•distributive quantum computing

1

QuantumQuantumComputing ?Computing ?

Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr.Dept. of Comp. Dept. of Comp. SciSci. & Electrical Engineering. & Electrical Engineering

University of Maryland Baltimore CountyUniversity of Maryland Baltimore CountyBaltimore, MD 21250Baltimore, MD 21250

Email: Email: [email protected]@UMBC.EDUWebPageWebPage: : http://http://www.csee.umbc.edu/~lomonacowww.csee.umbc.edu/~lomonaco

LL--OO--OO--PP

Quantum ComputingQuantum Computing

• The Defense Advance Research ProjectsAgency (DARPA) & Air Force Research

Laboratory (AFRL), Air Force Materiel Command,USAF Agreement Number F30602-01-2-0522.

• The National Institute for Standards and Technology (NIST)

• The Mathematical Sciences Research Institute (MSRI).

• The L-O-O-P Fund.

• The Institute of Scientific Interchange

LL--OO--OO--PP

This work is supported by:This work is supported by: OverviewOverviewFour TalksFour Talks

•• A Rosetta Stone for Quantum ComputationA Rosetta Stone for Quantum Computation

•• Quantum Algorithms & BeyondQuantum Algorithms & Beyond

•• Distributive Quantum ComputingDistributive Quantum Computing

•• Topological quantum Computing and theTopological quantum Computing and theJones PolynomialJones Polynomial

•• A Quantum Computing Knot Theoretic MysteryA Quantum Computing Knot Theoretic Mystery---- Can be found on my webpage.Can be found on my webpage.

These talks These talks

are available at:are available at:

http://http://www.csee.umbc.edu/~lomonacowww.csee.umbc.edu/~lomonaco

and othersand others

OverviewOverviewFour TalksFour Talks

•• A Rosetta Stone for Quantum ComputationA Rosetta Stone for Quantum Computation

•• Quantum Algorithms & BeyondQuantum Algorithms & Beyond

•• Distributive Quantum ComputingDistributive Quantum Computing

•• Topological quantum Computing and theTopological quantum Computing and theJones PolynomialJones Polynomial

2

Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr.Dept. of Comp. Dept. of Comp. SciSci. & Electrical Engineering. & Electrical Engineering

University of Maryland Baltimore CountyUniversity of Maryland Baltimore CountyBaltimore, MD 21250Baltimore, MD 21250

Email: Email: [email protected]@UMBC.EDUWebPageWebPage: : http://http://www.csee.umbc.edu/~lomonacowww.csee.umbc.edu/~lomonaco

A A Rosetta StoneRosetta Stone

forforQuantum ComputationQuantum Computation

LL--OO--OO--PP

Adami, Barencol, Benioff, Bennett, Brassard, Calderbank, Chen, Crepeau, Deutsch,

DiVincenzo, Ekert, Einstein, Feynman, Grover, Heisenberg,

Jozsa, Knill, Laflamme, Lloyd,

Peres, Popescu,

Preskill, Podolsky,Rosen, Schumacher, Shannon, Shor,Simon, Sloane,

Schrodinger, Townsend, Unruh, von Neumann, Vazirani,

Wootters, Yao, Zeh, Zurek

Lomonaco,

& many more

Lomonaco, Samuel J., Jr., Lomonaco, Samuel J., Jr., A Rosetta stone A Rosetta stone for quantum mechanics with an introduction to for quantum mechanics with an introduction to quantum computationquantum computation, in AMS PSAPM/58, , in AMS PSAPM/58, (2002), pages 3 (2002), pages 3 –– 65.65.

Quantum Computation and InformationQuantum Computation and Information,, Samuel J. Samuel J. Lomonaco, Jr. and Howard E. Brandt (editors),Lomonaco, Jr. and Howard E. Brandt (editors), AMS AMS CONM/305, (2002). CONM/305, (2002).

? ? ? Why ? ? ?? ? ? Why ? ? ?QuantumQuantum

ComputationComputation• Limits of small scale integrationLimits of small scale integration

technology to be reached 2010technology to be reached 2010--20202020

• No Longer !No Longer ! MooreMoore’’s Laws Law, i.e., , i.e., every year, double the computing powerevery year, double the computing powerat half the price. at half the price. No Longer !No Longer !

• A whole new industry will be built aroundA whole new industry will be built aroundthe new & emerging quantum technologythe new & emerging quantum technology

CollisionCollision CourseCourse

QuantumQuantumComputationComputation

MultiMulti--DisciplinaryDisciplinary

Math

CompSci

EE

Physics

3

TheTheClassicalClassical

WorldWorld

ClassicalClassicalShannonShannon

BitBit

0 or 1

DecisiveIndividual

CopyingMachine

OutIn

ClassicalClassical BitsBits CanCan BeBe CopiedCopied

TheTheQuantumQuantum

WorldWorld

IntroducingIntroducingthethe QubitQubit

? ? ?? ? ?

Quantum BitQuantum BitQubitQubit

IndecisiveIndividual

Can be both 0 & 1at the same time !!!

Quantum Representations Quantum Representations of of QubitsQubits

ExampleExample 11.. A spinA spin-- particleparticle12

Spin Up Spin Down11

00

4

Quantum Representations Quantum Representations of of QubitsQubits (Cont.)(Cont.)

ExampleExample 22.. Polarization States of a PhotonPolarization States of a Photon

1 = 0 = ↔

1 = 0 =

or,

,

H =Where does a Where does a QubitQubit live ?live ?

HomeDef. A Hilbert Space is a vector space over together with an inner product such that

H, :− − × →H H

The elements of will be called The elements of will be called ketskets, and , and will be denoted bywill be denoted by label

H

1) & 1 2 1 2, , ,u u v u v u v+ = + 1 2 1 2, , ,vu u v u vu+ = +2) , ,u v u vλ λ=3) , ,u v v u=4) Cauchy seq in , ∀ 1 2, ,u u … H lim nn

u→∞

∈ H

A A QubitQubit is a is a quantum quantum systemsystem whose whose statestate is is represented by a represented by a KetKetlying in a 2lying in a 2--D Hilbert D Hilbert SpaceSpaceH

Superposition of StatesSuperposition of States

A typical Qubit is ???Inde

cisive

0 10 1α α= +

where 2 20 1 1α α+ =

The above Qubit is in a SuperpositionSuperposition of statesand

It is simultaneously both and !!!0 1

10

““CollapseCollapse”” of the Wave Functionof the Wave Function

0 10 1α α+ =

Observer

Qubit

Whoosh !!!

i

Prob

= |a i|

2

KetsKets as Column Vectors overas Column Vectors overLet be a 2-D Hilbert space with orthonormal basis H

0 , 1

In this basis, each ket can be thought of as a column vector. For example,

10

0 =

01

1 =

and

And in general, we have

0 10 1

1 0a

a b a bb

ψ = + = + =

5

Tensor Product of Hilbert SpacesTensor Product of Hilbert SpacesThe tensor product of two Hilbert spaces and is the “simplest” Hilbert space such that the map

is bilinear, i.e., such that

HK

( ),h k h k⊗× → ⊗H K H K

( )( )

( ) ( )

1 2 1 2

1 2 1 2

h h k h k h kh k k h k h kh k h kλ λ

+ ⊗ = ⊗ + ⊗ ⊗ + = ⊗ + ⊗ ⊗ = ⊗

We define the action of on as

( ) ( ) ( )h k h k h kλ λ λ⊗ ⊗ = ⊗⊗H K

In other words,In other words,

is constructed in the simplest non-trivial way such that:

⊗H K

( )( )

( ) ( ) ( )

1 2 1 2

1 2 1 2

,

h h k h k h kh k k h k h kh k h k h kλ λ λ λ

+ ⊗ = ⊗ + ⊗ ⊗ + = ⊗ + ⊗ ⊗ = ⊗ ⊗ ∀ ∈

KroneckerKronecker (Tensor) Product of Matrices(Tensor) Product of Matrices

and11 12

21 22

a aA

a a =

11 12

21 22

b bB

b b =

The Kronecker(tensor) product is defined as:1 1 1 2 1 1 1 2

1 1 1 22 1 2 2 2 1 2 2

1 1 1 2 1 1 1 22 1 2 2

2 1 2 2 2 1 2 2

b b b ba a

b b b bA B

b b b ba a

b b b b

⊗ =

11 11 11 12 12 11 12 12

11 21 11 22 12 21 12 22

21 11 21 12 22 11 22 12

21 21 21 22 22 21 22 22

a b a b a b a ba b a b a b a ba b a b a b a ba b a b a b a b

=

So So ……

1 001 0 1 0 1

0 10 0

11 1

000

01

= = ⊗ = ⊗

= =

i

i

Representing Integers in Quantum ComputationRepresenting Integers in Quantum Computation

Let be a 2-D Hilbert space with orthonormal basis

2H0 , 1

Then is a 2n-D Hilbert space with induced orthonormal basis

10 2n−= ⊗H H

0 00 , 0 01 , 0 10 , 0 11 , 1 11…

where we are using the convention

1 2 1 0 1 2 1 0n n n nb b b b b b b b− − − −= ⊗ ⊗ ⊗ ⊗

Representing Integers in Quantum ComputationRepresenting Integers in Quantum Computation

So in the 2n-D Hilbert space with induced orthonormal basis

H

0 00 , 0 01 , 0 10 , 0 11 , 1 11…we represent the integer with binary expansion

m1

02 , 0 1,n jj jj

m m m or j−

== = ∀∑

as the ket1 2 1 0n nm m m m m− −=

For example,23 010111=

6

Indexing Convention for MatricesIndexing Convention for Matrices

The indices of matrices start at 0, not 1.For example, in 2 2 2⊗ ⊗H H H

0 00 10 2

0 1 0 0 35 101

1 0 1 0 41 50 60 7

indexindexindexindexindexindexindexindex

← = ← =

← = ← = = = ⊗ ⊗ = ← = ← = ← = ← =

The The QubitQubit VillageVillage

Each inEach in

KetsKetsQubitvilleQubitville

1 2, , nΨ Ψ Ψ

1 2, , , nH H H• The Qubits in Qubit Village collectively live in

1 21

n

n jj =

⊗ ⊗ ⊗ = ⊗H H H H

• The populace of Qubit Village is

1 21

n

n jj

P opu lace=

= Ψ ⊗ Ψ ⊗ ⊗ Ψ ∈ ⊗ H

• Other names for the populace of Qubit Village

1 2 1 2n nPopulace = Ψ Ψ Ψ = Ψ Ψ Ψ

Massive ParallelismMassive Parallelism

Example. For , let

Then

1,2, ,j n= …0 1

2j+Ψ =

1 21

0 12

n

nj=

+ Ψ Ψ Ψ = ⊗

Therefore, the n-qubit register contains n-bit binary numbers simultaneously !

all

2 1

0

12

nn

aa

−

=

= ∑

( ) ( ) ( )1 0 1 0 1 0 12

n = + + +

( )1 00 0 00 1 11 12

n = + + +

… … … …

But But ! ! !! ! !1 2 nΨ Ψ Ψ

a

Whoosh

Whoosh!!

Prob

Prob=1

/2=1

/2nn ObserverObserver

U

1Ψ

Activities in Quantum VillageActivities in Quantum Village

All activities in Quantum Village are All activities in Quantum Village are UnitaryUnitarytransformationstransformations

At timeAt timet=0t=0

At timeAt timet=1t=1

HH HHU

T T

U U I UU= =

0Ψ

where a where a unitaryunitary transformation is one such thattransformation is one such that

MeasurementMeasurementConnectingConnecting

Quantum VillageQuantum Villageto theto the

Classical WorldClassical World

7

Another Activity in Quantum Village:Another Activity in Quantum Village:

MeasurementMeasurement

Group of Friendly PhysicistsGroup of Friendly Physicists

Another Activity in Quantum Village:Another Activity in Quantum Village:

MeasurementMeasurementMeasurementMeasurement

Group of Group of AngryAngry PhysicistsPhysicists

ObservablesObservables

What does our observer What does our observer actually observe ?actually observe ?

??????

Observables = Observables = HermitianHermitian OperatorsOperatorsA

→O

H H

TA A=O O

wherewhere

, and let , and let denote the corresponding denote the corresponding eigenvalueseigenvaluesLet be the Let be the eigenketseigenkets of of

Observables (Cont.)Observables (Cont.)

What does our observer actually What does our observer actually observe ?observe ?

??????

iϕ AO

A i i iaϕ ϕ=O

ia, i.e., , i.e.,

CaveatCaveat:: We only consider observables whoseWe only consider observables whoseeigenketseigenkets form an orthonormal basis of form an orthonormal basis of H

Observables (Cont.)Observables (Cont.)

What does our observer observe ?What does our observer observe ?

??????

So with probability , the observer So with probability , the observer observes the observes the eigenvalueeigenvalue , and, and

The state of an The state of an nn--QubitQubit register can register can be written in the be written in the eigenketeigenket basis asbasis as

i iiα ϕΨ =∑

ia2

i ip α=

iϕ Whoosh !

Whoosh !

Example: Example: PauliPauli Spin MatricesSpin MatricesConsider the following observables, called the Consider the following observables, called the PauliPauli Spin Spin matricesmatrices::

1 2 3

0 1 0 1 0, ,

1 0 0 0 1i

iσ σ σ = = = − −

which can readily be checked to be which can readily be checked to be HermitianHermitian..

E.g., E.g., *

†2 2

0 0 00 0 0

T Ti i ii i i

σ σ− − = = = = −

The respective The respective eigenvalueseigenvalues and and eigenketseigenkets of these matrices are of these matrices are listed in the table belowlisted in the table below

EigenvalueEigenvalue

( )0 1 / 2+ ( )0 1 / 2i+ 0

( )0 1 / 2− ( )0 1 / 2i− 1

1σ 2σ3σ

1+

1−

8

Measurement ExampleMeasurement Example

Consider a 2Consider a 2--D quantum system in stateD quantum system in state, where , where 0 1a bψ = + 2 2 1a b+ =

What happens if we measure What happens if we measure w.r.tw.r.t. observable ?. observable ?1σFirst express in terms of the First express in terms of the eigenketeigenket basis ofbasis of 1σψ

Thus, if is observed Thus, if is observed w.r.tw.r.t. , either. , either1σ

EigenvalueEigenvalueis meas.is meas.

Possibility0Possibility0

0 1a = +( )0 1 / 2ψ +

2Prob / 2a b= +

0 1 0 12 2 2 2

a b a bψ + −+ − = + ψ

EigenvalueEigenvalueis meas.is meas.

Possibility1Possibility12Prob / 2a b= −

1 1a = −( )0 1 / 2ψ −

oror

ψ

Important Feature ofImportant Feature ofQuantum MechanicsQuantum Mechanics

It is important to mention that:It is important to mention that:

We cannot completelyWe cannot completelycontrol the outcome of control the outcome of quantum measurementquantum measurement

CopyingMachine

OutIn

CloningCloningThe NoThe No-- TheoremTheorem

DieksDieks, , WoottersWootters, , ZurekZurek

The No Cloning TheoremThe No Cloning Theorem

DefinitionDefinition.. Let be a Hilbert space. Then Let be a Hilbert space. Then a a quantumquantum replicatorreplicator consists of an consists of an auxiliary Hilbert space , a fixed state auxiliary Hilbert space , a fixed state

(called the (called the initialinitial statestate of of the the replicatorreplicator), and a unitary transformation ), and a unitary transformation

AH

H

# Aψ ∈ H

: A AU ⊗ ⊗ → ⊗ ⊗H H H H H H

states , where states , where (called the (called the replicatorreplicator statestate after replication ofafter replication of

) depends on .) depends on .

such that, for some fixed state ,such that, for some fixed state ,blank ∈H

# aU a blank a aψ ψ=a ∈H a Aψ ∈H

a a

forfor allall


•• Cloning is:Cloning is:

( ) ( )( )# @0 1 0 1 0 1a b blank a b a bψ ψ+ + +

•• Cloning isCloning is NOTNOT::

( ) ( )# @0 1 00 11a b blank a bψ ψ+ +


•• CloningCloning is inherently is inherently nonnon--linearlinear

•• Quantum mechanicsQuantum mechanics is inherently is inherently linearlinear

•• ErgoErgo, , quantum quantum replicatorsreplicators do not existdo not exist

Key IdeaKey Idea

9

IntroductionIntroductiontoto

Quantum EntanglementQuantum Entanglement

A Illustration of the A Illustration of the of Quantum Mechanicsof Quantum Mechanics

WeirdnessWeirdness

QubitsQubits

•• Not EntangledNot Entangled

•• SeparateSeparate

•• EntangledEntangled

•• Not Separate Not Separate !!

02

1 01⊗ − ⊗0 0⊗

⊗U

UnitaryUnitaryTransfTransf

Entangled Entangled

Observing Entangled Observing Entangled QubitsQubits

02

1 01⊗ − ⊗Observe OnlyObserve Onlythe Blue the Blue QubitQubit

0 1⊗ 1 0⊗

Prob

Prob

=1/2

=1/2 Prob

Prob=1/2=1/2

•• No Longer EntangledNo Longer Entangled

•• Separate IdentitySeparate Identity

Whoosh !Whoosh !

Hidden Variable Theory Hidden Variable Theory vsvs Quantum MechanicsQuantum Mechanics

EPR PairEPR Pair0

21 01⊗ − ⊗

EinsteinEinsteinPodolskyPodolsky

RosenRosen

Bah ! Humbug !Bah ! Humbug !Something is Missing from Quantum Mechanics.Something is Missing from Quantum Mechanics.

There Must Exist Hidden VariablesThere Must Exist Hidden Variables

Hidden Variable Theory Hidden Variable Theory vsvs Quantum MechanicsQuantum Mechanics

EPR PairEPR Pair0

21 01⊗ − ⊗


RosenRosen

Something is Missing from Quantum Mechanics.Something is Missing from Quantum Mechanics.There Must Exist Hidden VariablesThere Must Exist Hidden VariablesBell InequalitiesBell Inequalities

Aspect ExperimentAspect Experiment

Score So FarScore So Far

•• HVT Score = 0HVT Score = 0

•• QM Score = 1QM Score = 1

10

Why didWhy did


RosenRosen

Object So Object So Vehemently ?Vehemently ?

Forces of Nature Are Local InteractionsForces of Nature Are Local Interactions

All the forces of nature (i.e., gravitational, All the forces of nature (i.e., gravitational, electromagnetic, weak, & strong forces) are electromagnetic, weak, & strong forces) are local interactions.local interactions.

•• Mediated by another entity, e.g., gravitons,Mediated by another entity, e.g., gravitons,photons, etc. photons, etc.

•• Propagate no faster than the speed of lightPropagate no faster than the speed of light cc

•• Strength drops off with distanceStrength drops off with distance

SpacelikeSpacelike DistanceDistanceHello !Hello ! CanCan’’t Heart HearYou !! ??You !! ??

1P 2P( ), , ,x y z t ( ), , ,X Y Z T

SpacelikeSpacelike DistanceDistance

( )1 2,Dist P P c T t> −

No signal can travel between No signal can travel between spacelikespacelike regions of spaceregions of space

Ergo, Ergo, spacelikespacelike regionsregions of space are of space are physically physically independentindependent, i.e., one cannot influence the other., i.e., one cannot influence the other.

•• The forces of nature are localThe forces of nature are localinteractionsinteractions

•• SpacelikeSpacelike regions of space areregions of space arephysically independentphysically independent

The EPR PerspectiveThe EPR Perspective

All perfectlyAll perfectlyreasonablereasonable

assumptions !assumptions !

Alpha CentauriAlpha CentauriEarthEarth

Instantly,Instantly,Both Both QubitsQubits

Are Determined !Are Determined !

SpacelikeSpacelike DistanceDistance

0 11( 20 ) /−

0 1⊗ 1 0⊗

Prob

Prob

=1/2

=1/2 Prob

Prob=1/2=1/2

MeaurementMeaurement of EPR Pairof EPR PairBlue Blue QubitQubit Red Red QubitQubit

Meas. BlueMeas. BlueQubitQubit

No Local Interaction !No Local Interaction !

•• No force of any kindNo force of any kind-- Not mediated by anythingNot mediated by anything

•• Acts instantaneouslyActs instantaneously-- Faster than lightFaster than light

•• Strength does not drop off with distanceStrength does not drop off with distance-- Full strength at any distanceFull strength at any distance

Yet, still consistent with General Relativity !Yet, still consistent with General Relativity !

11

Quantum Entanglement Quantum Entanglement Appears to Pinpoint Appears to Pinpoint the of the of Quantum MechanicsQuantum Mechanics

WeirdnessWeirdness

Properties of Properties of QubitsQubits

•• Properties of StatesProperties of States•• QubitsQubits can exist in a superposition ofcan exist in a superposition ofstatesstates

•• QubitsQubits can be entangledcan be entangled

•• Actions on StatesActions on States•• QubitsQubits ““collapsecollapse”” upon measurementupon measurement

•• QubitsQubits are transformed by unitaryare transformed by unitarytransformationstransformations

Properties of Quantum Computer DataProperties of Quantum Computer Data

Quantum Computer InstructionsQuantum Computer Instructions

UsefulUsefulfor Quantum Computationfor Quantum Computation

QuantumQuantumTeleportationTeleportation

An Application of Quantum EntanglementAn Application of Quantum EntanglementTeleportationTeleportation:: TransferingTransfering an object between an object between two locations by a process of:two locations by a process of:•• Dissociation to obtain infoDissociation to obtain info

-- Scanned to extract Scanned to extract suffsuff. Info. to. Info. torecreate originalrecreate original

•• Information TransmissionInformation Transmission

-- Exact replica is reExact replica is re--assembled at destinationassembled at destinationout of locally available materialout of locally available material

•• Reconstruction from infoReconstruction from info

NetNet EffectsEffects::•• Destruction of original objectDestruction of original object•• Creation of an exact replica at theCreation of an exact replica at theintended destination.intended destination.

Teleportation ?Teleportation ?Oxford Unabridged DictionaryOxford Unabridged Dictionary

Asked Scotty about TeleportationAsked Scotty about Teleportation

Beam me up, Scotty !Beam me up, Scotty !

Aye, Aye, Captain !Aye, Aye, Captain !

II’’m just a wee bit busy. m just a wee bit busy.

13

Skip to the DeutschSkip to the Deutsch--JozsaJozsaAlgorithmAlgorithm

Skip to More Skip to More DiracDirac Notation and the Notation and the Density OperatorDensity Operator

The The DeutschDeutsch--JozsaJozsa

AlgorithmAlgorithm

DD--J AlgorithmJ Algorithm

DefinitionDefinition. A . A coincoin is is fairfair (or (or balancedbalanced) if it has ) if it has heads on one side and tails on the other side. It is heads on one side and tails on the other side. It is unfairunfair (or (or constantconstant) if either it has tails on both ) if either it has tails on both sides, or heads on both sides.sides, or heads on both sides.

HH

HHHH HH

TT TTTT

TT

Side1Side1 Side2Side2

Fair (Balanced)Fair (Balanced)


Fair (Balanced)Fair (Balanced)


Unfair (Constant)Unfair (Constant)


Unfair (Constant)Unfair (Constant)

ObservationObservation

ObservationObservation.. In the classical world, we need In the classical world, we need to observe both sides of the coin to determine to observe both sides of the coin to determine whether or not it is fair ?whether or not it is fair ?

But what about in But what about in the quantum world ?the quantum world ?

We represent a coin mathematically as a We represent a coin mathematically as a Boolean function: Boolean function:

{ } { }: 0, 1 0, 1f →

Side1Side1 Side2Side2 HH TT

14

The Unitary Implementation of The Unitary Implementation of

( )

fU

x y x f x y⊕→H H

Let be the unitary transformation Let be the unitary transformation fU

then then

x0 1

2−

( ) ( ) 0 112

f x x −−fU

f Moreover, Moreover,

0

1 1fU

HH

HH

HH

HH

( ) ( )(0) (1)1 1 02

f f − + −

( ) ( )(0) (1)1 1 12

f f − − −

+

( ) ( )(0) (1)1 1 0 1 0 1 1 0 12

Outputf f − + −

= + =±

i

( ) ( )(0) (1)1 10 0 1 1 1Outpu 1t 12

f f − − −= + = ±

i

CaseCase 11. is . is fairfair, i.e., , i.e., balancedbalancedf

CaseCase 11. is . is unfairunfair, i.e., , i.e., constantconstantf

So So ……

If we only make one observation, i.e., if we If we only make one observation, i.e., if we observe the left register, then we can observe the left register, then we can determine whether or not is fair or determine whether or not is fair or unfair.unfair.

f MoreMoreDiracDirac

NotationNotation

Skip Skip DiracDirac Notation and the Density Notation and the Density OperatorOperator

More More DiracDirac NotationNotation

LetLet ( )* ,Hom=H H

Hilbert SpaceHilbert Spaceof morphismsof morphismsfrom tofrom toH

We call the elements of We call the elements of BraBra’’s, and s, and denote them asdenote them as

*H

label

More More DiracDirac NotationNotation

There is a There is a dualdual correspondencecorrespondence between and between and *H H

KetKetBraBra

There exists a bilinear mapThere exists a bilinear mapdefined bydefined by

which we more which we more simpysimpy denote by denote by

* × →H H( )( )1 2ψ ψ ∈

1 2|ψ ψ

†

ψ ψ↔KetKet Bra

Bra

BraBra--cc--KetKet

15

BraBra’’s as Row Vectors over s as Row Vectors over

Let be a 2Let be a 2--D Hilbert space with D Hilbert space with orthonormal basis orthonormal basis 0 , 1H

( )* ,Hom=H H

0 , 1

and letand let

be the corresponding dual Hilbert space with be the corresponding dual Hilbert space with corresponding dual basiscorresponding dual basis

( ) ( )and0 1,0 1 0,1= =

( )0 1 ,a b a b+ =

Then with respect to this basis, we haveThen with respect to this basis, we have

BraBra’’s & s & KetKet’’ss as as AdjointsAdjoints of One Another of One Another

The dual correspondence The dual correspondence

is given byis given by

and is called the and is called the adjointadjoint

†*↔H H

( )†

0 1 0 1 ,a

a b a b a bb = + + =

↔

1

2

0 10 1a bc d

ψψ

= + = +

( )( )

( )1 2| 0 1 0 1

,

a b c d

ca b ac bd

d

ψ ψ = + +

= = + i

IfIf

then the bracket product becomesthen the bracket product becomes

as a Matrix as a Matrix OuterproductOuterproduct1 2ψ ψ

1

2

0 10 1a bc d

ψψ

= + = +

1 2

1 2 |

ψ ψ

ψ ψ ψ ψ→H H

IfIf

then is the linear transformation then is the linear transformation 1 2ψ ψ

which, when written in matrix notation, becomes the which, when written in matrix notation, becomes the matrix matrix outerproductouterproduct

( )1 2 ,a ac ad

c db bc bd

ψ ψ = = i

Let be an NLet be an N--D Hilbert space with orthonormal D Hilbert space with orthonormal basis basis

If we use the convention that If we use the convention that matrix indices begin matrix indices begin atat 00, then the matrix of the linear transformation, then the matrix of the linear transformation

is an is an NxNNxN matrix consisting of all zeroes with the matrix consisting of all zeroes with the exception of entry exception of entry ((m,km,k)) which is which is 11

For example if For example if N=4N=4, then , then

H0 , 1 , , 1N −…

m k

0 0 0 00 0 0 0

2 30 0 0 10 0 0 0

=

Entry Entry (2,3)(2,3)

Density OperatorsDensity Operators&&

Mixed EnsemblesMixed Ensembles

16

Two Ways to Represent Quantum StatesTwo Ways to Represent Quantum States

Density OperatorsDensity Operators

ρKetsKets

ψ &&


ExampleExample.. We have seen We have seen pure ensemblespure ensembles, i.e., , i.e., pure states, such aspure states, such as

ProblemProblem.. Certain types of quantum states Certain types of quantum states are difficult to represent in terms of are difficult to represent in terms of ketskets

ProbProb

KetKet ψ1


ExampleExample.. Consider the following state for Consider the following state for which we have incomplete knowledge, called a which we have incomplete knowledge, called a mixed ensemblemixed ensemble::

wherewhere

ProbProb

KetKet1ψ1p

2ψ kψ2p kp

1 2 1kp p p+ + + =…

All unit LengthAll unit Length& not & not necnec.. ⊥

ProbProb

KetKet 1ψ1p

2ψ kψ2p kp


Johnny von Neumann suggested that we use the Johnny von Neumann suggested that we use the following operator to represent a state:following operator to represent a state:

1 1 1 2 2 2 k k kp p pρ ψ ψ ψ ψ ψ ψ= + + +…is called a is called a density operatordensity operator. It is a . It is a HermitianHermitian

positive semipositive semi--definite operator of tracedefinite operator of trace 11..ρ

MixedMixedEnsembleEnsemble

For the pure ensemble , For the pure ensemble ,

11ProbProb

KetKet ψ 1ρ ψ ψ= i


If for example,If for example,

0 1a bψ = +

wherewhere2 2 1a b+ =

thenthen( )( )

( )2

2

0 1 0 1a b a b

a abaa b

b ba b

ρ = + +

= =

3/8 3 /80 1 0 13 11 14 4 3 /8 5/82 2

ii ii

ρ − + = + = −


On the other hand, On the other hand,

is the mixed ensembleis the mixed ensemble

ProbProb

KetKet ( )1 0 1 / 2iψ = − 2 1ψ =34

14

1ψ 1ψ1p 2p2 2ψ ψ

17

Quantum Mechanics from the Two PerspectivesQuantum Mechanics from the Two Perspectives

ObservationObservation

UnitaryUnitaryEvolutionEvolution

SchroedSchroed..EqEq..

Density OpsDensity OpsKetsKetsψ ρ

i Ht

ψ ψ∂ =∂

| |A Aψ ψ= ( )A trace Aρ=

[ ],i Htρ ρ∂ =

∂

Uψ ψ †U Uρ ρ

•• We now have a more powerful way ofWe now have a more powerful way ofrepresenting quantum states.representing quantum states.

•• Density operators are absolutelyDensity operators are absolutelycrucial when discussing and dealingcrucial when discussing and dealingwith quantum noise and quantumwith quantum noise and quantumdecoherence.decoherence.

Density OperatorsDensity Operators

Weird !Weird !Measurement RevisitedMeasurement Revisited

InIn OutOut

jλ

jj

j

PPψ

ψψ ψ

=ψ

O

BlackBoxBlackBox

MacroWorldMacroWorld

QuantumQuantumWorldWorld

EigenvalueEigenvalueObservableObservable

Q. Sys.Q. Sys.StateState

Q. Sys.Q. Sys.StateState

Pr job Pψ ψ=

j jjPλ= ∑Owherewhere Spectral DecompositionSpectral Decomposition

PhysicalPhysicalRealityReality

PhilosopherPhilosopherTurfTurf

•distributive quantum computing

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