1

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: Lomonaco@UMBC.EDULomonaco@UMBC.EDUWebPageWebPage: : http://http://www.csee.umbc.edu/~lomonacowww.csee.umbc.edu/~lomonaco

Quantum ComputingQuantum Computing

OverviewOverviewFour TalksFour Talks

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

•• Three Quantum AlgorithmsThree Quantum Algorithms

•• Quantum Hidden Subgroup AlgorithmsQuantum Hidden Subgroup Algorithms

•• An Entangled Tale of Quantum EntanglementAn Entangled Tale of Quantum Entanglement

ElementaryElementary

AdvancedAdvanced

AnAnEntangled TaleEntangled Tale

Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr.Dept. of Computer Science & Electrical EngineeringDept. of Computer Science & Electrical Engineering

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

Email: Email: Lomonaco@UMBC.EDULomonaco@UMBC.EDUWebPage: WebPage: http://www.csee.umbc.edu/~lomonacohttp://www.csee.umbc.edu/~lomonaco

Quantum EntanglementQuantum Entanglementofof

Lecture 4Lecture 4

This work is supported by:This work is supported by:

•• The Defense Advance Research Projects The Defense Advance Research Projects Agency (DARPA) & Air Force Research Agency (DARPA) & Air Force Research Laboratory (AFRL), Air Force Materiel Command, Laboratory (AFRL), Air Force Materiel Command, USAF Agreement Number F30602USAF Agreement Number F30602--0101--22--0522.0522.

•• The National Institute for Standards The National Institute for Standards and Technology (NIST).and Technology (NIST).

•• The Mathematical Sciences The Mathematical Sciences Research Institute (MSRI).Research Institute (MSRI).

•• The LThe L--OO--OO--P Fund.P Fund.LL--OO--OO--PP

NielsenNielsenPeresPeres

PlenioPlenioPopescuPopescu

SchumacherSchumacherSudberrySudberry

TerhalTerhalWallachWallach

BennetBennetBrassardBrassard

BrylinskiBrylinskiHorodeckiHorodecki’’ss

Jonathan Jonathan JozsaJozsa

LindenLinden

MeyerMeyer

Talk based on the work of many peopleTalk based on the work of many people

LomonacoLomonaco

& Many Others& Many Others

2

Lomonaco, Samuel J., Jr., Lomonaco, Samuel J., Jr., An Entangled Tale An Entangled Tale of Quantum Entanglementof Quantum Entanglement, in AMS PSAPM/58, , in AMS PSAPM/58, (2002), pages 305 (2002), pages 305 –– 349.349.

PreamblePreamble

Physics Quantum Entanglement LaboratoryPhysics Quantum Entanglement Laboratory Quantum Entanglement is Quantum Entanglement is ……

( ) ( ) ( )2 2Adn nL n u u× →

Big AdjointBig AdjointActionAction

( ) ( )(2 )nn Vec uΩ →

InfinitesimalInfinitesimalActionAction

LiftLift

Over the 20Over the 20--th century, the scientific communityth century, the scientific community’’s s view of view of Q.E.Q.E. has dramatically changed.has dramatically changed.

Initially, Initially, Q.E.Q.E. was viewed as an unnecessary and was viewed as an unnecessary and unwanted ugly unwanted ugly WartWart on Quantum Mechanics.on Quantum Mechanics.••

EPR tried to surgically remove it.EPR tried to surgically remove it.

Bell and Aspect showed that Bell and Aspect showed that surgery can not be performed surgery can not be performed w/o destroying the very life of w/o destroying the very life of physical reality.physical reality.

••••

QuantumQuantumMechanicsMechanics

TodayToday, , Q.E.Q.E. is viewed as a useful resource is viewed as a useful resource in Q.M. It is viewed as a in Q.M. It is viewed as a commoditycommodity to be to be traded and utilized.traded and utilized.

Q.E. SavingsQ.E. Savings& Loan& Loan

Q.E. NASDAQQ.E. NASDAQExchangeExchange

3

••

••

Q.E.Q.E. appears to be an important appears to be an important resource for quantum computationresource for quantum computation

Many claim that it is Many claim that it is Q.E.Q.E. that that somehow enables us to harness the somehow enables us to harness the vast parallelism of Quantum vast parallelism of Quantum Mechanics.Mechanics.

Quantum EntanglementQuantum Entanglement&&

Quantum ComputationQuantum Computation WhatWhatisisQ.E. ?Q.E. ?

??? Questions ?????? Questions ???

•• How do we How do we Q.E.Q.E. ??•• MeasureMeasure•• QuantifyQuantify•• ClassifyClassify

When is the When is the Q.E.Q.E. of two quantum of two quantum systems the same? Different? systems the same? Different?

When is the When is the Q.E.Q.E. of one quantum of one quantum systems greater than another ? systems greater than another ?

••••

Answers to the above questions are expected to Answers to the above questions are expected to have a profound impact on the development of have a profound impact on the development of Quantum ComputationQuantum ComputationFinding answers to these questions isFinding answers to these questions is

•• ChallengingChallenging•• IntriguingIntriguing•• Very HabitVery Habit

FormingForming

••••

OverviewOverview

Lie algebra ofLie algebra of

Unitary GroupUnitary Group

Lie algebra ofLie algebra of

Local Unitary GroupLocal Unitary Group( ) ( )1

2n

L n S U= ⊗

1

( ) ( 2 )n

n s u= +

( )2nU

( )2nu

( )L n

( )2nU

( ) (2 ) (2 )Adn nL n u u× →Density Ops Density Ops ““LiveLive”” HereHere

Big Adjoint actionBig Adjoint actionof of L(n)L(n) on on u(2u(2nn))

The problem ofThe problem ofQ.E. Q.E. ““LivesLives”” HereHere

Overview (Cont.)Overview (Cont.)

( ) (2 ) (2 )Adn nL n u u× →

•• We study We study Q.E.Q.E. by lifting the above by lifting the above action to the induced infinitesimal action to the induced infinitesimal action action

( ) ( )(2 )nn Vec uΩ →

•• We use the induced infinitesimal We use the induced infinitesimal action to quantify and classify action to quantify and classify Q.E.Q.E.by constructing a by constructing a complete set ofcomplete set ofQ.E.Q.E. invariantsinvariants. .

Chapter 1Chapter 1

A Story of Two QubitsA Story of Two Qubits

oror

How Alice & BobHow Alice & BobLearn to Live with Q.E. Learn to Live with Q.E.

& Love it& Love it

4

Bob Pops the Big QuestionBob Pops the Big Question

AliceAliceBobBob

What is What is Q.EQ.E. ???. ???

Gee, I donGee, I don’’t t know ???know ???

A Story of Two QubitsA Story of Two QubitsOur saga continues Our saga continues

•• Alice & Bob visit the local Alice & Bob visit the local Q.E.Q.E. wholesale wholesale outlet outlet

•• They find on one of the shelves a box labeled They find on one of the shelves a box labeled as follows: as follows:

Two Qubit Quantum SystemTwo Qubit Quantum System

Consisting of QubitsConsisting of Qubitsand and

Q.E., Inc.Q.E., Inc.

ABQ

AQ BQ

(*) Not legally responsible for the effects of decoherence.(*) Not legally responsible for the effects of decoherence.(*) Not legally responsible for the effects of decoherence.(*) Not legally responsible for the effects of decoherence.

The content label required by federal law reads The content label required by federal law reads

StateStateSpaceSpace

Unitary Unitary Transf.Transf.

StateStateHilb.Hilb.SpaceSpace

Q SysQ Sys

ABQ

AQ

BQ

ABH

AH

BH

ρ

ABρ

Aρ

Bρ

( )22AB

U

( )2A

U

( )2B

U

2(2 )ABu

(2)Au

(2)Bu

1/ 2 0 0 1/ 2

0 0 0 0

0 0 0 0

1/ 2 0 0 1/ 2

ABρ

− = −

AQUniv. ofUniv. of

MelbourneMelbourne

A Story of Two QubitsA Story of Two Qubits

•• Alice & Bob purchase the two qubit system .Alice & Bob purchase the two qubit system .ABQOutside the store, they open the box. Alice Outside the store, they open the box. Alice takes qubit . Bob takes qubit .takes qubit . Bob takes qubit .AQ BQ••

•• They then separate with their respective qubits. They then separate with their respective qubits. Alice flies to the University of Melbourne. Bob Alice flies to the University of Melbourne. Bob flies to Vancouver, British Columbia to attend flies to Vancouver, British Columbia to attend the University of British Columbia.the University of British Columbia.

Univ. ofUniv. ofB.C.B.C.

BQAlic

eAlic

e BobBob

A Story of Two QubitsA Story of Two Qubits

After they arrive, Alice has second thoughts After they arrive, Alice has second thoughts about their purchase. She phones Bob, and about their purchase. She phones Bob, and rattles off in rapid succession the following rattles off in rapid succession the following two questions:two questions:

•• Did we get our moneyDid we get our money’’s worth of s worth of Q.E.Q.E. ??

•• How much How much Q.E.Q.E. did we actually purchase ?did we actually purchase ?

Bob immediately hangs up, and phones the Bob immediately hangs up, and phones the Q.E.Q.E. Consumer Protection agency, which refers Consumer Protection agency, which refers him to the National Institute of him to the National Institute of Q.E.Q.E.Standards & Technology (NIStandards & Technology (NIQEQEST) in ST) in Gaithersburg, MD.Gaithersburg, MD.

? ? ?? ? ?

•• After receiving their two boxes, Alice and Bob After receiving their two boxes, Alice and Bob open them, take out their respective qubits open them, take out their respective qubits and read the enclosed manuals.and read the enclosed manuals.

•• After a long distance (After a long distance ($$$$$$) phone conversation, ) phone conversation, NIQEST agrees to send Alive & Bob their NIQEST agrees to send Alive & Bob their STANDARDS Q.E. KITSTANDARDS Q.E. KIT..••

The NIQEST rep. Takes a The NIQEST rep. Takes a STANDARDSTANDARD Q.E.Q.E. two two qubit quantum system off the self, places qubit quantum system off the self, places qubit together with a qubit together with a STANDARDS STANDARDS MANUALMANUAL in a Box A. He/She also takes qubit in a Box A. He/She also takes qubit together with a together with a STANDARDS MANUALSTANDARDS MANUAL in Box B. in Box B. He/She then sends the two boxes by overnight mail He/She then sends the two boxes by overnight mail ($$$)($$$) respectively to Alice and Bob.respectively to Alice and Bob.

' ''A BQ''AQ

''BQ

5

The The STANDARDS MANUALSTANDARDS MANUAL reads as follows:reads as follows:

QQ..EE.. YardstickYardstick 11. and possess the . and possess the samesame Q.E.Q.E. if it is possible for Alice and Bob if it is possible for Alice and Bob to use their own local reversible operations to use their own local reversible operations (either individually or collectively) to (either individually or collectively) to transform and into one another. transform and into one another. If this is possible, then and are If this is possible, then and are of the of the same entanglement typesame entanglement type, written, written

ABQ

ABQ ' 'A BQ'ABQ ' 'A BQ'

' 'A BQ'

' 'AB A Bloc

Q Q'∼

The The STANDARDS MANUALSTANDARDS MANUAL reads as follows:reads as follows:

QQ..EE.. YardstickYardstick 22. possesses . possesses moremore Q.E.Q.E.thanthan if it is possible for Alice and Bob if it is possible for Alice and Bob (either individually or collectively) to apply (either individually or collectively) to apply their own reversible and irreversible local their own reversible and irreversible local operations to their local qubits to transformoperations to their local qubits to transform

into . In this case, we writeinto . In this case, we write

ABQ

ABQ

' 'A BQ'

' 'A BQ'

' 'AB A Bloc≥Q Q'

CAVEAT:CAVEAT: Q.E.Q.E. may be irrevocably may be irrevocably lost if lost if QQ..EE.. YardstickYardstick 22 is applied.is applied.

SummarySummary

•• QuestionQuestion.. What type of What type of Q.E.Q.E. do Alice do Alice and Bob collectively possess ?and Bob collectively possess ?

•• QuestionQuestion.. Is the Is the Q.E.Q.E. of greater of greater than the than the Q.E.Q.E. of ?of ?

' 'A BQ'ABQ

•• QuestionQuestion.. Is the Is the Q.E.Q.E. of the of the same as the same as the Q.E.Q.E. of ?of ?

ABQ' 'A BQ'

Chapter 2Chapter 2

Definition of the Definition of the Problem of Quantum Problem of Quantum

EntanglementEntanglement

Back to Alice & BobBack to Alice & Bob

Same story forSame story for

•• Alice, Bob, & Cathy and Alice, Bob, & Cathy and 33 qubitsqubits

•• nn people andpeople and nn qubitsqubits

•• Let be Let be nn qubits, and let qubits, and let denote their respectivedenote their respective

Hilbert spacesHilbert spaces

1 2, , , nQ Q Q…1 2, , , nH H H…

•• Let be the globalLet be the globalquant. sys. with Hilbert Space quant. sys. with Hilbert Space

1 2, , , n=Q Q Q Q…1

n

kk== ⊗H H

FundamentalFundamental ProblemProblem ofof QQ..EE. (. (FPQEFPQE).). Let Let and be density operators representing and be density operators representing two different states of . Is it possible two different states of . Is it possible to move from state to state by to move from state to state by applying only local moves ?applying only local moves ?

ρ

ρ

'ρ

'ρ

QQ

Question.Question. But what is a But what is a locallocal movemove ??

6

Local Moves ?Local Moves ?

1)1) ((StandardStandard) ) local unitarylocal unitary transformationstransformations

( )1 1

n n

k kk kU U= =∈⊗ ⊗ H

For example, For example, , ,A B A BU I I U U U⊗ ⊗ ⊗

2)2) Measurement ofMeasurement of ((StandardStandard) ) locallocal observablesobservables

( )1 1

n n

k kk kObservables

= =∈⊗ ⊗O H

For example, For example, , ,A B A BO I I O O O⊗ ⊗ ⊗

Local Moves ?Local Moves ?

3)3) ExtendedExtended local unitarylocal unitary transformationstransformations

( )1

nkkk

U= ⊗⊗ H H

where, are where, are distinct Hilbert spacesdistinct Hilbert spaces

1 21 2, , , , , , nnH H H H H H

4)4) Measurement ofMeasurement of ExtendedExtended locallocal observablesobservables

( )1

nkkk

Observables=

⊗⊗ H H

where, are where, are distinct Hilbert spacesdistinct Hilbert spaces

1 21 2, , , , , , nnH H H H H H

RestrictedRestricted FPQEFPQE ((RFPQERFPQE):): Given two density operators Given two density operators and and in the Lie algebra in the Lie algebra u(2u(2nn)) , does there exist , does there exist

a local move a local move UU ε ε L(n)L(n) s.t.s.t.

The Group The Group L(n)L(n) of Local Unitary Transformationsof Local Unitary Transformations

DefinitionDefinition.. The The groupgroup ofof locallocal unitaryunitarytransformationstransformations L(n)L(n) is the subgroup of is the subgroup of U(2U(2nn))defined bydefined by

1( ) (2)

nL n SU= ⊗

ConventionConvention.. From this point on, we consider only From this point on, we consider only the RFPQE. Thus, for the rest of the talk the RFPQE. Thus, for the rest of the talk

( ) 'UAd i iρ ρ=

Local Moves = Local Moves = L(n)L(n)

????( ) ( ) 1 'UAd i U i U iρ ρ ρ−= =

iρ 'iρ

Terminology Terminology

provided there exists a such that provided there exists a such that ( )U L n∈

( ) ( ) 1' Ui Ad i U i Uρ ρ ρ −= =

DefinitionDefinition.. Two elements Two elements ιριρ and and ιριρ ’’ in in u(2u(2nn)) are are said to be said to be locallylocally equivalentequivalent, written , written

'loc

i iρ ρ∼

The equivalence classThe equivalence class

[ ] ' : 'E loc

i i i iρ ρ ρ ρ= ∼

is called an is called an entanglemententanglement classclass (or (or orbitorbit). Finally, ). Finally, letlet (2 ) / ( )nu L ndenote the denote the setset ofof entanglemententanglement classesclasses..

Set of Entanglement Classes Set of Entanglement Classes u(2u(2nn)/L(n))/L(n)But What Is Q.E. ?But What Is Q.E. ?

Lest we forget, our objective is to measure, Lest we forget, our objective is to measure, quantify, classify quantify, classify Q.E.Q.E.

ObjectiveObjective 1.1. Find the dimension of each Find the dimension of each entanglement class. Given , findentanglement class. Given , findiρ [ ]( )E

Dim iρ

ObjectiveObjective 2.2. Find a Find a complete set of invariantscomplete set of invariantswhich classify all entanglement classes. In other which classify all entanglement classes. In other words, find a finite set words, find a finite set f f11,f,f22, , …… , , ffKK of real of real valued functions on valued functions on u(2u(2nn)) which distinguish all which distinguish all entanglement classes, i.e.,entanglement classes, i.e.,

( ) ( )' 'k kloci i f i f i kρ ρ ρ ρ⇔ = ∀∼

We will achieve the following two objectives:We will achieve the following two objectives:

7

Chapter 3Chapter 3

Applications Applications of of

Lie GroupLie Group

PerspectivePerspective

MathMathPhysicsPhysics

Observables:Observables:Density Ops:Density Ops:

HermitianHermitian OpsOps

Hilbert Space Hilbert Space

Unitary GroupUnitary GroupLie GroupLie Group

Observables:Observables:Density Ops:Density Ops:

Skew Skew HermitianHermitian OpsOps

where Lie algebra of where Lie algebra of

Dynamics via Dynamics via Dynamics via Dynamics via

wherewhereis the Big is the Big adjointadjoint rep. rep.

OρN N×

† TA A A= =

( )Dim N=HH

( )U N

iOiρ

N N× ( )u N∈( ) ( )† TiA iA iA= = −( )u N = ( )U N

( )U U N∈Uψ ψ

†U Uρ ρ

( )U U N∈Uψ ψ

( )Ui A d iρ ρ( ) ( ) 1

UAd i U i Uρ ρ −=

PauliPauli Spin MatricesSpin Matrices

1 2 31

0 1 0 1 0, ,

1 0 0 0 1

i

iσ σ σ

− = = = −

We also denote the We also denote the 2 x 22 x 2 identity matrix byidentity matrix by

0

1 0

0 1σ =

Basis for the Lie Algebra Basis for the Lie Algebra u(2)u(2)

••

If , then If , then (2)i uρ ∈0 0 1 1 2 2 3 3

0 0

i x x x x

x x

ρ ξ ξ ξ ξ

ξ ξ

= + + +

= + i

A vector space basis for A vector space basis for u(2)u(2) isis••0 0 1 1 2 2 3 3

1 1 1 1, , ,

2 2 2 2i i i iξ σ ξ σ ξ σ ξ σ =− =− =− =−

wherewhere

( )( )

0

31 2 3

1 2 3

if is a density 1

, ,

, , (2) (2) )

p

(2

ox i

x x x x

u u u

ρ

ξ ξ ξ ξ

= −

= ∈ = ∈ × ×

Coordinate Chart forCoordinate Chart for u(2)u(2)

( )4

0 1 2 3

: (2)

, , ,

u

i x x x x

πρ

→

Basis for the Lie Algebra Basis for the Lie Algebra u(2u(222))

00 0 0 01 0 1 33 3 3

1 1 1, , ,

2 2 2

1: , 0,1,2,3

2jk j k

i i i

i j k

ξ σ σ ξ σ σ ξ σ σ

ξ σ σ

=− ⊗ =− ⊗ =− ⊗

= =− ⊗ =

…

A basis for A basis for u(2u(222)) isis••

3

, 0 jk jkj ki xρ ξ

==∑

••

where if is a density operator where if is a density operator iρ00 1x = −

8

Coordinate Chart forCoordinate Chart for u(2u(222))

( )2 16

00 01 02 033 10 33

: (2 )

, , , , ,

u

i x x x x x x

πρ

→…

Basis for the Lie Algebra Basis for the Lie Algebra u(2u(2nn))

A basis for A basis for u(2u(2nn)) isis••

1 2 1 21 2

3

, , , 0 n nnj j j j j jj j j

i xρ ξ=

=∑ …

••

1 2 1 21

1: , , , 0,1,2,3

2n

n

j j j j ni j j jξ σ=

= − = ⊗ …

Coordinate Chart forCoordinate Chart for u(2u(2nn))

( )4

00 00 00 01 33 33

: (2 )

, ,

nnu

i x x x

πρ

→

… … ……

OverviewOverview

( ) ( )( ) ( ) ( ) ( )I Ru N T U N Vec U N Der C U N∞= = =

•• An An N x NN x N skew skew HermitianHermitian matrixmatrix

•• A tangent vector to A tangent vector to U(N)U(N) at at II

•• A right invariant vector field on A right invariant vector field on U(N)U(N)

is simultaneously each of the is simultaneously each of the following:following:

( )v u N∈

•• A derivation (directional derivative) onA derivation (directional derivative) on( ) ( ) : ( ) smoo h: t C U N f U N f∞ = →

Also RecallAlso Recall

•• In In U(2U(2nn)) , there are , there are 44nn independent independent directions to move in, namelydirections to move in, namely

1 2 1 21

1: , , , 0,1,2,3

2n

n

j j j j ni j j jξ σ=

= − = ⊗ …

•• For example, from we could moveFor example, from we could movefrom from gg in the direction in the direction

by following the curveby following the curve

from from t=0t=0 on.on.

( )2g U∈

2 2

12

iξ σ= −

2( ) tt e gξγ =

The Special Unitary Group The Special Unitary Group SU(N)SU(N)

( ) ( ) ( ) : det 1SU N U U N U= ∈ =

( ) ( ) ( ) : 0su N v u N trace v= ∈ =

•• A basis for A basis for su(2su(2nn)) is: is:

1 2

1 k

0,1,2

N

,31:

2 ot all are zero j n k

nk

j j j jk

j kiξ σ

=

∈ ∀ = −

⊗

•• Therefore Therefore Dim(SU(2Dim(SU(2nn))=4))=4nn--11

Lie Lie GroupGroup

Lie Lie AlgebraAlgebra

9

Big Big AdjointAdjoint & & little little adjointadjoint

•• BigBig AdjointAdjoint AdAd for global dynamics of for global dynamics of iiρρ

( ) ( ) ( ) 1

( ) ( ) ( )

,

Ad

U

U N u N u N

U i Ad i U i Uρ ρ ρ −

× →=

•• littlelittle adjointadjoint adad for infinitesimal dynamicsfor infinitesimal dynamicsof of iiρρ

( ) ( ) [ ]( ) ( ) ( )

, ,

ad

v

u N u N u N

v i ad i v iρ ρ ρ× →

=

where where [ ] ( ) ( ),v i v i i vρ ρ ρ= −

The Exponential MapThe Exponential Map

0

exp : ( ) ( )

!

kv

k

u N U N

vv e

k

∞

=

→

=∑

Moreover, , i.e.,Moreover, , i.e.,exp( ) exp( )v vAd ad=

( )

( )

( ) ( )

exp exp

( ) ( )

vad

Ad

U

v ad

u N End u N

U N Aut u N

U Ad

→↓ ↓

→

The Special Orthogonal Group The Special Orthogonal Group SO(3)SO(3)

•• The Lie group of rotations in isThe Lie group of rotations in is3

1(3) (3, ): d t( ) 1and eTSO A GL A A A−= ∈ = =

•• Its Lie algebra isIts Lie algebra is

and(3) (3, ): ( ) 0Tso A Mat v v trace v= ∈ =− =

•• And we haveAnd we have

( )

( )

(2) (2) (3)

exp exp

(2) (3) (3)

ad

Ad

u End u so

U Aut u SO

→ =↓ ↓

→ =

DiracDirac Belt TrickBelt Trick

Chapter 4Chapter 4

InvariantsInvariantsofof

Quantum Quantum EntanglementEntanglement

The Lie algebra of is the subThe Lie algebra of is the sub--Lie Lie algebra of algebra of u(2u(2nn)) generated by:generated by:

The Lie algebra of the Local GroupThe Lie algebra of the Local Group( )n ( )L n

1( ) (2) (2 )

n nL n SU U= ⊂⊗( ) (2 ) (2 )Adn nL n u u× →

LocalLocalGroupGroup

Big Big AdjointAdjointactionaction

( )n

1 2

, and

where exactly on

0,1,2,3:

e 0n

j

k k kj

k j

kξ

∈ ∀ ≠

( )L n

•• by by

•• is generated by is generated by (1) 1 2 3, ,ξ ξ ξ

•• by by (2)

(3)

01 02 03 10 20 30, , , , ,ξ ξ ξ ξ ξ ξ

001 002 003 010 020 030 100 200 300, , , , , , , ,ξ ξ ξ ξ ξ ξ ξ ξ ξ

The Lie algebra of the Local GroupThe Lie algebra of the Local Group( )n ( )L n

For example, For example,

10

What it is all aboutWhat it is all about

The essence of the RFPQE is to The essence of the RFPQE is to understandunderstandthe the BIG BIG AdjointAdjoint actionaction of the local group of the local group L(nL(n)) on the Lie algebra on the Lie algebra u(2u(2nn)) , namely, , namely, the actionthe action

What is a Q.E. invariant ?What is a Q.E. invariant ?

We seek We seek invariantsinvariants of the of the BIG BIG AdjointAdjointactionaction of the group of the group L(nL(n)) on the Lie algebra on the Lie algebra u(2u(2nn)).. ( ) (2 ) (2 )Adn nL n u u× →

iiρρ

iiρρ’’

AdAdUU

u(2u(2nn))

EntEnt. Class. Class

Fund. Fund. ObjObj..of Studyof Study

EssenceEssenceof of Q.E.Q.E.

Let Let

The Algebra of Q.E. InvariantsThe Algebra of Q.E. Invariants( ) ( )(2 )

L nnC u∞

( ) smo(2 ) : ( : ot2 h)n nC u f u f∞ = →

DefinitionDefinition.. The The algebraalgebra ofof QQ..EE. . invariantsinvariants, , denoted by , is defined asdenoted by , is defined as( ) ( )

(2 )L nnC u∞

( ) ( ) (2 ) : ( ) ( ) ( )nUf C u f Ad i f i U L Nρ ρ∞∈ = ∀ ∈

If , thenIf , then

A Complete Set of InvariantsA Complete Set of Invariants

( ) ( )(2 )

L nnf C u∞∈' ( ) ( ')

( ) ( ') 'loc

not loc

i i f i f i

f i f i i i

ρ ρ ρ ρ

ρ ρ ρ ρ

∴ ⇒ =

∴ ≠ ⇒

∼

∼

However, However, ?

( ) ( ') ~ 'loc

f i f i i iρ ρ ρ ρ= ⇒

In this case, we know nothing. The invariant is not In this case, we know nothing. The invariant is not enough to distinguish all Q.E. classes.enough to distinguish all Q.E. classes.

We seek We seek enough Q.E. invariantsenough Q.E. invariants to distinguish all to distinguish all Q.E. classes. Such a set of Q.E. invariants is calledQ.E. classes. Such a set of Q.E. invariants is called

A Complete Set of InvariantsA Complete Set of Invariants

How do we find Q.E. invariants ?How do we find Q.E. invariants ?

We seek which are invariant We seek which are invariant under the BIG under the BIG AdjointAdjoint actionaction

: (2 )nf u →

( ) (2 ) (2 )Adn nL n u u× →

In other words, we seek In other words, we seek such thatsuch that

( ) ( )( ) , (2 ), U L(n)nUf Ad i f i i uρ ρ ρ= ∀ ∈ ∀ ∈

OurOur ApproachApproach:: Lift the problem to the Lie Lift the problem to the Lie algebra where it becomes a linear algebra where it becomes a linear problem.problem.

( )n

f

The BIG The BIG AdjointAdjoint action of action of on on induces an induces an infinitesimalinfinitesimal actionaction

( )( ) (2 )nn Vec uΩ →

Let . We define the vector field Let . We define the vector field by constructing the tangent vector by constructing the tangent vector for each . Let be a for each . Let be a smooth curve in defined bysmooth curve in defined by

( )v n∈ ( )vΩ( ) iv ρΩ

(2 )ni uρ ∈ ( )v tγ(2 )nu

( )exp( )( )v tvt Ad iγ ρ=

Then passes through at time, . Then passes through at time, . Define to be the tangent vector to Define to be the tangent vector to at time .at time .

( )v tγ iρ 0t =( )

iv ρΩ ( )v tγ0t =

(2 )nu( )L n

11

The Infinitesimal ActionThe Infinitesimal Action

( ) vi iv adρ ρΩ =

( )exp( )( )v tvt Ad iγ ρ=

What is the meaning of the infinitesimal action ?What is the meaning of the infinitesimal action ?

( )( ) (2 )nn Vec uΩ →

Each is a direction in from whichEach is a direction in from which

we can move w/o leaving the Q.E. class .we can move w/o leaving the Q.E. class .

Movement in all directions not in Movement in all directions not in

will force us to immediately leave .will force us to immediately leave .

( )i

v ρΩ (2 )nu[ ] Eiρ

( )Im ( )i

v ρΩ

[ ] Eiρ

( ) vi iv adρ ρΩ =

[ ] ( ) ( )Im (2 ) (2 )n ni iE i i

T i Vec u T uρ ρρ ρρ = Ω ⊂ =

ObjectiveObjective 1. Find (Achieved !!! )1. Find (Achieved !!! )[ ] EDim iρ

ConsiderConsider ( ) ( )1 1( ) Im (2 )nn Vec uΩ− → Ω ⊂

ThenThen

[ ] [ ] ( )Imi iE EDim i DimT i Dimρ ρρ ρ= = Ω

Hence, Hence,

Tangent SpaceTangent Spaceto at to at [ ] E

iρ iρ Tangent SpaceTangent Spaceto at to at (2 )nu iρ

ObjObj. 2. Find a complete set of Q.E. invariants. 2. Find a complete set of Q.E. invariants

Achieved !!Achieved !!

( ) ( )1 1( ) Im( ) (2 ) (2 )n nn Vec u Der C uΩ ∞− → Ω ⊂ =

Recall that consists of all directions Recall that consists of all directions in that we can move in w/o leaving a in that we can move in w/o leaving a Q.E. class that we are presently in. IfQ.E. class that we are presently in. If

( )Im Ω(2 )nu

( ) ( )(2 )

L nnf C u∞∈then will not change if we move in any then will not change if we move in any direction in .direction in .( )Im Ω

f

TheoremTheorem.. Let be a vector Let be a vector space basis of the Lie algebra . Then space basis of the Lie algebra . Then

1 2 3, , , , nv v v…( )n

( ) ( )( )(2 ) 0

L nnjf C u v f j∞∈ ⇔ Ω = ∀

Hence, finding Q.E. invariants is equivalent Hence, finding Q.E. invariants is equivalent to solving the above system of linear partial to solving the above system of linear partial differential equations.differential equations.

The above approach leads to the following The above approach leads to the following theorem:theorem:

Chapter 5Chapter 5

Q.E. Invariants Q.E. Invariants forfor

n=1n=1QubitsQubits

12

Q.E. Invariants for Q.E. Invariants for n=1n=1 QubitsQubits

•• This is a trivial, but instructive caseThis is a trivial, but instructive case

•• No Q.E.; but many Q.E. classes !!!No Q.E.; but many Q.E. classes !!!∃

For , basis For , basis 31 21 2 3, ,

2 2 2ii i σσ σξ ξ ξ = =− =− =−

(1)

For , basis For , basis 0 1 2 3, , ,ξ ξ ξ ξ=(2)u

3(1) (2)su= =••4(2)i uρ ∈ =••

(1) (2) -sph e3 erL SU= =••

Coordinate Coordinate Chart forChart for u(2)u(2)( ) ( )4

3

0 1 2 3 00

( 2 )

, , , ,j jj

u

i x x x x x x x

π

ρ ξ=

→

= =∑

( )0 1 2 3

bas, , , (is 2) for Vec ux x x x∂ ∂ ∂ ∂

∂ ∂ ∂ ∂

Coordinate ChartCoordinate Chartfor for u(2)u(2)

CurveCurve

( )1 0 1 2 3( ) , , ,t x x t x xγ = +

π4

iρ

TangentTangentVectorVector 1/ x∂ ∂

(2)u

( )0 ,x x

CurveCurve( )1

1π γ−

We identify via the chart We identify via the chart 4(2)u = π( )0 0 1 1 2 2 3 3 0 ,i x x x x x xρ ξ ξ ξ ξ∴ = + + + =

0 00

0k kk

ad LLξ

∴ = ⊕ =

wherewhere

1 2 3

0 0 0 0 0 1 0 1 0

0 0 1 , 0 0 0 , 1 0 0

0 1 0 1 0 0 0 0 0

L L L

− = − = = −

are the infinitesimal generators of the Lie algebra are the infinitesimal generators of the Lie algebra of the special orthogonal groupof the special orthogonal group(3)so (3)SO

Hence, Hence, ( ) 00 0

0k kk

xad i L x

L xξ ρ

= =

and soand so

( ) ( ) ( )0

1

2

3

/

/

/

/

k

T

k k

x

xad i L x

x

x

ξξ ρ

∂ ∂ ∂ ∂ Ω = = ∇∂ ∂ ∂ ∂

i i

wherewhere

( )1 1 1/ , / , /T

x x x∇ = ∂ ∂ ∂ ∂ ∂ ∂

Thus, is spanned by Thus, is spanned by ( )Im Ω

( )

( )

( )

1 3 22 3

2 1 33 1

3 2 11 2

x xx x

x xx x

x xx x

ξ

ξ

ξ

∂ ∂Ω = − ∂ ∂∂ ∂Ω = − ∂ ∂

∂ ∂Ω = − ∂ ∂

( )if2 0

Im0if0

i

xDim

xρ

≠ Ω = =

Hence, Hence,

The Bloch The Bloch ““SphereSphere””

13

So span theSo span thetangent space to attangent space to at

1 2 3( ) , ( ) , ( )i i iρ ρ ρξ ξ ξΩ Ω Ω

[ ] Eiρ iρ

Moreover,Moreover,

is the normal vector field to is the normal vector field to

1 2 31 2 3

x x xx x x∂ ∂ ∂+ +

∂ ∂ ∂

[ ] Eiρ

Finally, a Complete Set of InvariantsFinally, a Complete Set of Invariants

The solution of The solution of

1

2

3

( ) 0

( ) 0

( ) 0

f

f

f

ξξξ

Ω =Ω =Ω =

isis

2( )f i xρ =

Chapter 6Chapter 6

Q.E. Invariants Q.E. Invariants forfor

n=2n=2QubitsQubits

Q.E. Invariants for Q.E. Invariants for n=2n=2 QubitsQubits

For , basis For , basis : , 0,1,2,3jk j kξ= ∈2(2 )u

6(2) (2) (2)su su= + =••16(2 2)i u Hρ ∈ =••

(2) (2) (2)L SU SU= ⊗••

For , basis For , basis (2) 01 02 03 10 20 30, , , ,ξ ξ ξ ξ ξ ξ=

wherewhere

( ) ( )A B A I I B+ ≡ ⊗ ⊕ ⊗

KroneckerKronecker SumSum

Coordinate Chart for Coordinate Chart for u(2u(222))

( )

( )

2 16

3

00 01 03 10 33, 0

0*000* *0 **00

*0 **

(2 )

, , , , ,

, , ,

jk jkj k

T

u

i x x x x x x

x xx x x x

x x

π

ρ ξ=

→

=

= =

∑ …

wherewhere

11 12 13

** 21 22 23

31 32 33

x x x

x x x x

x x x

=

( )0* 01 02 03, ,x x x x=

( )** 11 12 13 21 22 23 31 32 33, , , , , , , ,x x x x x x x x x x=

( )0* 10 20 30, ,x x x x=

A basis for isA basis for is( )2(2Vec u

: , 0.1,2,3jk

j kx

∂ = ∂

14

( ) ( )

( ) ( )0

0

4 4

4 4

0 0

0 0

k k

k k

k

k

ad ad I L I

ad I ad I L

ξ ξ

ξ ξ

= ⊕ ⊗ = ⊕ ⊗

= ⊗ ⊕ = ⊗ ⊕

So, So,

( ) ( )( )0

00

0*

4*0

**

0k

T

Tk

T

x

xad i L I

x

x

ξ ρ

= ⊕ ⊗

A similar formula holds for A similar formula holds for ( )0k

ad iξ ρ

Hence, Hence, ( ) ( )( ) ( )

0

0

0

0

k

k

k

k

ad i

ad i

ξ

ξ

ξ ρ

ξ ρ

Ω = ∇

Ω = ∇

i

i

wherewhere

0* *0 0* **

, , ,x x x x

∂ ∂ ∂ ∂ ∇ = ∂ ∂ ∂ ∂

( )

( )

( )

01 02 03 2 303 02 3 2

02 03 01 3 101 03 1 3

03 01 02 1 202 01 2 1

j jj j j

j jj j j

j jj j j

x x x xx x x x

x x x xx x x x

x x x xx x x x

ξ

ξ

ξ

∂ ∂ ∂ ∂Ω = − + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂Ω = − + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂Ω = − + − ∂ ∂ ∂ ∂

∑

∑

∑

( )

( )

( )

10 20 30 2 330 20 3 2

20 30 10 3 110 30 1 3

30 10 20 1 220 10 2 1

j jj j j

j jj j j

j jj j j

x x x xx x x x

x x x xx x x x

x x x xx x x x

ξ

ξ

ξ

∂ ∂ ∂ ∂Ω = − + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂Ω = − + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂Ω = − + − ∂ ∂ ∂ ∂

∑

∑

∑

( )Im Ω is spanned byis spanned by It follows that, It follows that, ( ) a e.Im .6i

Dim ρ Ω =

In other words, almost all density operatorsIn other words, almost all density operatorsbelong to a belong to a 6 dimensional Q.E. class6 dimensional Q.E. class..iρ

But But …… there are some important there are some important exceptionsexceptions..

Please notePlease note that all four 2that all four 2--qubit Bell states lie qubit Bell states lie in the same Q.E. class. Because of this, in the same Q.E. class. Because of this, teleportation is possible.teleportation is possible.

Consider the Bell stateConsider the Bell state

( )

1

01 100 11

02 2

1

ψ

= − = −

1 0 0 1

0 0 0 012 0 0 0 0

1 0 0 1

ρ ψ ψ

− = = −

So, So,

In terms of the basis , this becomesIn terms of the basis , this becomes jkξ

( ) ( ) ( ) ( )[ ]00 11 22 33

11 1 1 1

4iρ ξ ξ ξ ξ= − + − + + + +

Thus, in this case is spanned byThus, in this case is spanned by( )ImiρΩ

( )( )( )

0 1 3 2 2 3

0 2 1 3 3 1

0 3 2 1 1 2

/ /

/ /

/ /

i

i

i

x x

x x

x x

ρ

ρ

ρ

ξ

ξ

ξ

Ω = ∂ ∂ + ∂ ∂ Ω = − ∂ ∂ + ∂ ∂ Ω = − ∂ ∂ − ∂ ∂

( )( )( )

1 0 2 3 3 2

2 0 3 1 1 3

3 0 1 2 2 1

/ /

/ /

/ /

i

i

i

x x

x x

x x

ρ

ρ

ρ

ξ

ξ

ξ

Ω = ∂ ∂ + ∂ ∂ Ω = − ∂ ∂ + ∂ ∂ Ω = − ∂ ∂ − ∂ ∂

15

Hence, the dimension of the Bell state Hence, the dimension of the Bell state entanglement class is an entanglement class is an exceptional exceptional 33, i.e., , i.e.,

[ ]Bell Eiρ

[ ] ( )Im 3Bell

Bell E iDim i Dim ρρ = Ω =

Linden, Linden, PopescuPopescu, , SudberrySudberry

A Complete Set of Q.E. Invariants for A Complete Set of Q.E. Invariants for n=2n=2 QubitsQubits

Let us use the above mentioned chart to make Let us use the above mentioned chart to make the identification the identification

0*00

*0 **

T

x xi

x xρ

=

π

and let and let

** **TZ x x=

A Complete Set of Q.E. Invariants for A Complete Set of Q.E. Invariants for n=2n=2 QubitsQubits

The following set of The following set of 99 algebraically independent algebraically independent polynomial functions form a polynomial functions form a basicbasic setset ofof polynomialpolynomialQQ..EE. . invariantsinvariants::

But But …… they do not form a complete set of Q.E. they do not form a complete set of Q.E. invariants !!!invariants !!!

A tenth polynomial , A tenth polynomial , which is algebraically dependent on the above which is algebraically dependent on the above polynomials is need to determine the sign of the polynomials is need to determine the sign of the components of and to form a components of and to form a completecomplete setset ofofQQ..EE. . invariantsinvariants. .

( ) ( )0* 0* 0*T Tx Zx Zx×i

iρ

( )trace Z ( )2trace Z ( )**det x

0* 0*Tx x 0* 0*

Tx Zx 20* 0*

Tx Z x

0* ** *0Tx x x 0* ** *0

Tx Zx x 20* ** *0

Tx Z x x

Chapter 7Chapter 7

ConclusionConclusion

The RFPQE The RFPQE ““LivesLives”” in the Followingin the FollowingMathematical StructureMathematical Structure

( ) ( ) ( )2 2Adn nL n u u× →

Big AdjointBig AdjointActionAction

( ) ( )(2 )nn Vec uΩ →

InfinitesimalInfinitesimalActionAction

LiftLift

In the paper In the paper

it is shown how the general problem of finding it is shown how the general problem of finding a a complete set of Q.E. invariantscomplete set of Q.E. invariants for for nnqubitsqubits can be reduced to the task of finding a can be reduced to the task of finding a complete set of solutions to the system of complete set of solutions to the system of 3n3nPDEsPDEs listed on the following slide: listed on the following slide:

The General Problem of Finding a Complete The General Problem of Finding a Complete Set of Q.E. Invariants for Set of Q.E. Invariants for nn QubitsQubits

““An Entangled Tale of Quantum EntanglementAn Entangled Tale of Quantum Entanglement,,””

16

For a Complete Set of Q.E. Invariants For a Complete Set of Q.E. Invariants Solve the Following System of Solve the Following System of 3n3n PDEsPDEs

1 2 1

1 2 1 0 1 2 1

1 2 1

1 2 1 0 1 2 1

1 2 1

1 2 1 0 1 2 1

3

*, , , *

3

*, , , *

3

*, , , *

0

0

0

n

n n

n

n n

n

n n

j j jj j j j j j

j j jj j j j j j

j j jj j j j j j

fx

x

fx

x

fx

x

−− = −

−− = −

−− = −

∂ × = ∂ ∂× = ∂ ∂× = ∂

∑

∑

∑

…

…

…

General General nn QubitQubit CaseCaseWhere, for example, Where, for example,

1 2 1

1 2 1

**

n

n

j j jj j j

fx

x−

−

∂×∂

denotes the denotes the vectorvector crosscross productproduct of the of the two vectorstwo vectors

1 2 1* nj j jx−

1 2 1* nj j j

fx

−

∂∂andand

ProblemProblem::TheThe Number of Q.E. Invariants Grows Number of Q.E. Invariants Grows Exponentially with the Number n of Exponentially with the Number n of QubitsQubits !!

24324312124454549933

99662244nn--11--3n3n3n3n

QubitsQubitsnn

[ ] EMax Dim iρ # CSQEI

10081008151555

The RFPQE Is Difficult for The RFPQE Is Difficult for QubitsQubits3n ≥

Some progress has been made for Some progress has been made for 33and and 44 qubitsqubits. For example, Meyer & . For example, Meyer & WallachWallach have recently been able to have recently been able to count the number of entanglement count the number of entanglement classes for classes for n=4n=4 qubitsqubits..

For example, the mathematical model of the For example, the mathematical model of the Big Big AdjointAdjoint action needs to be extended to action needs to be extended to capture such other physical effects of Q.E. as:capture such other physical effects of Q.E. as:

There is More to Q.E. Than the RFPQEThere is More to Q.E. Than the RFPQE

•• The effects of classical communication The effects of classical communication (LOCC)(LOCC)

•• The distillation of entangled statesThe distillation of entangled states

•• And much more And much more ……

The Big The Big AdjointAdjoint action does not mathematically action does not mathematically fully captures all of the physical phenomenon of fully captures all of the physical phenomenon of Q.E.Q.E.

Thinking Inside or Outside the Box ?Thinking Inside or Outside the Box ?

Thinking InsideThinking Insidethe Boxthe Box

??

Thinking OutsideThinking Outsidethe Boxthe Box

??

oror

17

22--D Physical Euclidean GeometryD Physical Euclidean Geometry33--D Physical Euclidean GeometryD Physical Euclidean Geometry

CC

AA

BBBB

AA

CC

≅??

Thinking Inside or Outside the Box ?Thinking Inside or Outside the Box ?

ReallyReally

The EndThe End

Quantum Computation:Quantum Computation: A Grand Mathematical Challenge A Grand Mathematical Challenge for the Twentyfor the Twenty--First Century and the Millennium,First Century and the Millennium,Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr. (editor),(editor), AMS PSAPM/58, AMS PSAPM/58, (2002). (2002).

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