Entanglement and Bell's inequality

MotivationMermin’s essay

The GHZ experimentSummary

Measuring the MetaphysicalImplications Of The Bell’s Inequality

Gaurav Gautam

Department of PhysicsIndian Institute of Technology, Kanpur

Communications Lab, 2013-14

Gaurav Gautam Measuring the Metaphysical



Outline

1 MotivationThe metaphysical-Quantum Theory and The EPRParadoxThe measurable-EPR paradox and Bell’s inequality

2 Mermin’s essayBell’s idea in disguise: The two flashersBell’s inequality in its full gloryLooking back at the flashers in the new light

3 The GHZ experimentWhat and whyThe three flashersQuantum Mechanical explanation




The metaphysical-Quantum Theory and The EPR ParadoxThe measurable-EPR paradox and Bell’s inequality

Outline








Wave-functions Are Too Peculiar to Be Real

Quantum description of state - |Ψ〉Physical interpretation is onlyprobabilisticWave function can be asuperposition of states.EPR asked if this is a truedescription of nature, or is Quantumtheory insufficientFor many years this question wasphilosophical





Then EPR Claimed Quantum Theory is Incomplete

Consider the by now familiar state|Ψ〉 = | ↑↓〉 − | ↓↑〉QM claims that, e− and e+ haveopposite spins, but neither’s spincan be ’predicted’This gives rise to two views

The realist view-this inability is ashortcoming of QM.The orthodox view-this is the waynature is

The EPR claim: Only realist view iscorrect, because the universe islocal





Outline








Bell Came Up With a Test For EPR’s Theory

Acc. EPR, since no signal couldhave reached the positron and thedata are perfectly correlated. Theonly solution is that the states werepredefined, and QM insufficient.However there was no way to testthisBell was the first to formulate asituation where, the presence orabsence of predefined states wouldlead to different resultsAnd the results agreed with QM.




Bell’s idea in disguise: The two flashersBell’s inequality in its full gloryLooking back at the flashers in the new light

Outline








Mermin’s Way To Describe Bell’s Inequality

The flasher has three settings 1,2and 3, and two lights Red andGreen.Switches are set randomly, andthere is no connectionUpon running we see—

Same switches (a) yield same ledsall the timeDifferent switches (b) yield sameleds a quarter of the times anddifferent leds in the rest





The EPR Theory Fails

There is no connection betweenflashers, except the particles theyreceiveCase (a) must mean that theparticles carry the same informationto the two flashersPerfect, but from the figurealongside, the above situation willmake case (b) impossibleSo the particles are not carryingfixed information. How then, doescase (a) happen?





Inside the Flashers: Making Sense of the Mess

The flashers have Stern-Gerlachmagnets. When the flasher 1,detects | ↑〉 it flashes red and greenotherwise. The other flasher followsthe opposite convention.The magnets are aligned at 120◦

with each otherThe probability of detection of upspin along another direction (acc.QM) is cos2 ( θ

2

), which for 120◦ is 1

4

Explanation will be given later.





Outline








Bell’s Inequality For Physicists

Bell suggested measuring theaverage value of the product ofspins along two directions a and bIf a theory can predict which spinwould be detected then—

For one detector- A(λ, a) = ±1For other detector- B(λ, b) = ±1The anti-correlation observed inthe data means A(λ, a) = −B(λ, a)Represent the average spinproducts as P(a, b)

According to QM P(a, b) = −a · bProof if time permits





Bell’s Inequality For Physicists: Continued

In the framework of a hidden variable theory—P(a, b) =

∫ρ(λ)A(λ, a)B(λ, b)dλ

So, P(a, b)− P(a, c) =

−∫ρ(λ)

[A(λ, a)A(λ, b)− A(λ, a)A(λ, c)

]dλ

After some manipulation, and using the fact that in thistheory [A(λ, r)]2 = 1, it can be shown that—

Bell’s Inequality

|P(a, b)− P(a, c)| ≤ 1 + P(b, c)

This is the famous Bell’s inequality. Note that it was derivedusing a general hidden variable theory framework.





Demonatrating The Violation of Bell’s Inequality

Quantum Mechanics

P(a, b) = −a · b = 0

P(a, c) = P(b, c) = −0.707

Hidden variables-In order to satisfyBell’s inequality we must have

0.707 ≤ 1− 0.707

Therefore, hidden variable theoriesand quantum mechanics areincompatible.





Outline








The Inadequacy of Classical Description

Now we can see what is happeningwith the flashersin case a the detectors areanti-aligned. And you can ponderover the possibility of prewritteninstruction sets.in case b however, there is noclassical way of thinking. The resultemerges naturally if we look at howspin measurement works.





How QM Describes Everything Perfectly

Define the spin operator Sz witheigenvalues ±1, to be the operatorcorresponding to the spin along zaxis with the eigenstates | ↑〉 and| ↑〉.Then the operator corresponding tothe spin along an arbitrary axis at anangle θ with z is Sr , witheigenstates—

|+〉 = cos(θ

2

)| ↑〉+ eiφsin

(θ

2

)| ↓〉





How QM Describes Everything Perfectly: Continued

And the second eigenstate—

|−〉 = eiφsin(θ

2

)| ↑〉+ cos

(θ

2

)| ↓〉

So, if the electron is measured in|+〉, state, the state of positronbecomes |−〉. And the probability toget the state | ↓〉 becomes cos2 ( θ

2

).

In case of our flashers, θ = 120◦,and so cos2( θ2) = 1

4 .




What and whyThe three flashersQuantum Mechanical explanation

Outline








GHZ Experiment: Who, What and Why

What-Till now there were twoentangled particles. The GHZexperiment deals with threeentangled particlesWhy -The Bell’s test works with alarge set of data and statisticalmeasurements.

The two types of runs (a) and (b),don’t admit a common explanationin terms of predetermined states.Here the discrepancy can beshown with a single run.





Outline








Describing the GHZ Experiment

This time there are three flasherswith two settings eachWe look at a subset of thepossibilities, those in which an oddnumber of flashers are set to 1.

If only one flasher at one-an oddnumber of red flashesIf all three are at 1, odd number ofred flashes is never seen

Again since there is no connection,the particles must be telling theflashers what to do





Trying to Handle GHZ With Predetermined States

Let us look at what can be thepredetermined information beingpassed (equivalent of (a))Observe that every kind of runrequires an odd number of redflashes if the flashers are at 111.But this is never observed.Every single run demonstrates whythere can’t be predeterminedmessages.





Outline








Inside the Flashers And the Entangled State

The flashers are againStern-Gerlach magnets. The z-axesfor the particles are along thetrajectoriesThe detectors measure spins alongthe x and y axesThe spin operators are-σx , σy ANDσz .We consider three commutinghermitian operators-σa

xσbyσ

cy ,σa

yσbxσ

cy

and σayσ

byσ

cx .

They have common eigenstates,withλ = ±1.





QM Description of the Mermin Flashers

The state |Ψ〉 = | ↑↑↑〉 − | ↓↓↓〉 has eigenvalue +1.+1 flashes red and -1 flashes green, therefore there mustbe an even number of green lights, because whichevermeasurement we make– σa

xσbyσ

cy ,σa

yσbxσ

cy or σa

yσbyσ

cx we

must get +1 for |Ψ〉.Now if we go back to the setting 111, the correspondingoperator will be σa

xσbxσ

cx .

It is easily seen that —

σaxσ

bxσ

cx = −(σa

xσbyσ

cy )(σa

yσbxσ

cy )(σa

yσbyσ

cx )





QM Description of the Mermin Flashers: Continued

The state |Ψ〉 = | ↑↑↑〉 − | ↓↓↓〉, is thus an eigenstate for theconfiguration 111 with eigenvalue -1. Since, -1 flashesgreen and +1 flashes red, this predicts an even number ofred flashes.Thus, QM predicts the outcomes of the experimentcorrectly (even red flashes), but the assumption that thereare predefined states runs into contradictions.




Summary

EPR claimed that only the realist view is sustainable,based on grounds of locality.Bell’s inequality provides a framework in which the realistand orthodox views give different results, making theQuantum theory incompatible with the local hidden variabletheories. A better test is given by the GHZ experiment.We live in a universe where weird things like particles insuperposition of states exist.


Appendix For Further Reading

For Further Reading I

D.J.GriffithsIntroduction to Quantum Mechanics.Pearson,2012

N. David MerminBringing home the atomic world:Quantum mysteries foranybodyAmerican Journal of Physics(940),Oct 1981

N.David MerminQuantum mysteries revisitedAmerican Journal of Physics(731),Aug 1990


Entanglement and Bell's inequality

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