Short Course in Quantum Information Lecture 3web.cecs.pdx.edu/~mperkows/CLASS_FUTURE/video-q/QI_Lecture3.pdf · I. H. Deutsch, University of New Mexico Short Course in Quantum Information

I. H. Deutsch, University of New Mexico

Short Course in Quantum Information

Short Course in

Quantum Information

Lecture 3

Entanglement



Course Info

• All materials downloadable @ websitehttp://info.phys.unm.edu/~deutschgroup/DeutschClasses.html

• Syllabus

Lecture 1: Intro

Lecture 2: Formal Structure of Quantum Mechanics

Lecture 3: Entanglement

Lecture 4: Qubits and Quantum Circuits

Lecture 5: Algorithms

Lecture 6: Error Correction

Lecture 7: Physical Implementations

Lecture 8: Quantum Cryptography



Postulates of QM

• In a closed system the state dynamics is determined by the

Schrödinger equation ---> Unitary map that preserves in product.

�

! • A (pure) state of the system describing our knowledge of the

system is given by a vector in Hilbert space, .

�

a{ }

�

ˆ A • A physical observable is a Hermitian linear operator whose (real)

eigenvalues determine the possible measurement outcomes.

• Given state , and measurement of , the probability offinding eigenvalue a is given by,

where is the eigenvector of .�

!

�

ˆ A

�

pa |! = a !2

�

ˆ A

�

a

�

! "a

a• Upon finding value a the state “collapses”,



Example: Photon Polarization (I)

�

ˆ Z !

= +1!

ˆ Z +1!

= cos2! " sin

2! = cos(2!)

�

r ! = cos"e

H+ sin"e

V= cos" H + sin" V =

cos"

sin"

#

$ %

&

' (

�

ˆ Z

�

H

�

V

�

!"

�

pH = H +1!2

= cos2!

�

pV = V +1!2

= sin2!

�

ˆ Z =1 0

0 !1

"

# $

%

& '

Eigenvalues/vectors

�

!1" V =0

1

#

$ % &

' (

�

+1! H =1

0

"

# $ %

& '



Example: Photon Polarization (II)

• State: Normalized complex polarization vector ,

�

r !

�

r ! "#r ! = 1

• Orthonormal bases:

�

eH,e

V{ }

�

eD+,e

D!{ }

�

eR,e

L{ }Let us call {H,V}

the “standard basis”

�

eH

=1

0

!

" # $

% &

�

eV

=0

1

!

" # $

% &

�

eD+

=1

2

1

1

!

" # $

% &

�

eD! =

1

2

1

!1

"

# $

%

& '

�

eR

=1

2

1

i

!

" # $

% &

�

eL

=1

2

1

!i

"

# $

%

& '

• Three (incompatible) observables:

(H,V)(D+,D-)(R,L)

�

X =0 1

1 0

!

" #

$

% &

�

Y =0 !i

i 0

"

# $

%

& '

�

Z =1 0

0 !1

"

# $

%

& '

Pauli matrices with

eigenvalues ±1



Eigenstates of General

Linear Polarization Analyzer

�

ˆ ! " # cos2" ˆ Z + sin2" ˆ X = cos2"1 0

0 $1%

& '

(

) * + sin2"

0 1

1 0

%

& '

(

) * =

cos2" sin2"

sin2" $cos2"%

& '

(

) *

�

+1! =cos!

sin!"

# $

%

& '

�

!1" =! sin"

cos"#

$ %

&

' (

�

ˆ ! " ±1" = ±1 ±1"

�

e! = cos! H + sin! V = +1!

Define:

Eigenvectors/values

Every linear polarization is +1 eigenvector of some

�

ˆ ! "

�

0 ! " < #



Basic Measurement Statistics (I)

�

r ! = cos"e

H+ sin"e

V

�

ˆ ! "

�

+1!

�

p+1 = +1! +1"

2

= cos2 ! # "( )

�

p!1 = !1" +1#2

= sin2 " ! #( )

�

ˆ ! " #= +1# ˆ ! " +1# = cos

2 " $ #( ) $ sin2 " $ #( ) = cos 2 " $ #( )[ ]

�

+1!

�

!1"



Course Info

• All materials downloadable @ websitehttp://info.phys.unm.edu/~deutschgroup/DeutschClasses.html

• Syllabus

Lecture 1: Intro

Lecture 2: Formal Structure of Quantum Mechanics

Lecture 3: Entanglement

Lecture 4: Qubits and Quantum Circuits

Lecture 5: Algorithms

Lecture 6: Error Correction

Lecture 7: Physical Implementations

Lecture 8: Quantum Cryptography



Joint Probabilities for Multiple Events

Example: Spontaneous Parametric Downconversion

BBO

Coincidence

Counter

PHH=?

Correlations: Joint Probability



Multipartite Systems and Tensor Product

Multiple Degrees of Freedom

Consider a physical system with many degrees of freedom

(e. g. many particles.)

Pure states of the ith subsystem is described by a vector in a Hilbert

space hi , .

Joint state of whole system is a vector in the tensor product space:

Example: Bipartite System of Two Photons

�

! "H = h1 #h2 #L#hn

�

!i"h

i

�

H = C2!C

2= C

2( )!2

Four dimensional

�

!1

="

#$

% &

'

( ) , * 2 =

+

,$

% & '

( )

�

!1" #

2=

$

%&

' (

)

* + "

,

-&

' ( )

* + =

$,

$-

%,

%-

&

'

( ( ( (

)

*

+ + + +



Tensor Product: Formal Structure

�

ei A

i = 1,K,dA{ }

�

f jBj = 1,K,dB{ }

�

Ei, jAB

= ei A ! f j B= ei f j = ei, f j = ij{ }

General state vector:

�

!AB

= cij ei A "ij

# f jB

�

cij = ei A ! f j B( ) " AB = ij " AB

�

HAB = hA !hB

Orthonormal basis for

�

hAOrthonormal basis for

�

hB

Orthonormal “product basis” for joint space

�

p(i and j) = ij !AB

2

= cij2

Joint Bipartite Hilbert space:



Uncorrelated Probabilities

Consider a “product state” in the joint Hilbert Space HAB

Product state !" Statistically Uncorrelated Events

�

P(A = i and B = j) = ei A ! f j B( ) " AB2

= ei # A

2

f j $B

2

= P(A = i)P(B = j)

Joint Probability of Measurement

“Separable State”

�

!AB

= "A# $

B



Entangled States

BBO

“Quantum Correlated” events = Superposition of joint processes.

Feynman: Add probability amplitudes for indistinguishable processes.

Pump

Signal

Idler

“Type II” Downconversion: Signal and Idler have opposite polarization.

But which? Process does not distinguish them --> superposition.

�

!si

=1

2H

s" V

i# V

s" H

i( ) $ %

s" &

i

Bob

Alice



Entanglement and correlated collapse

Suppose a measurement of the signal photon’s polarization is made in

the H-V basis and the result “H” is found.

What is the post-measurement state?

�

!si" H

sH

s!

si=1

2H

sH

sH

s

1

1 2 4 3 4 # V

i$ H

sH

sV

s

0

1 2 4 3 4 # H

i

%

& ' '

(

) * * " H

s# V

i

The state of the idler photon “collapses” due to measurement of the signal.

What Alice knows about the Bob’s photon is effected by her measurement

because she knows the photons are correlated.



Classical Correlation




I got red.

Alice must have gotten green.I got green.

Bob must have gotten red.




I got green.

Alice must have gotten red.I got red.

Bob must have gotten green.

Note: Alice and Bob’s results are random, but perfectly correlated.



“Singlet”: Anticorrelated in any basis

BBO

Pump

Signal

Idler

(D +,D -

)

(D+ ,D

- )



“Singlet”: Anticorrelated in any basis

Proof:

�

!si" 1#

s

1# !si

=1

21#

s

1# Hs

cos#1 2 3

$ Vi% 1#

s

1# Vs

sin#1 2 3

$ Hi

&

'

( (

)

*

+ +

" 1#s

$ cos# Vi% sin# H

i( ) = 1#

s

$ %1#i

If signal photon is found linear along #, idler is found in

the orthogonal polarization.

�

!si

=1

2H

s" V

i# V

s" H

i( )

Entangled state of joint system

Measure signal photon

�

+1! =cos!

sin!"

# $

%

& '

�

!1" =! sin"

cos"#

$ %

&

' (

�

ˆ ! " eigenvectors




I got red.






I got blue.

Alice must have gotten orange.I got orange.

Bob must have gotten blue.

But....

(R,L) (H,V)(D+,D-)

�

X =0 1

1 0

!

" #

$

% &

�

Y =0 !i

i 0

"

# $

%

& '

�

Z =1 0

0 !1

"

# $

%

& '

INCOMPATIBLE.

Cannot be measured

simultaneously.




I got red.






I got purple.

Alice must have gotten yellow.I got yellow.

Bob must have gotten purple.




Results are uncorrelated.

I got red.

Alice must have gotten green.I got yellow.

Bob must have gotten purple.

(random result)



EPR ParadoxCan Quantum-Mechanical Description of Physical Reality Be Considered Complete?

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935)

.

If, without in 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 physics quantity

EPR argue that, by their definition of “realistic properties”, quantum

mechanics “incomplete” as it cannot give definite predictions of

measurement results that have some definite value (hidden variables).



The EPR Argument(Version due to Bohm, 1951)

• Consider entangled state,

�

!AB

=1

2H

A" V

B# V

A" H

B( )

•If Alice were to measure on her photon, she can, without in any way

effecting Bob, determine whether he will find H or V should he perform a

measurement. Bob’s value of is an “element of reality”.

�

ˆ Z

�

ˆ Z

�

ˆ Z

•If Alice were to measure on her photon, she can, without in any way

effecting Bob, determine whether he will find D+ or D- should he perform a

measurement. Bob’s values of is an “element of reality”.

�

ˆ X

�

ˆ X

�

ˆ X

• Quantum mechanical states cannot give a simultaneous definite

value of both and since these operators don’t commute.

�

ˆ Z

�

ˆ X

Quantum mechanical sates are not a “complete” description of the

physical world and can be completed by some “hidden variables”.



John Bell:

Putting Hidden Variables to the Test

Bell took EPR seriously, 30 years

after they published their original

paper and asked when the EPR

assumption had any measurable

significance. Amazingly...YES!

Bell’s Inequality.

J.S. Bell, Physics 1 195 (1964).



Formal Statement of EPR

Consider measurement of, linear polarization at angle #.

�

ˆ ! "

According to EPR, the value that Alice measures is a function of

her polarizer setting #a and the “realistic hidden variables” $.

Similarly for Bob, and his polarizer setting #b.

Measured values:

�

A(!a,") = ±1

�

B(!b,") = ±1

The crucial assumption is that A is not a function of #b, and B is

not a function of #a. “Local hidden variable theory”.

These functions show produce the same statistics as quantum

mechanics for some suitable distribution of $. They should reproduce

the quantum mechanical results.

�

ˆ ! "a

= P(#)$ A("a,#)



Hidden Variable Model for Single Photon

Linear Polarization Measurement

�

+!

�

!(",#)

�

p!1 =" /2 !#($,%)

" /2

�

p+1

=!(",#)

$ /2

$ is a unknown unit vector with

probability uniformly distributed

in the blue 1/4-wedge.

�

+!

Choose:

+1 if $ is in pink 1/4-wedge.

-1 if $ is not.

�

!(",#) =$

2cos

2 # % "( )

Deterministic binary choice, given $:



Correlation functions

For quantum mechanical singlet state

�

ˆ ! "a

# ˆ ! "b

= E("a,"

b)

= $AB

ˆ ! "a

# ˆ ! "b

$AB

= d% PAB

(%) A("a,%)B("

b,%)&

�

ˆ ! "a

# ˆ ! "b

= $cos 2 "a$"

b( )[ ]

Averages of joint observables

�

!AB

=1

2H

AV

B" V

AH

B( )

Can the local hidden variable theory mimic the QM prediction?

(Expectation value)



Measuring Correlation Functions

BBO

Pump

Signal

Idler(D

+ ,D- )

(H,V)

�

ˆ ! "a

# ˆ ! "b

�

= C(1!a ,1!b )

�

+C(!1"a ,!1"b )

�

!C(1"a ,!1"b )

�

!C(!1"a ,1"b )

C

Coincidence counting



Bell/CHSH Inequality(Clauser-Horne-Shimony-Holt version, Phys. Rev. Lett. 23 880 (1969))

Alice and Bob can choose to measure in one of two local basis

�

!a,! " a { }

�

!b,! " b { }

Consider the following correlation joint observable

�

ˆ S = ˆ ! a" ˆ ! # b $ ˆ ! b( ) + ˆ ! # a " ˆ ! # b + ˆ ! b( )

ˆ S = E(%a,% # b ) $ E(%a,%b ) + E(% # a ,% # b ) + E(% # a ,%b )

The value of S! in a local hidden variable model

�

S! = A("a,!) B(" # b ,!) $ B("b,!)[ ] + A(" # a ,!) B(" # b ,!) + B("b ,!)[ ]

�

S!

= ±2

�

!2 " SLHV

" 2



QM Violates LHV

�

ˆ S QM

= !cos 2 "a !" # b ( )[ ] + cos 2 "a !"b( )[ ]! cos 2 " # a !" # b ( )[ ]! cos 2 " # a !"b( )[ ]

In singlet state:

�

ˆ S QM

= 3cos2! " cos(6!)

Consider case where the polarizer directions are equally spaced as follows:

�

!

�

!

�

!

a

a’

b’

b

�

2 2

�

!2 2

�

22.5°

�

67.5°

Quantum correlations stronger

than classical correlations.

Max violation:

• Alice measures or

• Bob measures or

�

ˆ X

�

ˆ Z

�

ˆ X + ˆ Z

2

�

ˆ X ! ˆ Z

2



Experimental Verification

• First definitive test:

Aspect et al. 1982, Phys. Rev. Lett. 49 1804.

• Modern tests:

P.G. Kwiat et al. 1995, 75 4337

100 standard

deviations in



Implications

• Quantum mechanics cannot be described by a local realistic theory.

• Nonlocal hidden variables?

• No “realistic properties” of observed quantities.

• Entanglement CANNOT be used to communicate faster than the speed

of light.

• Alice and Bobs results are random but correlated.

• Entanglement as a resource.

• Quantum correlations are special and destroyed by “eavesdropper”.

• Communication tasks aided by shared entanglement:

• Quantum dense coding.

• Teleportation.

• Distributed computation (e.g. appointment problem).

Short Course in Quantum Information Lecture 3web.cecs.pdx.edu/~mperkows/CLASS_FUTURE/video-q/QI_Lecture3.pdf · I. H. Deutsch, University of New Mexico Short Course in Quantum Information

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