an introduction to quantum information...

©© CCllaauuddee CCrrééppeeaauu 22000022--22000088 11

CCllaauuddee CCrrééppeeaauu

SScchhooooll ooff CCoommppuutteerr SScciieenncceeMMccGGiillll UUnniivveerrssiittyy

an Introduction toQuantum

Information Processing



�1


((11))QQuuaannttuumm IInnffoorrmmaattiioonn

�2


Photons

�3


Photons

�4


Photons

�5


q +=

BBiittss && QQuuBBiittss

00::

11::

Cosq Sinq

+= C0 C1|YÒ0000::

0011::

1100::

1111::

+= C00|YÒ+C01+C10

C11

Ci,CijŒCC_

�6


standard notationsBBaassiiss vveeccttoorrss:: ||00ÒÒ aanndd ||11ÒÒ

�7



AArrbbiittrraarryy ssttaatteess:: ||yyÒÒ==aa||00ÒÒ++bb||11ÒÒssuucchh tthhaatt ||aa||22++||bb||22==11

a,b ŒCC_

�8



AArrbbiittrraarryy ssttaatteess:: ||yyÒÒ==aa||00ÒÒ++bb||11ÒÒssuucchh tthhaatt ||aa||22++||bb||22==11

AArrbbiittrraarryy mmuullttii--ssttaatteess::||yyÒÒ==aa||0000ÒÒ++bb||0011ÒÒ++dd||1100ÒÒ++gg||1111ÒÒ

ssuucchh tthhaatt ||aa||22++||bb||22++||dd||22++||gg||22==11

a,b,d,gŒCC_

�9


((22))QQuuaannttuumm MMeeaassuurreemmeennttss

�10


Polarizing Filterand photodetectors

qq ccooss22 qq

ccooss qq

qq ssiinn22 qq

ssiinn qq

Photo-multipliers

�11


Polarizing Filteroptical axis

�12


Polarizing Filterand photodetectors

qq ccooss22 qq

ccooss qq

qq ssiinn22 qq

ssiinn qq

�13


Photons

�14


Calcite Crystal

optical axis

optical axis

�15


ccooss qq

ssiinn qq

Calcite Crystal

qq

�16


Photons

�17


+= C00|YÒ +C01 +C10 C11

|YÒ

|C00|2

Projective Measurements

|C01|2

|C10|2

|C11|2S|Cij|2=1

�18


((33))QQuuaannttuumm CCoommppuuttaattiioonnss

�19


+C0 C1

|YÒ U |Y'Ò

Quantum Evolution:Unitary Operators

U |Y0ÒU |Y1ÒU +C0 C1|Y0Ò |Y1Ò

�20


standard notationsUUnniittaarryy OOppeerraattoorrss:: UU††UU==UUUU††==II

�21



lliinneeaarr:: UU((aa||00ÒÒ++bb||11ÒÒ)) == aaUU||00ÒÒ++bbUU||11ÒÒ

�22



lliinneeaarr:: UU((aa||00ÒÒ++bb||11ÒÒ)) == aaUU||00ÒÒ++bbUU||11ÒÒ

pprreesseerrvveess ssccaallaarr pprroodduuccttss::

((··yy||UU††))((UU||ffÒÒ))== ··yy||UU††UU||ffÒÒ == ··yy||II||ffÒÒ == ··yy||ffÒÒ

�23


example: Hadamard

||00ÒÒ H (( ||00ÒÒ++||11ÒÒ))//√22

||11ÒÒ H (( ||00ÒÒ-- ||11ÒÒ))//√22

H == 11//√22 (( ))11 1111 --11

�24


ccooss qq

ssiinn qq

Calcite Crystal

qq|0⟩|1⟩

©©CCllaauuddeeCCrrééppeeaauu22000022--220000882266

CalciteCrystal

opticalaxis

opticalaxis

|0⟩

�25


||00ÒÒ

||00ÒÒ

example: Control-NOT||00ÒÒ

||00ÒÒ

||00ÒÒ

||11ÒÒ

||00ÒÒ

||11ÒÒ

||11ÒÒ

||00ÒÒ

||11ÒÒ

||11ÒÒ

||11ÒÒ

||11ÒÒ

||11ÒÒ

||00ÒÒ

==(( ))1 0 0 00 1 0 00 0 0 10 0 1 0

�26


((44))GGeenneerraall MMeeaassuurreemmeennttss

�27


Calcite Crystaland photodetectors

qq

ssiinn22 qq

ccooss22 qq

ccooss qq

ssiinn qq

�28


1/2

Calcite crystal& Photodetection

1/2

�29


1/2


1/2

�30


POSSIBLE!

qq

�31


q qq

qq

IMPOSSIBLE!

�32


QQuuaannttuumm

CCllaassssiiccaall & Quantum Information0000111100111111000000111100 CCllaassssiiccaall

CCooppyyiinngg:: YYeess NNOO

MMeeaassuurriinngg:: YYeess ppaarrttiiaall

BBrrooaaddccaassttiinngg:: YYeess NNOO

SSuuppeerrppoossiinngg:: NNOO YYeess

IInntteerrffeerriinngg:: NNOO YYeess

�33

©© CCllaauuddee CCrrééppeeaauu 22000022--22000055 11�34


((33))

QQuuaannttuumm KKeeyy DDiissttrriibbuuttiioonn

�35


1/2


1/2

�36


1/2


1/2

�37


0 1

+

¥

Ambiguous Coding Scheme

((33..11..11)) KKeeyy ddiissttrriibbuuttiioonn

�38


0V

400V

800V

Pockel Cells

�39


LED Pinhole

Filter Pockel Cells

0/800V 0/400V

SENDER

�40


LED Pinhole

Filter Light source:

~ 1/10 photon per pulse

n = #photons per pulse follows

a Poisson distribution

Pr(n≤x) = 1-e-x/10

Problem:

• may transmit multiple

correlated polarized photons

�41


0/400V

Pockel Cell Calcite Crystal

Photo Multipliers

RECEIVER

�42


00

00 00 00 111111000000 00 00001100

00 ææææ 00 ææ 11 ææææ 11 ææ 00 ææææææææ 11 11ææææ ææ 00 00 00¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++

¥¥ ¥¥00 00 11 11 00 11 11 00 00 00

BB:: ++ ++ ¥¥ ++ ++ ++ ¥¥ ++ ++ ¥¥ ¥¥ ¥¥ ++ ¥¥ ¥¥ ¥¥ ++ ++ ¥¥ ++ ¥¥ ++

AA::BB::

¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥11 11 11 11 11 11 11 11 11 1111

++AA:: 00 00 00 00 00 00 00 00 00 00 00 00 00

++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++

0000 00 11 11 00 11 11 00 00 00BB::11 11 11 11 11AA:: 00 00 00 00 00 00

11 11AA:: 00 00 00

ππ== == == ==BB::00 11 11 11 00 00BB::

11 11 11AA:: 00 00 00

2200%%

Q-distributionof keys

BBeennnneetttt-- BBrraassssaarrdd

»

�43


¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥11 11 11 11 11 11 11 11 11 1111

++AA:: 00 00 00 00 00 00 00 00 00 00 00 00 00

++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++


»

�44


00 00 00 111111000000 00 00001100¥¥ ¥¥00 00 11 11 00 11 11 00 00 00

BB:: ++ ++ ¥¥ ++ ++ ++ ¥¥ ++ ++ ¥¥ ¥¥ ¥¥ ++ ¥¥ ¥¥ ¥¥ ++ ++ ¥¥ ++ ¥¥ ++¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥

11 11 11 11 11 11 11 11 11 1111++

AA:: 00 00 00 00 00 00 00 00 00 00 00 00 00++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++


»

�45


00 00 00 111111000000 00 00001100¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++

¥¥ ¥¥00 00 11 11 00 11 11 00 00 00

BB:: ++ ++ ¥¥ ++ ++ ++ ¥¥ ++ ++ ¥¥ ¥¥ ¥¥ ++ ¥¥ ¥¥ ¥¥ ++ ++ ¥¥ ++ ¥¥ ++

AA::

¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥11 11 11 11 11 11 11 11 11 1111

++AA:: 00 00 00 00 00 00 00 00 00 00 00 00 00

++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++


»

�46


00

00 00 00 111111000000 00 00001100

00 ææææ 00 ææ 11 ææææ 11 ææ 00 ææææææææ 11 11ææææ ææ 00 00 00¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++

¥¥ ¥¥00 00 11 11 00 11 11 00 00 00

BB:: ++ ++ ¥¥ ++ ++ ++ ¥¥ ++ ++ ¥¥ ¥¥ ¥¥ ++ ¥¥ ¥¥ ¥¥ ++ ++ ¥¥ ++ ¥¥ ++

AA::BB::

¥¥ ¥¥ ¥¥¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥ ¥¥11 11 11 11 11 11 11 11 11 1111

++AA:: 00 00 00 00 00 00 00 00 00 00 00 00 00

++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++

0000 00 11 11 00 11 11 00 00 00BB::


»11 11 11 11 11AA:: 00 00 00 00 00 00

�47


0000 00 11 11 00 11 11 00 00 00BB::11 11 11 11 11AA:: 00 00 00 00 00 00


»

�48


0000 00 11 11 00 11 11 00 00 00BB::11 11 11 11 11AA:: 00 00 00 00 00 00

11 11AA:: 00 00 00

≠≠== == == ==BB:: 2200%%


»

≠

�49


00 11 11 11 00 00BB::11 11 11AA:: 00 00 00

2200%%


»

≠≠== == == ==BB:: ≠

�50


00 11 11 11 00 00BB::11 11 11AA:: 00 00 00

2200%%


»

�51



•• •• •• •• ••

• Produces raw classical key

• Observed error rate indicates amount

of eavesdropper information

• Error-correction is used to fix errors

• Random hash function is used to distill

a smaller secret classical key

•• •• •• •• ••

»

�52


Information <--> Errors

(intercept resend)

¥¥ ¥¥ ++¥¥ ¥¥++ ++ ++ ++ ++

¥¥ ¥¥ ++¥¥ ¥¥++ ++ ++ ++ ++�53


Mostly Identical

Partly Secret

String

X X'

E(qq)

�54


≥d

(classical) error-

correcting codes

�55


Identical

Partly Secret

String

X X'

E:=E(qq)+W

W:=C+X

C

W

C':=W+X'C

C'X:=C+W

�56


Identical

Partly Secret

String

X X

E

�57


in BB84, Eve’s information at thispoint (before privacy amplification)contains:

• results of eavesdropping

• multi-photon pulses

• error correction

�58


Identical Secret Shorter String

through Privacy Amplification

XÆÆh(X)

h

XÆÆh(X)

EÆÆh(E)

�59


BBCM

H( G(X) | E=e,G ) > |G(X)| - 2(|G(X)|-R(X|E=e))

k:=g(X)

g

k:=g(X)

???????????k':=g(E)??????????

�60


((55))

PPrrooooffss ooff sseeccuurriittyy

�61


Biham, Boyer, Boykin, Mayers, Oppenheim - 2000 -

Shor, Preskill - 2000 -

Mayers - 2001 -

Gottesman, Lo - 2003 -

Renner - 2006 -

TTiimmee--lliinnee

�62


((77))QQuuaannttuumm FFaaccttoorriinngg

�64


Quantum Superposition

++

++

+

++

+

++

+

++

+

+

+

�65


Quantum Superposition+

++

+

++

+

++

+

+

+

F F( ) F F( )

+

++

+

++

+

++

+

+

+

F( )F( )F( )F( )F( )F( )F( )

F( )F( )F( )F( )

�66


QQuuaannttuumm FFaaccttoorriinngg

GGiivveenn nn==ppqq aanndd ff((nn))==((pp--11))((qq--11))iitt iiss eeaassyy ttoo ffiinndd pp aanndd qq bbeeccaauussee

ff((nn))== ((pp--11))((qq--11))==ppqq--pp--qq++11==nn--pp--((nn//pp))++11

pp++((nn//pp))++nn--ff((nn))++11==00

pp22++((nn--ff((nn))++11))pp++nn==00 hhaass ssoolluuttiioonnss::pp== ((--bb±±√√44aacc))//22aa ((aa==11,, bb==nn--ff((nn))++11,, cc==nn))p=−b±√(b2−4ac)/2a (a=1, b=n−φ(n)+1, c=n)�

67



EEuulleerr’’ss tthheeoorreemm

aaff((nn)) mmoodd nn == 11

--------------------------------------------------------------

FF((aa,,rr,,nn))::==((aarr mmoodd nn,,rr,,nn))FF((aa,,rr,,nn))::==(( ,,rr,,nn))

FF((aa,,kkff((nn)),,nn))::==(( ,,kkff((nn)),,nn))�68


QQuuaannttuumm FFaaccttoorriinnggFF((aa,,rr,,nn))::==((aarr mmoodd nn,,rr,,nn))FF((aa11,,rr,,nn))::==(( ,,rr,,nn))FF((aa22,,rr,,nn))::==(( ,,rr,,nn))

......FF((aaii,,rr,,nn))::==(( ,,rr,,nn))

bbuutt ..FF((aa11,,kkff((nn)),,nn))::==(( ,,kkff((nn)),,nn))FF((aa22,,kkff((nn)),,nn))::==(( ,,kkff((nn)),,nn))

......FF((aaii,,kkff((nn)),,nn))::==(( ,,kkff((nn)),,nn))

�69


QQuuaannttuumm FFaaccttoorriinnggFF((aa,,rr,,nn))::==((aarr mmoodd nn,,rr,,nn))FF((**,,rr,,nn))::==(( ,,rr,,nn)).. ==(( ,,rr,,nn))

bbuutt ..FF((**,,kkff((nn)),,nn))::==(( ,,kkff((nn)),,nn))FF((**,,kkff((nn)),,nn))::==(( ,,kkff((nn)),,nn))

using Quantum Fourrier Transform

�70


QQuuaannttuumm FFaaccttoorriinngg(( ,,11,,nn))(( ,,22,,nn))(( ,,33,,nn))......

(( ,,ff((nn))--11,,nn))(( ,,ff((nn)),,nn))(( ,,ff((nn))++11,,nn))

......(( ,,kkff((nn)),,nn))(( ,,ff((nn))++11,,nn))

......�71



�72


CCoonnssttrruucctt tthhee ssuuppeerrppoossiittiioonnffoorr aallll rr aanndd aallll aa ooff

((aarr mmoodd nn,,rr,,nn))

mmeeaassuurree rr,, aanndd wwiitthh hhiigghhpprroobbaabbiilliittyy rr==kkff((nn))ffoorr ssoommee iinntteeggeerr kk

rreeppeeaatt ttoo oobbttaaiinn rr’’==kk’’ff((nn))ggccdd((rr,,rr’’)) == ff((nn))..


�73


CCllaauuddee CCrrééppeeaauu

SScchhooooll ooff CCoommppuutteerr SScciieenncceeMMccGGiillll UUnniivveerrssiittyy





�74

an introduction to quantum information...

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