quantum braids mosaics - inspiring innovationlomonaco/powerpoint... · wilczek, f., fractional...
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Samuel LomonacoUniversity of Maryland Baltimore County (UMBC)
Email: [email protected]: www.csee.umbc.edu/~lomonaco
Quantum Braids&&
Mosaics
This workis in collaboration with
Louis Kauffman
LomonacoLomonaco LibraryLibrary
Two papers on Two papers on Quantum Knots Quantum Knots can be found in can be found in this book.this book.
This talk was motivated by:
Kitaev, Alexei Yu, Fault-tolerant quantum computation by anyons, arxiv.org/abs/quant-ph/9707021
WilczekWilczek, F., , F., Fractional statistics and Fractional statistics and anyonanyonsuperconductivitysuperconductivity, World Scientific Press, , World Scientific Press, (1990).(1990).
RasettiRasetti, Mario, and , Mario, and TullioTullio ReggeRegge,, Vortices in Vortices in He II, current algebras and quantum knots,He II, current algebras and quantum knots,PhysicaPhysica 80 A, North80 A, North--Holland, (1975), 217Holland, (1975), 217--2333.2333.
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This talk is based on the paper:This talk is based on the paper:Lomonaco and Kauffman,Lomonaco and Kauffman, Quantum Braids and Quantum Braids and Other Mathematical Structures: The General Other Mathematical Structures: The General Quantization Procedure,Quantization Procedure, This SPIE This SPIE Proceedings, (2011).Proceedings, (2011).
Lomonaco and Kauffman,Lomonaco and Kauffman, Quantum Knots and Quantum Knots and Lattices,Lattices, AMS PSAPM/68, (2010), 209AMS PSAPM/68, (2010), 209--276276
The above paper distills the ideas found in The above paper distills the ideas found in the following two papers into a general the following two papers into a general quantization procedure.quantization procedure.
Lomonaco and Kauffman,Lomonaco and Kauffman, Quantum Knots and Quantum Knots and Mosaics,Mosaics, Journal of Quantum Information Journal of Quantum Information Processing, vol. 7, Nos. 2Processing, vol. 7, Nos. 2--3, (2008), 853, (2008), 85--115. 115.
All the above papers can be found on the All the above papers can be found on the ArKivArKiv and on the website:and on the website:
PowerPoint slides can be found at: PowerPoint slides can be found at: www.csee.umbc.edu/~lomonaco/Lectures.htmlwww.csee.umbc.edu/~lomonaco/Lectures.html
www.csee.umbc.edu/~lomonaco
This general mathematical procedure can be used to quantize:
• Knots, Graphs, & Braids
• Algebraic Varieties
• Groups
• Categories
• Topological & Differential Manifolds
• And more
• Each particular application of this general procedure creates a blueprint for a physically implementable quantum system.
• These quantum systems are physically implementable in the same sense as Shor’s quantum factoring algorithm is physically implementable
Outline of General Quantization Procedure
Step 1. Mathematical construction of a Symbolic Motif System S
Step 2. Mathematical construction of a Quantum Motif System Q based on S
MathematicalStructure
QuantumMechanics
GroupRepresentation
Theory=
FormalRewritingSystem
=
FormalRewritingSystem
= GroupRepresentation
OutlineOutline
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Thinking Outside the BoxThinking Outside the Box
Mathematical StructureMathematical Structureof your choiceof your choice
Quantum MechanicsQuantum Mechanics
is a tool for exploringis a tool for exploring
We will illustrate the general quantization procedure by showing how it can be used to quantize braids.
Please keep in mind that the same general quantization procedure can be applied to many other mathematical structures.
Quantum BraidsQuantum Braids
Braiding Naturally Arise in the Braiding Naturally Arise in the QuantumQuantum World as Dynamical ProcessesWorld as Dynamical Processes
Examples of dynamical knots and braids naturally Examples of dynamical knots and braids naturally occur in quantum physics as occur in quantum physics as Quantum Quantum VorticesVortices::
•• In In supercooledsupercooled helium IIhelium II
•• In the BoseIn the Bose--Einstein CondensateEinstein Condensate
•• In the Electron fluid found within the In the Electron fluid found within the fractional quantum Hall effectfractional quantum Hall effect
Reason for current intense interest:Reason for current intense interest:Topology Is a Natural Obstruction to Topology Is a Natural Obstruction to DecoherenceDecoherence
•Our ultimate objective is to create and to investigate mathematical objects that can be physically implemented in a quantum physics lab.
• Our objective is to do mathematics in such a way that it is intimately related to quantum physics
ObjectivesObjectives
•• We seek to define a quantum braid in We seek to define a quantum braid in such a way as to represent the state of such a way as to represent the state of braided pieces of rope, i.e., the particular braided pieces of rope, i.e., the particular spatial configuration.spatial configuration.
•• We also seek to model the ways of We also seek to model the ways of moving the braid around (without cutting the moving the braid around (without cutting the rope, and without letting it pass through rope, and without letting it pass through itself.)itself.)
•• We seek to create a quantum system We seek to create a quantum system that simulates braided physical pieces of that simulates braided physical pieces of rope.rope.
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Rules of the GameRules of the Game
Find a mathematical definition of a quantum Find a mathematical definition of a quantum braid that isbraid 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.
AspirationsAspirations
We would hope that this definition will be We would hope that this definition will be useful in modeling and predicting the useful in modeling and predicting the behavior of vortices that actually occur in behavior of vortices that actually occur in quantum physics such asquantum physics such as
•• In supercooled helium IIIn supercooled helium II
•• In the BoseIn the Bose--Einstein CondensateEinstein Condensate
•• In the Electron fluid found within the In the Electron fluid found within the fractional quantum Hall effectfractional quantum Hall effect
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
0 i n 1n i or Move 3RThe Reidemeister Moves
3
2 6 21 6
2 5 16 5
# 2 3 8 4
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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