper was initially a shock to the physics communityand the world (the New York Times carried an article entitled ‘Einstein at-tacks quantum theory’ in May 1935). Schrodinger applauded the paper andfollowed it with his series of articles published in Die Naturwissenschaften,entitled ‘The present situation of quantum mechanics’. In these, he intro-duced the term entanglement to describe the type of correlations shared bytwo-particle systems that share the types of correlations that EPR intro-duced, as well as his famous ‘cat’ thought experiment to bring to light thedifficulties the measurement collapse hypothesis introduces for macroscopicsystems.

Einstein believed that quantum mechanics was an incomplete descrip-tion of the natural world, in much the same way that the classical statisticalmechanics of Boltzmann is incomplete. That is to say that he believedquantum probabilities describe our ignorance of the actual properties (orreality) of physical systems and that there is a deeper, more fundamentaltheory yet to be discovered in which all physical properties of a systemare well defined and evolve deterministically in time. The assumption wasthat quantum theory was incomplete and there were “hidden variables” thatwere not experimentally accessible at the time, and with improved exper-imental precision, such hidden variables would be discovered. He thoughtthat the wave function interpretation of Schrodinger’s cat thought exper-iment could not actually apply to a single cat in a box, but rather mustdescribe an ensemble (or large set) of identically prepared cats in boxes, inwhich the actual physical state of each cat is well defined, but unknown tous. The motivation of Einstein and Schrodinger was not to derail quantumtheory’s successes, but to note its shortfalls, particularly when pertainingto the implications the standard interpretation has on the nature of ‘reality’.

A key aspect of the approach taken by Einstein relies on his definitionof reality, in which physical properties of a system are localized in spacewith the system. So a measurement on one particle far from its correlatedtwin, should not be able to affect the physical state of its twin. This ‘local-realistic’ view of the world seems perfectly natural with our classical waysof thinking. However, as we will see, quantum theory forces us to choosebetween local-reality, that is well-defined physical properties attached to aparticle, and the completeness of quantum theory.

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These challenges raised in the early days of quantum mechanics werethought to reside in the realm of philosophy and were largely neglected onthe grounds that one could not perform experiments on individual quantumsystems, but only collections of many quantum systems. For these reasons,most practitioners of quantum mechanics were happy to accept the Copen-hagen interpretation and just calculate predictions of experiments basedupon the standard approach. However, further developments, both theoret-ical and experimental, over several decades began to shed light on the EPRthought experiment. In particular, David Bohm’s reformulation of the EPRexperiment in terms of two correlated two-level systems (Bohm discussedpairs of correlated spin-1/2 systems) brought the EPR issue back to the sur-face of physics research. He further introduced the a theoretical frameworkthat re-instilled reality to the wave function, at a cost of non-local inter-actions between localized particles. However, in 1964, John Bell put forthan argument that pitted all possible local-hidden-variable theories againstquantum mechanics. He derived an expression for measurement outcomesbased on any local-hidden-variable model proposed to replace quantum the-ory and showed that the predicted measurement outcomes differ from whatquantum mechanics predicts. This was the first point at which the choicebetween interpretations of quantum theory was no longer a matter of taste,but of correctness. It was not until the early 1980s that the first experimen-tal demonstration of the correctness of quantum theory, and ruling out of alllocal hidden-variable models, was performed by Alain Aspect and co-workersin Paris.

In the following sections we discuss Bohm’s version of the EPR thoughtexperiment in more detail and the apparent ‘paradox’ that arises. Thenwe introduce the ideas of Bell that compare local-hidden-variable modelswith quantum predictions. Lastly, we discuss the Aspect experiments anddescribe how these show the correctness of quantum theory.

7.2 Bohm version of EPR

Bohm’s version of the EPR thought experiment involves measurement notof the positions and momenta of correlated particles, but the spin projectiononto different axes. For our purposes we will discuss not spins, but photonpolarizations. Suppose we have a source of photon pairs in which the photonsare emitted in opposite directions (±x-directions, which we call photons 1and 2, respectively) with either right- or left-circular polarization (R or L),but it is not known which polarization each photon has. Such a source canbe realized from an atomic system in which the photon pair is emitted in a

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Figure 7.3: David Bohm

cascaded process. The quantum state of the photon pair can then be written

|Φ−〉 =1√2

(|R,L〉1,2 − |L,R〉1,2) , (7.1)

where the notation |R,L〉1,2 implies that photon 1 has R polarization andphoton 2 has L polarization, that is, the left label inside the Dirac branotation corresponds to photon 1 and the right label to photon 2. Now,if we choose to measure the polarization of photon 1 in the RL-basis andobtain R, then we know that a measurement on particle 2 will yield L.Alternatively, we could choose to measure photon 1 in the HV-basis. If wemeasure photon 1 to have H polarization, what can we say about photon 2?To determine this, we need to write the state in Eq. (7.1) in terms of H andV polarizations. In the previous chapter, we wrote down the relationshipbetween the HV, ±, and RL bases for a single photon. Substituting theexpressions for R and L in terms of H and V and + and − into Eq. (7.1),we find that the photon pair state in the HV and ± bases is

|Φ−〉 =−i√

2(|H,V〉1,2 − |V,H〉1,2) ,

=i√2

(|+,−〉1,2 − |−,+〉1,2) . (7.2)

Thus we see that the polarization state of the photon pair is correlated in anyand all bases! In his series of articles in 1935 Schrodinger introduced the termentanglement to describe the strong correlations displayed by such states.This type of correlation does not exist in classical physics. Mathematically,an entangled state of two systems labelled 1 and 2, cannot be written as a

39

product of wave functions for the subsystems,

Ψ1,2:entangled 6= ψ1φ2, (7.3)

or in Dirac notation,

|Ψ1,2:entangled〉 6= |ψ1〉|φ2〉. (7.4)

For a pair of two-level systems, there are four possible entangled states oftencalled the Bell states, which for photon polarization can be expressed as

|Ψ+〉 =1√2

(|HH〉+ |VV〉) , (7.5)

|Ψ−〉 =1√2

(|HH〉 − |VV〉) , (7.6)

|Φ+〉 =1√2

(|HV〉+ |VH〉) , (7.7)

|Φ−〉 =1√2

(|HV〉 − |VH〉) . (7.8)

Now, returning to our thought experiment, a measurement of photon 1 inthe HV basis yielding H, implies that a measurement of photon 2 would yieldV. However, as we saw in the previous chapter, quantum mechanics doesnot allow simultaneous measurement of HV, ±, or RL polarization becausethere is an uncertainty relation between these. Einstein argued that whilethis might be the case according to the orthodox quantum interpretation,whatever happened to particle 1 could have no immediate effect upon thedistant particle 2. The thinking behind EPR is that the spatial separationof 1 and 2 implies the independence of what happens at 1 and what happensat 2. If that is so, and if one can choose to measure either the HV, ±, orRL polarization at 1 and obtain absolute knowledge of the HV, ±, or RLpolarization of particle 2, then particle 2 must actually have these definitevalues of polarization, whether the measurements are made or not.

The problems raised by EPR, and Schrodinger arise from the followingaspects of quantum theory:

• Interpretation of quantum probabilities: Unlike probabilities in classi-cal physics that are associated with our ignorance of the intricacies ofsome underlying physical reality, quantum probabilities describe thelikelihood that a quantum system that interacts with a measurementdevice will produce a particular outcome.

• Wave-function collapse: The measurement induced collapse is simplynot described by quantum theory and must be posed as an additionalpostulate. The Copenhagen interpretation says nothing about thetime at which this collapse is to take place, leaving us to ponder thefate of cats in a superposition.

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• Entangled states, locality, and action at a distance: The essence of theEPR argument was a demonstration of the ‘spooky’ action at a dis-tance implied by the entanglement of two quantum particles that haveinteracted and moved apart. The Copenhagen interpretation assumesquantum theory is complete and therefore the two particles continueto be described by a single two-particle wave function no matter howfar apart they are separated. The particles have lost their individu-ality and locality in space-time. When we make a measurement onone particle, the state of the pair collapses instantaneously, forcingthe second particle to realize some specific state though it may be lo-cated half-way across the universe. EPR insists that ‘no reasonabledefinition of reality could be expected to permit this’.

The ‘EPR paradox’ is then associated with the idea that according toquantum theory, when we make a measurement on one particle that is en-tangled with a second particle located far from the measurement device, weaffect the physical state of the second system no matter how far away it islocated. This ‘spooky’ action at a distance troubled EPR and Schrodinger,who viewed physics as a description of reality, and the wave function shouldbe associated with some aspect of the physical state of the quantum sys-tems and that this reality should be localized with the particle. They be-lieved that this example showed that quantum theory had to be incom-plete and that there should be some yet-unknown underlying theory fromwhich quantum mechanics emerged. Such theories came to be known as“hidden-variable” theories. Many of those troubled with the Copenhageninterpretation worked to create a hidden-variable theories that could repro-duce quantum mechanical predictions. Indeed, Einstein devoted much of hislatter work to unified field theories that could eliminate what he saw as theincompleteness of quantum theory.

7.3 Bell-inequality and local realism

In 1932, John von Neumann published his textbook on quantum physics,Mathematical foundations of quantum mechanics, in which he presents an‘impossibility proof’ for hidden-variables. His conclusion was that hidden-variable models could not reproduce the same predictions of quantum the-ory, and therefore had to be ruled out as possible replacements for quantumtheory (as long as quantum mechanics accurately predicts the outcome ofexperiments). Many thought von Neumann had resolved the issue of hiddenvariables once and for all, and research efforts of physicists moved towarddeveloping and applying quantum theory in new regimes. For example,relativistic quantum theory in which the number of particles is no longerconserved, led to the development of quantum field theory. However, inthe 1950s and into the 1960s a few physicists began to question the proof’s

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Figure 7.4: John Bell

validity to describe all possible hidden-variable models. It slowly became ap-parent that hidden-variable models were not impossible, which re-awakenedthe question about the interpretation of quantum theory. We have alreadydiscussed the ideas of David Bohm, who examined the EPR argument andreframed it in terms of pairs of two-level systems. However, it was not until1964 that John Bell, a physicist from Northern Ireland, showed that onecould experimentally verify the validity of quantum theory versus all possi-ble local-hidden-variable models.

We will not go through the full derivation of Bell’s result, but give anillustrative example. Suppose that we split a pair of particles that haveinteracted and give one to Alice and the other to Bob. Each measures twodifferent observables A1, A2 and B1, B2, respectively. Each observable hastwo outcomes that we label +1 and −1. Consider the quantity

C = A1

(B1 + B2

)+ A2

(B1 − B2

). (7.9)

If we assume that the values of measurement outcomes are local propertiesof each particle, as in EPR, then the quantity B1 + B2 can only take on thevalues −2, 0, or +2 in a given measurement, and the corresponding quantityB1−B2 takes on the values 0, ±2, or 0 respectively. This implies that C canonly take on the value ±2 on a shot by shot basis. Thus any local-hidden-variable model will result in the following bound on the average value of theabsolute value of C ∣∣∣〈C〉∣∣∣ ≤ 2. (7.10)

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This inequality is known as a Bell inequality, and it sets an upper boundon the quantity |〈C〉| for measurement outcomes predicted by local-hidden-variable models.

Now let us examine what quantum theory predicts. Suppose that Aliceand Bob choose to measure photon polarization and that the state used isthe same Bell state as in Eq. (7.1), i.e.

|Φ−〉 =1√2

(|R,L〉1,2 − |L,R〉1,2) .

Let Alice measure the polarization in the ± basis for measurement 1

A1 = Sx

= |+〉〈+| − |−〉〈−|,= |H〉〈V|+ |V〉〈H|, (7.11)

and the polarization in the RL basis for measurement 2

A2 = Sy,

= |R〉〈R| − |L〉〈L|,= −i (|H〉〈V| − |V〉〈H|) (7.12)

while Bob performs measurements in elliptical polarizations given by mea-surement operators B1 = (Sx + Sy)/

√2 and B2 = (Sx − Sy)/

√2. We see

that the combinations of Bob’s measurements can be written

B1 + B2 =√

2Sx, (7.13)

andB1 − B2 =

√2Sy. (7.14)

Thus the quantity C can be expressed as

C =√

2(SxASxB + SyASyB

), (7.15)

where we have explicitly included the label on Alice and Bob’s measurementoperators. We can rewrite this in Dirac notation as

C = 2√

2 (|HV〉〈VH|+ |VH〉〈HV|) , (7.16)

where the state |HV 〉 implies Alice has H polarization and Bob has V po-larization. We have used the following simplification of the measurementoperators

SxASxB = (|H〉〈V|+ |V〉〈H|) (|H〉〈V|+ |V〉〈H|)= |HH〉〈VV|+ |HV〉〈VH|+ |VH〉〈HV|+ |VV〉〈HH|, (7.17)

43

and

SyASyB = − (|H〉〈V| − |V〉〈H|) (|H〉〈V| − |V〉〈H|)= − (|HH〉〈VV| − |HV〉〈VH| − |VH〉〈HV|+ |VV〉〈HH|) .(7.18)

For the Bell state in Eq. (7.1), the expectation value of C exceeds the valuegiven by any local hidden-variable model

〈C〉 = 2√

2〈Φ−| (|HV〉〈VH|+ |VH〉〈HV|) |Φ−〉,=

1

22√

2 (〈HV| − 〈VH|) (|HV〉〈VH|+ |VH〉〈HV|) (|HV〉 − |VH〉) ,

= −2√

2. (7.19)

Thus, we see that quantum theory predicts a very different outcome forsuch an experiment, i.e. |〈C〉| = 2

√2, which is significantly larger than that

predicted by any local hidden-variable theory.

7.3.1 Experimental evidence

It took nearly twenty years from the time that Bell made these predictionsto the point at which experimentalists could measure these kinds of corre-lations on individual pairs of quantum particles. The first comprehensiveexperiments to test the general form of Bell’s inequality were performedby Alain Aspect and his co-workers Philippe Grangier, Gerard Roger, andJean Dalibard at the Institut d’Optique Theoretique et Appliquee, Univer-site Paris-Sud in Orsay, in 1981 and 1982. They made use of cascadedemission of photon pairs from calcium atoms as a source of polarizationentangled photon pairs.

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Chapter 8

Quantum technologies

8.1 Introduction

Research in quantum physics over the past thirty years has continued toexamine its foundations, for example work aimed to quantify the amountof entanglement between two systems has led to the discovery of a differ-ent type of quantum correlations from entanglement itself called quantumdiscord. In addition to exploring the fundamental principles of the quan-tum theory, significant effort has been put into examining the implicationsthat quantum physics has on different applications. Broadly speaking, theaim of these quantum technologies or quantum applications is to utilize thenon-classical features of quantum physics to realize improved performanceof various real-world problems or tasks, or potentially develop technologieswith no classical counterpart. For example, quantum-enhanced sensing har-nesses the sensitivity of quantum superpositions to increase the precision ofmeasurements.

One can break these quantum technologies into four basic areas: Quan-tum simulation, quantum-enhanced sensing / metrology, quantum control,and quantum information processing. The goal of quantum simulation is tomodel the behavior of an experimentally challenging and computationallydifficult to model quantum system by using a quantum system that is ex-perimentally simpler to manipulate and measure.1 For example, quantumstates of photons are relatively easy to create, manipulate and measure com-pared to electron systems. So one could potentially use quantum photonicstates of light to simulate the behavior of many electrons. Quantum controluses auxiliary quantum systems and the ideas of quantum measurement, inwhich a measurement acts to modify the state of a quantum system afterthe measurement (the measurement collapse), to create and control quan-

1R. P. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys., 21,467–488, (1982)

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tum systems. One example of interest here is the use of “shaped” pulses oflight to drive chemical reactions that would otherwise not occur. Central toall quantum technologies is the role of quantum superpositions.

Much effort has been devoted to the understanding and application ofinformation theory for quantum systems, known collectively as quantum in-formation. Here researchers have explored the implications that quantumphysics has on information processing (quantum computing), and communi-cations (quantum communications). In computation, quantum informationenables efficient algorithms for factoring large numbers,2 which is believedto be difficult for classical computers (difficulty is measured in the time ittakes to perform the computation). Quantum computation also improvesthe efficiency of an unstructured database search,3 which makes it possibleto solve significantly larger optimization problems such as the schedulingand traveling salesman problems. In communications, quantum informa-tion provides a method for communicating in secret.4 The security such“quantum cryptography” schemes is guaranteed because any eavesdroppingattempts necessarily introduce disturbance to the exchanged quantum sys-tem that carries the message.

Here we will not discuss the details of these quantum technologies. Thereare several good books and articles devoted to quantum information.5 In-stead we aim to highlight two examples in the following sections. We willdiscuss two quantum applications, the quantum random number generatorand quantum cryptography. However, we first take a brief glance at classicalinformation theory.

2P. W. Shor, “Algorithms for quantum computation: Discrete logarithms and factor-ing,” In Proceedings of the 35th Annual Symposium on Foundations of Computer Science,pages 124–134, Los Alamitos, California, IEEE Press (1994) and P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,”SIAM J. Comput., 26, 1484–1509 (1997).

3L. K. Grover, “A fast quantum mechanical algorithm for database search,” In Pro-ceedings of the 28th Annual ACM Symposium on the Theory of Computation, pages212–219, New York, ACM press (1996).

4S. Wiesner, “Conjugate coding,” Sigact News, (original manuscript circa1969), 15,78–88, (1983) and C. H. Bennett, G. Brassard, S. Breidbart, and S. Wiesner, “Quantumcryptography, or unforgeable subway tokens,” In Advances in Cryptology: Proceedings ofCrypto82, pages 267–275, Plenum Press, (1982).

5See for example: M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quan-tum Information,” Cambridge University Press (2001), Valerio Scarani, “Quantum Infor-mation: Primitive Notions and Quantum Correlations,”http://arxiv.org/pdf/0910.4222,Todd Brun, “Lecture Notes on Quantum Information Processing,”http://almaak.usc.edu/~tbrun/Course/index.html,and E. Knill et al, “Introduction to Quantum Information Processing,”http://arxiv.org/pdf/quant-ph/020717v1.

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8.2 Information

Figure 8.1: Claude E. Shannon

The roots of modern information theory began with a paper writtenby the young American Claude E. Shannon in 1948 while working at BellLaboratories. His seminal article entitled A Mathematical Theory of Com-munication addressed the fundamental limits on signal processing operationssuch as data compression. One of the key developments was the introduc-tion of a measure of the amount of information in a message, known as theShannon entropy, which is usually expressed as the number of binary digits(bits) needed to store or communicate one symbol in a message. Typicallyone talks of a bit in terms of the two possible logical values ‘0’ and ‘1’, whichmust be encoded into a physical system. In modern computers such binaryvalues are often given by different voltage levels in an electronic circuit. Thisphysical encoding was stressed by Rolf Landauer in his 1993 paper entitledInformation is Physical.

A message x, consisting of n symbols from an alphabet with m possiblesymbols can be thought of as a sequence or list of symbols. For examplex = (aopiuenbpj) is a message with 10 symbols drawn from the Englishalphabet, which has 26 symbols. This is basically what a word is in everydaylanguage. The information contained in such a message is given by theShannon entropy

H(x) = −nm∑i=1

p(xi) log2 p(xi) (8.1)

where p(xi) is the probability that the ith symbol takes on that particular

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value xi. In many cases, the probability for any symbol is the same for allsymbols in the alphabet from which the message is drawn. For example, theEnglish alphabet contains 26 letters (or symbols), which implies a probabil-ity of p = 1/26 for any letter. The number of bits associated with a messagecomprised of say n letters from an alphabet consisting of m equally likelysymbols is thus given by

H(x(n)) = −nm∑i=1

p log2 p,

= −nmp log2 p,

= −nm 1

mlog2

(1

m

),

= −n(log2(1)− log2(m)),

= n log2(m), (8.2)

where in the going to the last line we used the fact that log2(1) = 0. Wesee that each symbol contributes mp log2 p = log2(m) information to themessage for equal weighting. Let us just double check that this makes sensein terms of a binary alphabet in which we expect one bit per symbol (bydefinition). The information per symbol in this case is log2(2) = 1, which isin line with the definition.

In quantum information, there is an analogous measure of informationquantified by what is called the von Neumann entropy. The natural unit ofquantum information is the quantum bit, or qubit. Following Landauer’snotion that information is physical, a qubit must be encoded in a two-levelquantum system. One of the key features of quantum bits is that not onlycan they take on the ‘logical’ values of |0〉 and |1〉, but any superposition aswell, e.g. |+〉 = (|0〉+ |1〉)/

√2.

8.3 Quantum random number generator

The ability to produce random numbers is necessary for several applications,ranging from modeling stochastic processes such as the stock market to cryp-tographic schemes. There are two basic approaches to generate a sequenceof random numbers: software and physical generators. Software-based gen-erators produce so-called pseudo-random numbers, because the computersthat run the algorithms are deterministic, given a certain input it will pro-duce the same output time and again, and therefore the number generatedis not truly random. Physical generators are based on random processesthat occur in nature. For example, one could potentially think of flipping acoin to generate a string of random binary digits (or bits). This process isnot truly random though since the evolution of the coin can in principle be

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determined if one knows the full initial conditions of the coin. To circumventthis quantum physicists have developed the concept of a quantum randomnumber generator that is based upon the random outcome of a measurement.

Figure 8.2: Quantum random number generator: Simplified experimentalsetup. A single photon source emits one photon at a time, which are directedto a 50:50 beam splitter. The photon will be either reflected or transmittedrandomly. Detection in the transmitted port of the beam splitter corre-sponds to a logical ‘0’, while detection in the reflected port corresponds toa logical ‘1’.

To generate a string of random bits, one can use a single-photon sourcethat emits one photon at a time. The photons are sent to a 50:50 beamsplitter where they can either be transmitted or reflected as depicted in Fig.8.2. One then detects the photon at the output of the beam splitter ineither the transmitted or reflected beam. A transmitted photon is assignedthe logical bit value ‘0’, while a reflected photon is assigned the logical bitvalue of ‘1’. The probability of each measurement outcome (transmittedor reflected) is 1/2, but the outcome of a given measurement is completelyrandom. Repeating this experiment n times gives a string of n randombinary digits (bits). It is really that simple. This quantum technology is oneof the most successful commercially available products.6 One of the largestusers of these devices are Internet gambling websites that must ensure theshuffling of their virtual card decks is truly random.

8.4 Quantum key distribution

In our modern, technology-based society in which commerce is based upon‘virtual transactions’ in which no money is exchanged, but rather a ledger,secure communications is extremely important. Cryptography, the studyand practice of techniques for secure communications, aims to develop pro-

6ID Quantique makes these as well as quantum cryptography systems:http://www.idquantique.com/true-random-number-generator/products-overview.

html

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tocols in which communicating parties can share information without anexternal party gaining any information about the message. Modern cryp-tography is primarily based on the assumption that factoring large numbersis a “computationally difficult” problem. Roughly speaking, if I increase thesize of the number to be factored, the time it takes a computer to factorthis number grows exponentially with the size of the number. However, aswe mentioned in the introduction, if a quantum computer is built, one of itskey applications is the factorization of numbers (Shor’s algorithm). I wouldnot be too alarmed at the moment, since the largest number to be factoredusing this quantum algorithm is 15 = 3× 5.

Even if a quantum computer were to be realized, all is not lost. Inthe early 1980s researchers developed a cryptographic scheme based uponthe incompatibility of quantum measurements in different bases. The basicscheme, known as BB84 (after its two authors Charles Bennett and GillesBrassard and the year it was published, 1984), involves two parties, Aliceand Bob who would like to share a secret message. To do this Alice canencrypt the message, i.e. converting the original message into apparentnonsense, and send it to Bob via an open public channel. To decrypt thetransmitted message requires the encryption key to be shared between thesender (Alice) and receiver (Bob), which is done first before the encryptedmessage is sent. To ensure security the key can be used only once and isthus often called a ‘one-time pad’. A simple encryption technique involvesperforming a exclusive-OR (XOR) operation between the original messageand the key. The XOR maps two binary digits A and B according to thefollowing truth table

A B XOR

0 0 0

0 1 1

1 0 1

1 1 0

in other words, if A and B are different it returns a logical 1, and if theyare the same it returns a logical 0. In Boolean logic, in which logical 0 isassociated with “False” and logical 1 is associated with “True”, the XORreturns True only when the inputs have one True and on False, but not bothTrue. The latter case occurs for the OR operation. As an example, supposeAlice wants to send the 8-bit message 10010100. She will need to share withBob an 8-bit key, which will be shared between them prior to sending theencrypted message. If the shared key is 01001100, then the encoded messagecan be found by applying the XOR truth table giving

10010100 original message01001100 key

11011000 XOR encrypted message

(8.3)

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To decrypt the message, Bob only needs to apply the XOR operation to theencrypted message using the same key that alice used to encode the originalmessage:

11011000 encrypted message at receiver01001100 same key

10010100 XOR decoded message

(8.4)

The success of quantum cryptography then lies in the ability to distribute(actually generate) the encryption key without an eavesdropper gaining anyinformation about the key. The sharing of a secret key between Alice andBob using quantum systems to carry the key is known as quantum key dis-tribution (QKD). In the BB84 approach to QKD, Alice sends a series ofsingle photons with well defined polarization in one of two mutually exclu-sive bases. For example, she sends in the horizontal-vertical basis, which wedenote ⊕, or the diagonal-anti-diagonal basis, denoted ⊗. Alice randomlychooses both the basis and state sent to Bob, and keeps track of both, either⊕ and H(V) or ⊗ and +(−). On the receiver end, Bob randomly chooses abasis in which to measure the photons sent to him, recording both the basisand the measurement outcome. They both agree in advance that in the HV(±) basis H(+) corresponds to a logical 0 and V(−) corresponds to a logical1. Alice and Bob then openly post via a ‘publicly accessible’ channel theirpreparation and measurement bases, respectively, as depicted in Fig. 8.3.Note that they do not discuss what state was prepared or measured, onlythe preparation and measurement bases.

If the channel that the photons propagate along is perfect, i.e. it doesnot add any noise, then Bob should measure exactly the state that Aliceprepared whenever the preparation and measurement bases are the same.However, when they differ, owing to the mutual exclusivity of the two bases,Bob gains no information about the polarization state of the photon that Al-ice prepared. For example, if Alice prepares a photon in the |V〉 polarizationstate and Bob measures in the ± basis, he has equal probability to measurethe photon in |+〉 and |−〉. However, he gains no information about the factthat the photon that Alice sent was prepared in the vertical state, since ahorizontal photon leads to exactly the same measurement outcomes in the± basis. For this reason, they keep only the data in which the photons wereprepared and measured in the same basis and discard the data in which thepreparation and measurement bases differ. Now, if an eavesdropper wereto try to measure the photon polarization in any way, it would inevitablydisturb the polarization state of the photon owing to the measurement col-lapse in quantum mechanics. This would appear as errors in the otherwiseperfect correlations between Alice and Bob’s prepared and measured statesin the same basis. To determine if it is ‘safe’ to use this data to establish akey, Alice and Bob then openly share a random portion of their remaining,

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!"#$%&'()!'

*+('

,*-./'"$',0-1/' ,*-./'"$',0-1/'

*+('

($&23$34"5',!&56&$/' 7&38#$&9&5:',;&%&<=&$/'

(#>?<%'%@355&?'

Figure 8.3: Polarization based quantum key distribution based upon theBB84 (prepare and send) protocol. The sender (Alice) prepares a particularpolarization state in either the HV or ± basis randomly using a half-waveplate after an ideal horizontally polarized single photon source. The receiver(Bob) randomly chooses to measure in either the HV or ± bases. Thecompare their bases via a public channel.

un-discarded data to determine if there was an eavesdropper trying to gainsome information. If they do not detect any disturbance, then they knowthat the channel was secure and can use the remaining data to establish theencryption key. This is depicted in the table below.

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Alice Bob

PB PS MB MS K E Key

⊕ H ⊕ H K *

⊗ + ⊗ + K 0

⊗ - ⊗ - K 1

⊗ - ⊗ - K 1

⊕ V ⊕ V K *

⊗ + ⊕ V

⊗ + ⊗ + K 0

⊕ H ⊗ -

⊗ - ⊕ H

⊗ - ⊗ - K *

⊕ V ⊗ +

⊗ - ⊕ V

⊕ H ⊕ H K 0

⊕ H ⊗ -

⊕ V ⊕ V K *

⊕ V ⊗ +

⊕ V ⊗ -

⊗ + ⊗ + K 0

⊗ - ⊕ V

⊗ - ⊕ H

⊕ H ⊕ H K 0

⊗ + ⊕ V

⊕ V ⊗ -

⊗ - ⊗ - K 1

⊗ + ⊗ + K *

⊕ H ⊕ H K 0

⊕ H ⊕ H K 0

⊗ - ⊕ V

⊕ V ⊗ -

In this table we show the preparation basis (PB) and prepared state (PS)sent by Alice, the measurement basis (MB) and measured state (MS) atthe receiver run by Bob. Alice and Bob only keep (K) the data in whichboth the preparation and measurement bases are the same. They choose asubset of the remaining data to share publicly to determine the presence ofan eavesdropper (E). Here there are no errors, and thus they can constructa secure key from the remaining data.

For an ideal channel, one expects to have perfect correlations in thetransmitted and detected states when the preparation and measurement

53

!"#$%&'()!'

*+('

,*-./'"$',0-1/' ,*-./'"$',0-1/'

*+('

($&23$34"5',!&56&$/' 7&38#$&9&5:',;&%&<=&$/'

(#>?<%'%@355&?'

A3=&86$"22&$'

Figure 8.4: Polarization based quantum key distribution based upon theBB84 (prepare and send) protocol. The sender (Alice) prepares a particularpolarization state in either the HV or ± basis randomly using a half-waveplate after an ideal horizontally polarized single photon source. The receiver(Bob) randomly chooses to measure in either the HV or ± bases. Thecompare their bases via a public channel. Any attempts by an eavesdropper(Eve the kitten looks so innocent) to measure the state of the photon willinevitably introduce noise that can be detected by Alice and Bob if theyshare a small set of their data over the public channel. If the detect aneavesdropper, they will not use channel to create the secret key.

bases are matched. The best strategy an eavesdropper (Eve) can employ toextract some information without being observed is to detect the photon andthen resend another photon in its place. This is known as the ‘intercept-resend’ attack. At best Eve can randomly choose to measure in the twobases (⊕ and ⊗) and resend the same state that was detected. Half thetime Eve will choose the same basis in which Alice and Bob do not discardtheir data, while the other half of the time she gets it wrong. Of the timesEve chooses the wrong basis, on average Bob will detect the ‘wrong’ state,since Eve resends the photon in the incommensurate basis. Thus 25% of thetime Bob will detect the ‘wrong’ state if Eve attempts to try and extractpart of the key. If Alice and Bob detect an error rate nearing 25%, thenthey known Eve is eavesdropping. Clearly, if the channel is not perfect (forexample there are losses, or it changes the polarization state of the photonsslightly), then there will be a reduction in the allowed error threshold.

This type of QKD system has moved from a laboratory setting intothe commercial market.7 However, due to the relatively low transmission of

7See the websites of the two leading manufacturers: ID Quantiquehttp://www.idquantique.com/

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single photons and low efficiencies of single photon detectors, this technologyis limited to short distance communications (on the order of a few tens ofkilometers). There are other approaches to QKD, but all are essentiallybased upon the idea that measurement and preparation in the same basislead to correlated results, while preparation and measurement in mutuallyexclusive bases give no information.

and MagiQhttp://www.magiqtech.com.

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