# Bell inequality & entanglement

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Bell inequality & entanglement

The EPR argument (1935)based on three premises:

1. Some QM predictions concerning observations on a certain type of system, consisting of two spatially separated particles, are correct.

2. A very reasonable criterion of the existence of ‘an element of physical reality’ is proposed: ’if, without any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity’

3. There is no action-at-a-distance in nature.

EPR paradox

EPR paradox

• Before making the measurement on spin 1 (in z direction) the state vector of the system is:

• After measurement on particle 1, (for argument’s sake say we measured spin down), the state of particle 2 is:

EPR paradox

• Since there is no longer an interaction between particle 1 and 2, and since we haven’t measured anything of particle 2, we can say that it’s state before the measurement is the same as after:

EPR paradox

• We could apply the same argument if we have measured the spin in the x direction and receive:

In other words: it is possible to assign two different state vectors to the same reality!

Bell’s theorem

• If premise 1 is taken to assert that all quantum mechanical predictions are correct, then Bell’s theorem has shown it to be inconsistent with premises 2 & 3.

Deterministic local hidden variables and Bell’s theorem

• Bohm’s theorem: spatially separated spin ½ particles produced in singlet state:

• All components of spin of each particle are definite, which of course is not so in QM description => HV theory seems to be required.

• The question asked by Bell is whether the peculiar non-locality exhibited by HV models is a generic characteristic of HV theories that agree with the statistical predictions by QM.

• He proved the answer was: YES.

)]2(ˆ)1(ˆ)2(ˆ)1(ˆ[2

1 nunununu

LHV and Bell’s theorem•Let be the result of a measurement of the spin component of particle 1 of the pair along the direction and the result of a measurement of the spin component of particle 2 of the pair along the direction•We denote a unit spin as hence , = +1•The expectation value of this observable is:

•When the analyzers are parallel we have:

•The EPR premise 2 assures us that if we measure A we know B

aAˆa

bB ˆ

b2/h aAˆbB ˆ

bababaE ˆˆˆˆ)]ˆ,ˆ([21

1)]ˆ,ˆ([ aaE

Local Hidden Variables defined.

• Since QM state does not determine the result of an individual measurement, this fact suggests that there exists a more complete specification of the state in which this determinism is manifest. We denote this state by

• Let be the space of these states• We represent the distribution function for these

states by

1 d

Bell’s definition:

• A deterministic hidden variable theory is local if for all and and all we have:

• The meaning of this is that once the state is specified and the particles have separated measurements of A can depend on and but not

• The expectation value is taken to be:

a b )()())(( ˆˆˆˆ

baba BABA

a

b

dBAbaEba )()(, ˆˆ

Proof of Bell’s inequality

1)]ˆ,ˆ([ aaE Holds if and only if )()( ˆˆ aa BA

Hence:

dAAAA

dAAAAcaEbaE

cbba

caba

)]()(1)[()(

)()()()()ˆ,ˆ()ˆ,ˆ(

ˆˆˆˆ

ˆˆˆˆ

Since A,B=+1

dAAcaEbaE cb )]()(1[)ˆ,ˆ()ˆ,ˆ( ˆˆ

Proof of Bell’s inequality

1 d

dBAbaEba )()(, ˆˆ

)()( ˆˆ aa BA

Using:

We have:

)ˆ,ˆ(1)ˆ,ˆ()ˆ,ˆ( abEcaEbaE

Violation of Bell inequality

• Taking to be coplanar with making an angle of with , and making an angle of

with both and then:

Which gives: and

cba ˆ,ˆ,ˆ c3/2 a b

3/ a c

2/1ˆˆ2/1ˆˆˆˆ

cacbba

1)ˆ,ˆ()ˆ,ˆ( caEbaE 2/1)ˆ,ˆ(1 abE

What is the meaning of violating the Bell inequality?

• No deterministic hidden variables theory satisfying the locality condition and

can agree with all of the predictions by quantum mechanics concerning spins of a pair of spin-1/2 particles in the singlet case.

In other words: once Bell’s inequality is violated we must abandon either locality or reality!

1)]ˆ,ˆ([ aaE

Requirements for a general experiment test

• Let us consider the following apparatus:

Experiment requirements

• The QM predictions take the following form:

im

iM

i

QM

QM

QM

fp

fp

nFgfp

2221

1111

212112112

21][

21][

)]cos([41)]([

iiM

im

if

g

F

Effective quantum efficiency of the detectormax & min transmission of the analyzersCollimator efficiency (probability that appropriate emission enters apparatus 1 or 2Conditional probability that if emission 1 enters apparatus 1 then emission 2 enters apparatus 2Measure of the initial state purity n=1 for fermions and n=2 for bosons

Experiment requirements

• Taking the following assumptions:

2]1)/(2[

4/,,

2

212121

Fg

nff

If the experimental values are within the domain of the above inequality, then we can distinguish between QM prediction

and inequalities.

Summery: for direct test of inequality the requirements are:

• A source must emit pairs of discrete-state systems, which can be detected with high efficiency.

• QM must predict strong correlations of the relevant observables of each pair, and the pairs must have high QM purity.

• Analyzers must have extremely high fidelity to allow transmittance of desired states and rejections of undesired.

• The collimators must have high transmittance and not depolarize the emissions.

• A source must produce the systems via 2-body decay, or else g becomes g<<1.

• For locality’s sake: Analyzer parameters must be changed while particles are in flight. (no information exchange between detectors.

CHSH

• Since no idealized system exists, one can abandon the requirement:

• CHSH arrived at the following inequality:

2)','()',(),'(),( baEbaEbaEbaE

1)]ˆ,ˆ([ aaE

Which was violated by Alain’s experiment =>Proof of non-local correlations occur on a time scale faster than the speed of light.

experiments

experiments• The third Alain Aspect experiment: faster than light correlation.• Using the following setup:

Crash course in information theory…

• Ensamble of quantum states with probability

i

ip

We can define the density operator as:

ii

iip

Crash course in information theory…

• A quantum system whose state is known exactly is said to be in pure state, otherwise it is said to be in mixed state.

• A pure state satisfies• A mixed state satisfies

1)( 2 tr

1)( 2 tr

Schmidt de-composition

Crash course in information theory…

• Shannon Entropy: quantifies how much information we gain, on average, on a random variable X, or the amount of uncertainty before measuring the value of X.

• If we know the probability distribution of X:

then the Shannon Entropy associated with it is: ni pp .....

Crash course in information theory…

• Definition of variable X obeys:

typical

Typical sources are sources which are highly likely to occur.ix

Von Neumann entropy

Entanglement distillation and dilution

• Suppose we are supplied not with one copy of a state , but with a large number of it.

Entanglement distillation is how many copies of a pure state we can convert into entangled Bell state. Entanglement dilution is the reverse process.

Entanglement distillation and dilution

• Defining a specific bell state as a ‘standard unit’ of entanglement, we can quantify entanglement.

• Defining an integer n which represents the number of Bell states, and an integer m representing the number of pure states that can be produced, then the limiting ratio n/m is the entanglement of formation of the state

Setting the limits

• Suppose an entangled state has a Schmidt decomposition

Setting the limits• An m-fold tensor product can be defined

Alice and Bob live by the limits

=> We have an upper limit for entanglement formation!

In a similar manner it was shown that there is a lower limit for entanglement distillation which is also

bibliography

• Quantum optics- an introduction/ M. Fox p.304-323• Quantum optics/ M.Scully & M.Tsubery p.528-550.• Bell’s theorem: experimental tests and implications, J.

Clauser & A. Shimony, Rep. Prog. Phys, Vol.41, 1978.• Experimetal tests of realistic local theories via Bell’s

theorem, PRL vol.47, nu.7, 1981 A.Aspect et al.• Quantum computation and quantum information,

M.Nielasen and I.Chuang, p.137,580, 607.