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Samuel LomonacoUniversity of Maryland Baltimore County (UMBC)

Email: Lomonaco@UMBC.eduWebPage: www.csee.umbc.edu/~lomonaco

Quantum Braids&&

Mosaics

This workis in collaboration with

Louis Kauffman

Rules of the GameRules of the Game

Find a mathematical definition of a quantum Find a mathematical definition of a quantum braid that is:braid that is:

•• Physically meaningful, i.e., physicallyPhysically meaningful, i.e., physicallyimplementable, andimplementable, and

•• Simple enough to be workable and Simple enough to be workable and useable.useable.

Quantum Topology Quantum Topology

My ultimate objective is to create and to investigate mathematical objects that can be physically implemented in a quantum physics lab.

Quantum Physics

My objective in this talk is to do topology in such a way that it is intimately related to quantum physics

Lomonaco and Kauffman, Quantum Knots and Mosaics, Journal of Quantum Information Processing, vol. 7, Nos. 2-3, (2008), 85-115. An earlier version can be found at: http://arxiv.org/abs/0805.0339

This talk is a continuation of a research program begun and outlined in:

Lomonaco, Samuel J., and Louis H. Kauffman, Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do Rope Tricks, AMS PSAMP, (2010) http://arxiv.org/abs/0910.5891

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Lomonaco, Samuel J., Jr., Lomonaco, Samuel J., Jr., “Quantum Information “Quantum Information Science & Its Contributions to Mathematics,Science & Its Contributions to Mathematics, AMS AMS PSAPM, (2010).PSAPM, (2010).

Quantum Information Science&

Its Contributions to Mathematics

American Mathematical SocietyShort CourseJanuary 3-4, 2009

Samuel J. Lomonaco, Jr.Editor

This talk was motivated by:

Kitaev, Alexei Yu, Fault-tolerant quantum computation by anyons,http://arxiv.org/abs/quant-ph/9707021

WilczekWilczek, F., , F., Fractional statistics and Fractional statistics and anyonanyon superconductivitysuperconductivity, World , World Scientific Press, (1990).Scientific Press, (1990).

What Is the Braid What Is the Braid Group BGroup Bnn ??????

Skip braid gp def

A BraidA Braid

Hat BoxHat Box

3 Strand braid3 Strand braid

Two Equal BraidsTwo Equal Braids

==

Two Unequal BraidsTwo Unequal Braids

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Shorthand NotationShorthand Notation

Hat BoxHat Box

Shorthand Shorthand NotationNotation

3 Strand braid3 Strand braid

Product of BraidsProduct of Braids

TimesTimes == ==

1 2 3

Inverse of of a BraidInverse of of a Braid

TimesTimes == ==1 1

To construct the inverse of a braid, take the mirrorTo construct the inverse of a braid, take the mirrorimage of each crossing, and then reverse the orderimage of each crossing, and then reverse the orderof the crossings.of the crossings.

The nThe n--Stranded Braid Group BStranded Braid Group Bnn

TheoremTheorem (Emil Artin).(Emil Artin). Under braid multiplication, Under braid multiplication, the the nn--stranded braids form a group stranded braids form a group BBnn, call the , call the nn--stranded braid group stranded braid group

There is a natural monomorphismThere is a natural monomorphism

1n nB B

'

Generators of the Braid Group BGenerators of the Braid Group Bnn

The braid group The braid group BBnn is generated byis generated by

1b 2b 1nb

Relations Among the Generators of BRelations Among the Generators of Bnn

1 1 1 , 1i i i i i ib b b b bb i n

, o 2f ri j j ib b b b i j

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1 1 1 , 1i i i i i ib b b b b b i n

==

Reidemeister 3 MoveReidemeister 3 Move

, o 2f ri j j ibb bb i j

==

Planar Planar IsotopyIsotopy MoveMove

A Presentation of the Braid Group BA Presentation of the Braid Group Bnn

1 1 1

1 2 1

, 1 1, , , :

, 1, 1 , 1

i i i i i i

n

i j j i

b b b b bb i nb b b

bb b b i j i j n

A Braid Is “Almost” a PermutationA Braid Is “Almost” a Permutation

1 1 1

1 2 1

, 1 1, , , :

, 1, 1 , 1

i i i i i i

n n

i j j i

b b b b bb i nB b b b

bb b b i j i j n

1 1 1

1 2 1

, 1 1, , , :

, 1, 1 , 1

i i i i i i

n n

i j j i

b b b b b b i nB b b b

b b b b i j i j n

NaturalNaturalEpimorphismEpimorphism

2 1, 1 1ib i n nS

Why is the Braid Why is the Braid Group Important ???Group Important ???

•• The braid group The braid group BBnn “sits above” the symmetric“sits above” the symmetricgroup group SSn n ,, i.e., there is a natural epimorphism i.e., there is a natural epimorphism

BBnn

SSnn•• Thus, new representations of the braid groupThus, new representations of the braid groupBBnn will give us new representations of the will give us new representations of the unitary group unitary group UU, i.e., , i.e., quantum gatesquantum gates

•• The representations of the Symmetric The representations of the Symmetric SSnn are are the basic building blocks for the representationsthe basic building blocks for the representationsof the unitary group of the unitary group UU used in quantum mechanics, used in quantum mechanics,

Why is the braid group important for Q Comp ? Why is the braid group important for Q Comp ?

•• ClaimClaim:: These quantum gates can be implemented in These quantum gates can be implemented in quantum systems that are quantum systems that are resistant to decoherence resistant to decoherence because of topological obstructionsbecause of topological obstructions, e.g., in terms , e.g., in terms of the of the fractional quantum Hall effect, anyonic systemsfractional quantum Hall effect, anyonic systems

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AnyonsAnyons are quantum systems that are are quantum systems that are confined to two dimensions. They were confined to two dimensions. They were first proposed by Nobel Laureate F. first proposed by Nobel Laureate F. Wilczek. See for example,Wilczek. See for example,

Anyons: Anyons: A Very Brief OverviewA Very Brief Overview

Wilczek, F., Wilczek, F., Fractional statistics and Fractional statistics and anyon superconductivityanyon superconductivity, World , World Scientific Press, (1990).Scientific Press, (1990).

Anyons can used to explain the Anyons can used to explain the fractional fractional quantum Hall effectquantum Hall effect

BBAA

A Braid Represents the Movement of n Holes A Braid Represents the Movement of n Holes in a Discin a Disc

This braiding can be used toThis braiding can be used torepresent Anyon exchangesrepresent Anyon exchanges

AnyonicAnyonic braiding corresponds to braiding corresponds to a Unitary transformationa Unitary transformation

Recall:Recall: Q.M.= Qroup Rep. TheoryQ.M.= Qroup Rep. Theory

BB

Anyons Can Also Fuse or SplitAnyons Can Also Fuse or Split

AA C C

Quantum Topology gives us the tools needed Quantum Topology gives us the tools needed to find to find new unitary representations based new unitary representations based on fusing and braidingon fusing and braiding

Recall:Recall: Q.M.= Qroup Rep. TheoryQ.M.= Qroup Rep. Theory

These new unitary transformations are These new unitary transformations are created with an object called a created with an object called a unitary unitary topological modular functortopological modular functor which we call which we call simply an simply an anyon modelanyon model..

Anyons: Anyons: A Very Brief Overview (Cont.)A Very Brief Overview (Cont.)

Braid Braid MosaicsMosaics

For each integer , let be the set of symbols

0n ( )nT2 1n

0 1b

1b 2b 1nb

1b 2b 1nb

called braid n-stranded tiles, or simply tiles, and also respectively denoted by 0 1 2 1, , , , nb b b b

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of braid n-tiles.

Def. A braid (n,l)- mosaic is a sequence of length l

(1) (2) ( ), , ,j j jb b b

Let be the set ofall braid (n,l)-mosaics.

( , )nB

The Set of Braid (n,l)-Mosaics( , )nB

Example: The braid (3,8)-mosaic

is an element of . (3,8)B

1 1 2 1 21 11b b b b b

1 111b 1b 2b 1b 2b

The Set of Braid (n,l)-Mosaics( , )nB

Observation: The cardinality of the set of braid (n,l)-mosaics is

( , )nB

2 1n

Braid Mosaic MovesBraid Mosaic Moves

Def. A braid move on a braid mosaic is a (cut & paste) operation that transforms into another braid ’ by replacing a submosaic of by another.

Braid Moves for the set of Braid (n,l)-Mosaics

( , )nB

Example:

=2

The location of the braid move is the location of the leftmost symbol in effected by the move.

The Planar Isotopy Moves

Move 1P 1 1i ib b

0 i n for

Observation: The number of moves is1P

2 1 1n

Example:

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The Planar Isotopy Moves

Move 2P i j j ib b b b

Observation: The number of moves is2P

1 2 6 1n n

Example:

0 ,i j n for 1i j &

The Reidemeister Moves

Move 2R 21i ib b

0 i n for

Observation: The number of moves is2R

2 1 1n

Example:

6

1 1 1 1i i i ii ib b b b b b

for

4

1 11 1i i i i iib b b b b b

2

1 11 1i i i i iib b b b b b

1 1 1i i i i i ib b b b b b

2

1 1 11i i i i i ib b b b b b

4

1 1 11i i i i i ib b b b b b

6

1 1 1 1i i i i iib b b b b b

4

1 1 1 1i i i i iib b b b b b

2

1 1 11i i i i i ib b b b b b

1 1 1i i i i i ib b b b b b

2

1 1 11i i i i i ib b b b b b

4

1 1 11i i i i iib b b b b b

0 i n 1n i or Move 3RThe Reidemeister Moves

3

2 2 6 21 6

2 2 5 16 5

# 2 2 3 8 4

2 2 2 3

0 3

n n if

n n if

R Moves n n if

n n if

if

The Reidemeister Moves

Observation: The number of moves is3R

The Reidemeister Moves

Examples:

3R

The Ambient GroupThe Ambient Group

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Braid Mosaic Moves Are PermutationsBraid Mosaic Moves Are Permutations

Each braid mosaic move acts as a local Each braid mosaic move acts as a local transftransfon an braid on an braid ((n, n, ll))--mosaic whenever its conditions mosaic whenever its conditions are met. If its conditions are not met, it acts are met. If its conditions are not met, it acts as the identity transformation. as the identity transformation.

Ergo, each Ergo, each braid mosaic movebraid mosaic move is a is a permutationpermutationon the set of all braid on the set of all braid (n, l)--mosaicsmosaics

In In fact,eachfact,each braid mosaic movebraid mosaic move, as a , as a permutation, is a permutation, is a productproduct ofof disjointdisjointtranspositionstranspositions..

( , )nB

We define the ambient group A(n,l) as the subgroup of the group of all permutations of the set generated by the all braid (n,l)-moves.

The Ambient Group The Ambient Group A(n,l)

( , )nB

Braid TypeBraid Type

The Braid Mosaic InjectionThe Braid Mosaic Injection

We define the We define the braidbraid mosaicmosaic injectioninjectionasas

( , ) ( , 1)

(1) (2) ( ) (1) (2) ( )' 1

n n

j j j j j jb b b b b b

B B

( , ) ( , 1): n n B B

( , ) ( , 1): n n B B

Mosaic Braid TypeMosaic Braid Type

~ 'n

providedprovided therethere exists an element of the ambient exists an element of the ambient group group A(n,l) that transformsthat transforms intointo ’’ . .

DefDef.. Two braid Two braid (n,l)--mosaics mosaics and and ’’ are of are of the samethe same braidbraid (n,l)--mosaicmosaic typetype, , writtenwritten

'k

k k

ni

Two Two (n,l)--mosaics mosaics and and ’’ are of the same are of the same braidbraid typetype if there exists a nonif there exists a non--negative negative integer integer k such thatsuch that

Quantum BraidsQuantum Braids&&

Quantum Braid SystemsQuantum Braid Systems

Part 2Part 2

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Let be the Let be the 2n-1 dimensional Hilbert space dimensional Hilbert space with orthonormal basis labeled by the tiles with orthonormal basis labeled by the tiles

We define the We define the HilbertHilbert spacespace ofof braid braid (n,l)-- mosaicsmosaics as as

( , ) ( )

1

n n

k B H

This is the Hilbert space with induced This is the Hilbert space with induced orthonormal basisorthonormal basis

( )1: ( )j kk

b n j k n

The Hilbert Space of The Hilbert Space of (n,l)--mosaics mosaics ( , )nB

( )nH

( , )nH

0 1 2 ( 1), , , , nb b b b

is identified with braid is identified with braid (3,4)-mosaic labeled mosaic labeled ketket

For example, in the braid For example, in the braid (3,4)-mosaic Hilbertmosaic Hilbertspace , the basis space , the basis ketket

We identify each basis We identify each basis ketket withwitha a ketket labeled by a braid labeled by a braid (n,l)--mosaic mosaic . .

( )1 j kkb

2 1 2 0b b b b

The Hilbert SpaceThe Hilbert Space of Braid of Braid (n,l)--MosaicsMosaics ( , )nB

(3,4)B

A quantum braid is an element of (3,4)B

An Example of a Quantum BraidAn Example of a Quantum Braid

2

A quantum braid (3,2)-mosaic

Since each element is a permutation, it is a linear transformation that simply permutes basis elements.

( , )g A n

The Ambient Group The Ambient Group A(n,l) as a as a UnitaryUnitary GroupGroup

We identify each element with the linear transformation defined by

( , )g A n

( , ) ( , )n n

g B B

Hence, under this identification, the ambientgroup becomes a discrete group of unitary transfs on the Hilbert space . ( , )n

B

( , )A n

An Example of the Group ActionAn Example of the Group Action

2R

( , )A n

2

2R

2

A (3,2)-move

The Quantum Braid System

Def. A quantum braid system is a quantum system having as its state space, and having the Ambient group as its set of accessible unitary transformations.

( , )A n

The states of quantum system are quantum braids. The elements of the ambient group are quantum moves.( , )A n

( , )nB

( , ), ( , )

nA nB

( , ), ( , )

nA nB

( , ), ( , )

nA nB

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Quantum Braid Type

Def. Two quantum braid (n,l)-mosaics and are of the same braid (n,l) -type, written

12

1 2 ,n

provided there is an element s.t. ( , )g A n 1 2g

They are of the same braid type, written

1 2 ,

1 2

mm m

n

provided there is an integer such that 0m

2R

2

2R

2

A (3,2) move

Two Quantum Braids of the Same Braid TypeTwo Quantum Braids of the Same Braid Type

HamiltoniansHamiltoniansof theof the

GeneratorsGeneratorsof theof the

Ambient Group Ambient Group

Hamiltonians for Hamiltonians for ( )A n

Each generator is the product of disjoint transpositions, i.e.,

( , )g A n

1 1 2 2, , ,g K K K K K K

11 2 3 3 1, , ,g K K K K K K

Choose a permutation so that

Hence, 1

11

1

2n

g

I

00

1

0 11 0

, where

0

1 00 1

Also, let , and note that

For simplicity, we always choose the branch . 0s

0 1 1

2 2

00 02 n n

I

1 1lngH i g

Hamiltonians for Hamiltonians for ( , )A n

1 0 1ln 2 12

,i s s ObservablesObservableswhich arewhich are

Quantum BraidQuantum BraidInvariants Invariants

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Observable Q. Braid Invariants Observable Q. Braid Invariants

Question. What do we mean by a physically observable braid invariant ?

Let be a quantum braid system. Then a quantum observable is a Hermitian operator on the Hilbert space .

( , ), ( , )

nA nB

( , )nB

Observable Q. Braid Invariants Observable Q. Braid Invariants

Question. But which observables are actually braid invariants ?

Def. An observable is an invariantof quantum braids provided for all

1U U ( , )U A n

( , )n

jj W B

be a decomposition of the representation ( , ) ( , )

( , )n n

A n B B

Observable Q. Knot Invariants Observable Q. Knot Invariants

Question. But how do we find quantum braid invariant observables ?

Then, for each j, the projection operator for the subspace is a quantum braid observable.

jPjW

into irreducible representations .

Theorem. Let be a quantum braid system, and let

( , ), ( , )

nA nB

Observable Q. Braid Invariants

1( , ) :St U A n U U

Then the observable

1

( ) /U A n StU U

is a quantum braid invariant, where the above sum is over a complete set of coset representatives of in . St ( , )A n

Let be the stabilizer subgroup for , i.e.,

Theorem. Let be a quantum braid system, and let be an observable on

.

( , ), ( , )

nA nB

St ( , )n

B

Future DirectionsFuture Directions&&

Open QuestionsOpen Questions

Future Directions & Open QuestionsFuture Directions & Open Questions

• Presentation of the ambient group A(n,l)

• How is the ambient group A(n,l) related to the homology group of the braid group?

• Can quantum braids be used to simplify the search for unitary representations of the braid group?

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Future Directions & Open QuestionsFuture Directions & Open Questions

The Yang-Baxter relation “lives” in the ambient group A(n,l) . Can it be lifted to the Lie algebra of the unitary group

? ( ,nBU

If so, the search for unitary reps of the braid group reduces to the task of associating Hamiltonians with the generators of the braid group.

Future Directions & Open QuestionsFuture Directions & Open QuestionsIf so, we could choose an assignment of Hamiltonians

j jH gwhich is consistent with the Yang-Baxter relation.

These Hamiltonians determine a unitary evolution of Schroedinger’s equation, which is a unitary representation of the braid group.

Future Directions & Open QuestionsFuture Directions & Open Questions

As an example, we have found Hamiltonians that produce the Fibonnacci representation.

Question: Can we find a general way to lift the Yang-Baxter relation?

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