USEQIP BellInequalities Resch

7/27/2019 USEQIP BellInequalities Resch

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USEQIP June 2009 1

Bells inequalities and quantum

optics

Kevin Resch

Institute for Quantum Computing

Dept. of Physics & Astronomy

University of Waterloo

USEQIP June 2009 2

Outline

The Einstein-Podolsky-Rosen paradox

Bohms refinement

The CHSH Bell inequality

Measuring the CHSH Bell parameter in the

lab

Further suggested reading: experiments,

loopholes

USEQIP June 2009 3

EPR

USEQIP June 2009 4

EPR


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EPR

They showed that the two-particle state:

(x1, x2) =Z

e

i(x1x2+x0)p

~

dp

has perfect correlations in both position

and momentum.

It is a simultaneous eigenstate of the

operators

x1 x2 p1+p2and

USEQIP June 2009 6

EPRs argument

If we separate the two particles far apart,

then a measurement on particle 1 cannot

disturb particle 2 If we randomly decide at the last moment

to measure the position of particle 1 (and

find outcome ), then we know with

certainty the position of particle 2,

Thus the position of particle 2 is an

element of physical reality

x0

x0 +x0

USEQIP June 2009 7

EPRs argument

We could also have randomly decided at

the last moment to measure the

momentum of particle 1 (and find

outcome, ) then we would have knownwith certainty the momentum of particle 2,

Thus the position of particle 2 is also an

element of physical reality

p0

p0

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EPRs argument

But this is a contradiction with quantum

mechanics, since the uncertainty principle

says that a quantum state cannot have

both a well-defined position andmomentum simultaneously.

Therefore, EPR argued, quantum

mechanics must be an incomplete theory.

A more complete theory would be able to

describe both the position and momentum

of the particles.


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USEQIP June 2009 9

Bohm and Aharonovs refinement

Converted the EPR argument to spin

degrees of freedom, and

D.Bohm and Y. Aharonov, Phys. Rev. 108, 1070 (1957).

Sx Sz

Pairs of spin operators obey commutationrelations:

[Sx, Sz ] = i~Sy (+cyc.) Using Robertsons inequality

H.P. Robertson, Phys. Rev. 34, 163 (1929).

(A)2(B)2

1

2i

h[A, B]i2

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We see:

For spin-1/2 particles, a consequence of

this set of uncertainty relations is that it is

not possible for both

(Sx)2(Sz)

2

~2

4

hSyi2 (+cyc.)

Sx = 0 Sz = 0and

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Now we can repeat the EPR argument

with the singlet state:

|i= 1

2(|i|i

|i|i)

Noting that,

|i= 1

2(|+xi| xi | xi|+xi)

To avoid potential pitfalls of dealing with x

and p, we focus on spin/qubits

USEQIP June 2009 12

Bells inequality

contrary to the EPR argument, Bells [paper] isnot about quantum mechanics. Rather it is ageneral proof, independent of any specificphysical theory, that there is an upper limi t to

the correlation of distant events, if one justassumes the validi ty of local causes. Thisprinciple (also called Einstein locality) assertsthat events occurring in a given spacetimeregion are independent of external parametersthat may be controlled, at the same moment, byagents located in distant spacetime regions.

--Asher Peres, from QuantumTheory: concepts and methods:


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USEQIP June 2009 13

Bells inequality (CHSH)

The setup for a Bell experiment

J. Bell, Physics 1, 195 (1964)

J. Clauser, M.A. Horne, A. Shimony, R.A. Holt, PRL 23, 880 (1969) USEQIP June 2009 14


Measurement outcomes are correlated if A

and B measure the same thing i.e, both +1

or both -1. The measurement outcomesare anti-correlated if they measure

different things.

We define the degree of correlation, E

E=hABi= P+++P P+ P+

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We define a hidden variable, . This is

some property of the physical system that

we cant necessarily measure.

We assume that there is some normalizedprobability distribution over these hidden

variables, f()Z df() = 1

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Bell, and CHSH, imposed locality by

assuming that the correlation, for

measurement settings a and b, could be

written:

The outcome at A,A(a,), depends only

on the setting at a and the hidden variable

LOCALITY

E(a, b) =

Z df()A(a,)B(b,)


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Furthermore, they assumed that the

measurement simply revealed the

preexisting value of +1 or -1REALITY

Under these assumptions, one of

B(b,) +B(b0,)

B(b,)

B(b0,)

must be 0, the other +2 or -2

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We can now write the identity

A(a,) [B(b,) +B(b0,)]

+A(a0,) [B(b,) B(b0,)] =2 If we average over our probability

distribution , we are essentially

averaging a bunch of +2s and -2s, so

f()

2 Z df(){A(a,) [B(b,) +B(b0,)]+A(a0,) [B(b,) B(b0,)]} 2

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This expression can be simplified using

our definition of the correlation

|E(a, b) +E(a, b0) +E(a0, b)

E(a0, b0)|

2

This is the CHSH Bell inequality

USEQIP June 2009 20

Quantum mechanics and CHSH

A general basis for our spin-1/2 particle

can be defined along a direction, n

z

yx

n n = sin cosx

+sin siny

+cos z


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USEQIP June 2009 21


A general basis for our spin-1/2 particle

can be defined along a direction, n

|+ni = cos2

ei2 | i+ sin

2ei2 | i

| ni = sin 2

ei2 | i cos

2ei2 | i

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This state is special, since it has the same

form in any basis:

|i= 12

(|+ni| ni | ni|+ni)

|i= 12

(| i| i | i| i)

Again, consider the singlet state of two

spin-1/2 particles:

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If we make spin measurements along

directions , and , the singlet will exhibit

the correlation,

a b

E(a, b) = a b(prove this)

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We restrict ourselves to states on the

plane (i.e., ), then determines the

setting and

x-z= 0

E(a, b) = cos(a b)


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USEQIP June 2009 25


If we choose:a = 0

a0 = /2

b = /4

b0 =

/4

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The CHSH Bell inequality gives

|E(a, b) +E(a, b0) +E(a0, b)

E(a0, b0)|

2

For the settings weve chosen, the LHS is

which is clearly larger than 2.2

2

Quantum mechanics cannot be described

by a local realistic theory

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Testing Bells inequalities in the lab

Elements of an optical Bell experiment:

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Instead of spin-1/2, we will be using photon

polarization,

There is a subtlety. Orthogonal spins are 180

degrees apart whereas orthogonal (linear)

photon polarizations are 90 degrees apart.

All angles divided by two to go from Bloch sphere

to real space

| i |Hi| i |Vi


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USEQIP June 2009 33


After the polarizers

Single-photon counting APDs

Coincidence logic (window 4-5 ns)

Counting card and software

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E(0,67.5)

E(45,67.5)

E(45,22.5)

E(0,22.5)

135

45

90

0

157.567.5112.522.5

Setting 2 (H= deg)

Setting1(H=d

eg)

Num. violation

S (Poisson stat.)

S

a) Raw counts (counting time = s) b) Correlations

c) Measured Bell parameter

S = E(0, 22.5) +E(45, 22.5) +

E(45, 67.5)E(0, 67.5)

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USEQIP June 2009 36

Loopholes, Suggested reading

Bell inequalities theory

J. Bell, Physics 1, 195 (1964)

J. Clauser, M.A. Horne, A. Shimony, R.A. Holt, PRL

23, 880 (1969)

Bell inequalities early experiments

S.J. Freedman and J.F. Clauser, PRL 28, 938 (1972)

A. Aspect, P. Grangier, and G.Roger, PRL 47, 460

(1981); A. Aspect, J. Dalibard, and G. Roger, PRL 49,

1804 (1982)

Closing detection and locality loopholes (separately)

M.A. Rowe et al., Nature 409, 791 (2001)

Early work: Aspect (1982). Definitive test: G. Weihs,

PRL 81, 5039 (1998)

USEQIP BellInequalities Resch

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