Quantum Braids. Mosaics. Samuel Lomonaco 10/16/2011. This work is in collaboration with Louis Kauffman

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??? Quantum raids & Mosaics Samuel Lomonaco

Lomonaco@UMC.edu WebPage: www.csee.umbc.

collaboration with Louis Kauffman This talk was

1 ??? Quantum raids & Mosaics Samuel Lomonaco University of Maryland altimore County (UMC) WebPage: Lomonaco Library This work is in collaboration with Louis Kauffman This talk was motivated by: Two papers on Quantum Knots can be found in this book. Kitaev, Alexei Yu, Fault-tolerant quantum computation by anyons, arxiv.org/abs/quantph/97070 Wilczek, F., Fractional statistics and anyon superconductivity, World Scientific Press, (990). Rasetti, Mario, and Tullio Regge, Vortices in He II, current algebras and quantum knots, Physica 80 A, North-Holland, (975),

2 This talk is based on the paper: Lomonaco and Kauffman, Quantum raids and Other Mathematical Structures: The General Quantization Procedure, This SPIE Proceedings, (0). The above paper distills the ideas found in the following two papers into a general quantization procedure. Lomonaco and Kauffman, Quantum Knots and Mosaics, Journal of Quantum Information Processing, vol. 7, Nos. -3, (008), All the above papers can be found on the ArKiv and on the website: PowerPoint slides can be found at: Lomonaco and Kauffman, Quantum Knots and Lattices, AMS PSAPM/68, (00), This general mathematical procedure can be used to quantize: Knots, Graphs, & raids Groups Categories Algebraic Varieties Topological & Differential Manifolds 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 And more Outline of General Quantization Procedure Step. Mathematical construction of a Symbolic Motif System S Step. Mathematical construction of a Quantum Motif System Q based on S Mathematical Structure Formal Rewriting System Outline = = Formal Rewriting System Group Representation Quantum Mechanics = Group Representation Theory

3 Thinking Outside the ox Quantum Mechanics We will illustrate the general quantization procedure by showing how it can be used to quantize braids. is a tool for exploring Please keep in mind that the same general quantization procedure can be applied to many other mathematical structures. Mathematical Structure of your choice raiding Naturally Arise in the Quantum World as Dynamical Processes Quantum raids Examples of dynamical knots and braids naturally occur in quantum physics as Quantum Vortices: In supercooled helium II In the ose-einstein Condensate In the Electron fluid found within the fractional quantum Hall effect Reason for current intense interest: Topology Is a Natural Obstruction to Decoherence Objectives Our objective is to do mathematics in such a way that it is intimately related to quantum physics Our ultimate objective is to create and to investigate mathematical objects that can be physically implemented in a quantum physics lab. We seek to create a quantum system that simulates braided physical pieces of rope. We seek to define a quantum braid in such a way as to represent the state of braided pieces of rope, i.e., the particular spatial configuration. We also seek to model the ways of moving the braid around (without cutting the rope, and without letting it pass through itself.) 3

4 Rules of the Game Find a mathematical definition of a quantum braid that is Physically meaningful, i.e., physically implementable, and Simple enough to be workable and useable. Aspirations We would hope that this definition will be useful in modeling and predicting the behavior of vortices that actually occur in quantum physics such as In supercooled helium II In the ose-einstein Condensate In the Electron fluid found within the fractional quantum Hall effect A raid What Is the raid Group n??? Hat ox 3 Strand braid Skip braid gp def Two Equal raids Two Unequal raids = 4

5 Shorthand Notation Product of raids Hat ox Shorthand Notation Times = = 3 Strand braid 3 Inverse of of a raid The n-stranded raid Group n Times = = Theorem (Emil Artin). Under braid multiplication, the n-stranded braids form a group n, call the n-stranded braid group To construct the inverse of a braid, take the mirror image of each crossing, and then reverse the order of the crossings. There is a natural monomorphism n n ' Generators of the raid Group n Relations Among the Generators of n The braid group n is generated by b b bn b b in, i i i i i i, fr o i j i j j i 5

6 b b in, i i i i i i, fr o ij i j j i Reidemeister 3 Move Planar Isotopy Move = = A Presentation of the raid Group n A raid Is Almost a Permutation b, b,, b : n b b,in i i i i i i, i j, i, jn i j j i i ibibi i i,in n b, b,, bn : i j j i, i j, i, jn Natural Epimorphism b b, i n i i i i i i S n b, b,, bn : bi, i n i j j i, i j, i, j n Why is the braid group important for Q Comp? Why is the raid Group Important??? The representations of the Symmetric S n are the basic building blocks for the representations of the unitary group U used in quantum mechanics, The braid group n sits above the symmetric n group S n, i.e., there is a natural epimorphism S Thus, new representations of the braid group n n will give us new representations of the unitary group U, i.e., quantum gates Claim: These quantum gates can be implemented in quantum systems that are resistant to decoherence because of topological obstructions, e.g., in terms of the fractional quantum Hall effect, anyonic systems 6

7 Anyons: A Very rief Overview Anyons are quantum systems that are confined to two dimensions. They were first proposed by Nobel Laureate F. Wilczek. See for example, Wilczek, F., Fractional statistics and anyon superconductivity, World Scientific Press, (990). Anyons can used to explain the fractional quantum Hall effect A raid Represents the Movement of n Holes in a Disc This braiding can be used to represent Anyon exchanges A Anyonic braiding corresponds to a Unitary transformation Recall: Q.M.= Qroup Rep. Theory Anyons Can Also Fuse or Split Anyons: A Very rief Overview (Cont.) A C Quantum Topology gives us the tools needed to find new unitary representations based on fusing and braiding These new unitary transformations are created with an object called a unitary topological modular functor which we call simply an anyon model. Recall: Q.M.= Qroup Rep. Theory ( ) For each integer n 0, let T n be the set of n symbols b b bn raid Mosaics b0 b b b n called braid n-stranded tiles, or simply tiles, and also respectively denoted by b0, b, b,, bn 7

8 The Set ( n, ) of raid (n,l)-mosaics The Set ( n, ) of raid (n,l)-mosaics Def. A braid (n,l)- mosaic is a sequence of length l bj(), bj(),, bj( ) of braid n-tiles. Let all braid (n,l)-mosaics. ( n, ) be the set of Example: The braid (3,8)-mosaic b (3,8) is an element of. b b b b b b b Observation: The cardinality of the set of braid (n,l)-mosaics is n ( n, ) raid Mosaic Moves raid Moves for the set of raid (n,l)-mosaics ( n, ) 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. The location of the braid move is the location of the leftmost symbol in effected by the move. Example: = The Planar Isotopy Moves Move P b b i Observation: The number of Example: n i for P 0 i n moves is 8

9 The Planar Isotopy Moves The Reidemeister Moves Move P i j j i for 0 i, j n & i j Move R i i for 0 i n Observation: The number of nn6 P moves is Observation: The number of n R moves is Example: Example: Move R3 The Reidemeister Moves for 6 i i i b ib i i 4 i i i b i bi i i i i bi bi i b b i i i i i i i i i i i i b b 4 bi bi i ibib i 0 i n or n i 3 The Reidemeister Moves Observation: The number of n n 6 if 6 n n 5 6 if 5 # RMoves n n 3 8 if 4 R 3 moves is n n if 3 0 if 3 The Reidemeister R 3 Moves Examples: The Ambient Group 9

10 raid Mosaic Moves Are Permutations Each braid mosaic move acts as a local transf on an braid (n, l )-mosaic whenever its conditions are met. If its conditions are not met, it acts as the identity transformation. Ergo, each braid mosaic move is a permutation ( n, ) on the set of all braid (n, l)-mosaics The Ambient Group A(n,l) We define the ambient group A(n,l) as the subgroup of the group of all ( n, ) permutations of the set generated by the all braid (n,l)-moves. In fact,each braid mosaic move, as a permutation, is a product of disjoint transpositions. The raid Mosaic Injection : ( n, ) ( n, ) raid Type We define the braid mosaic injection ( n, ) ( n, ) : as ( n, ) ( n, ) b b b ' b b b j() j() j( ) j() j() j( ) Mosaic raid Type Def. Two braid (n,l)-mosaics and are of the same braid (n,l)-mosaic type, written ~ ' n provided there exists an element of the ambient group A(n,l) that transforms into. Two (n,l)-mosaics and are of the same braid type if there exists a non-negative negative integer k such that k k k i ' n Part Quantum raids & Quantum raid Systems 0

11 The Hilbert Space ( n) Let H be the n- dimensional Hilbert space with orthonormal basis labeled by the tiles We define the Hilbert space H (n,l)- mosaics as n H (, ) ( n) k This is the Hilbert space with induced orthonormal basis ( n, ) b, b, b,, b n 0 ( ) of (n,l)-mosaics ( n, ) b ( ) : n j( k) n k jk of braid The Hilbert Space ( n, ) We identify each basis ket b k j( k ) with a ket labeled by a braid (n,l)-mosaic. For example, in the braid (3,4)-mosaic Hilbert space (3,4), the basis ket b b b b 0 of raid (n,l)-mosaics is identified with braid (3,4)-mosaic labeled ket A quantum braid is an element of (3,4) An Example of a Quantum raid The Ambient Group A(n,l) as a Unitary Group A quantum braid (3,)-mosaic We identify each element g An (, ) with the linear transformation defined by ( n, ) ( n, ) g Since each element g An (, ) is a permutation, it is a linear transformation that simply permutes basis elements. Hence, under this identification, the ambient group An (, ) becomes a discrete group of unitary transfs on the Hilbert space ( n,. ) An Example of the R R An (, ) Group Action A (3,)-move The Quantum raid System Def. A quantum braid system is a quantum system having ( n, ) as its state space, and having the Ambient group An (, ) as its set of accessible unitary transformations. n (, ), (, ) The states of quantum system A n are quantum braids. The elements of the ambient group An (, ) are quantum moves. A n n (, ), (, ) A n n (, ), (, )

12 Quantum raid Type Def. Two quantum braid (n,l)-mosaics and are of the same braid (n,l) -type, written, n provided there is an element g An (, ) s.t. g They are of the same braid type, written, provided there is an integer m 0 such that m m m n Two Quantum raids of the Same raid Type R R A (3,) move Hamiltonians for An ( ) Hamiltonians of the Generators of the Ambient Group Each generator g An (, ) is the product of disjoint transpositions, i.e.,,,, g K K K K K K Choose a permutation so that,,, g K K K K K K Hence, 3 3 g 0 0 0, where 0 I n 0 Also, let 0 0, and note that i ln s 0, s For simplicity, we always choose the branch s. 0 H iln g g Hamiltonians for An (, ) Observables I nn which are Quantum raid Invariants

13 Observable Q. raid Invariants Observable Q. raid Invariants Question. What do we mean by a physically observable braid invariant? Question. ut which observables are actually braid invariants? ( n, ) Let, A ( n, ) be a quantum braid system. Then a quantum observable is a Hermitian operator on the Hilbert space ( n, ). Def. An observable is an invariant of quantum braids provided UU for all U A( n, ) Observable Q. Knot Invariants Question. ut how do we find quantum braid invariant observables? ( n, ) Theorem. Let, A ( n, ) braid system, and let ( n, ) W j j be a quantum be a decomposition of the representation ( n, ) ( n, ) A( n, ) into irreducible representations. Then, for each j, the projection operator for the subspace W j is a quantum braid observable. P j Observable Q. raid Invariants ( n, ) Theorem. Let, A ( n, ) be a quantum braid system, and let be an observable on ( n, ). Let St be the stabilizer subgroup for, i.e., St U A n U U (, ) : Then the observable UU UA( n)/ St is a quantum braid invariant, where the above sum is over a complete set of coset representatives of St in An (, ). Future Directions & Open Questions Future 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? 3

14 Future Directions & Open Questions The Yang-axter relation lives in the ambient group A(n,l). Can it be lifted to the Lie algebra of the unitary group ( n, U? 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 Questions If so, we could choose an assignment of Hamiltonians H j g j which is consistent with the Yang-axter relation. These Hamiltonians determine a unitary evolution of Schroedinger s equation, which is a unitary representation of the braid group. Future 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-axter relation? 4

Quantum Knots & Mosaics

Quantum Knots & Mosaics Quantum nots??? Quantum nots & Mosaics Samuel Lomonaco University of Maryland Baltimore County (UMBC) Email: Lomonaco@UMBC.edu Webage: www.csee.umbc.edu/~lomonaco L-O-O- This talk was motivated by a number