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

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1 0/6/0??? Quatum raids & Mosaics Samuel Lomoaco Uiversity of Marylad altimore Couty (UMC) WebPage: Rules of the Game This work is i collaboratio with Louis Kauffma Fid a mathematical defiitio of a quatum braid that is: Physically meaigful, i.e., physically implemetable, ad Simple eough to be workable ad useable. Quatum Topology Quatum Physics Topology My objective i this talk is to do topology i such a way that it is itimately related to quatum physics My ultimate objective is to create ad to ivestigate mathematical objects that ca be physically implemeted i a quatum physics lab. This talk is a cotiuatio of a research program begu ad outlied i: Lomoaco ad Kauffma, Quatum Kots ad Mosaics, Joural of Quatum Iformatio Processig, vol. 7, Nos. -3, (008), A earlier versio ca be foud at: Lomoaco, Samuel J., ad Louis H. Kauffma, Quatum Kots ad Lattices, or a lueprit for Quatum Systems that Do Rope Tricks, AMS PSAMP, (00)

2 0/6/0 Lomoaco, Samuel J., Jr., Quatum Iformatio Sciece & Its Cotributios to Mathematics, AMS PSAPM, (00). This talk was motivated by: Kitaev, Alexei Yu, Fault-tolerat quatum computatio by ayos, Quatum Iformatio Sciece & Its Cotributios to Mathematics America Mathematical Society Short Course Jauary 3-4, 009 Samuel J. Lomoaco, Jr. Editor Wilczek, F., Fractioal statistics ad ayo supercoductivity, World Scietific Press, (990). A raid What Is the raid Group??? Hat ox 3 Strad braid Skip braid gp def Two Equal raids Two Uequal raids =

3 0/6/0 Shorthad Notatio Product of raids Hat ox Shorthad Notatio Times = = 3 Strad braid 3 Iverse of of a raid The -Straded raid Group Times = = Theorem (Emil Arti). Uder braid multiplicatio, the -straded braids form a group, call the -straded braid group To costruct the iverse of a braid, take the mirror image of each crossig, ad the reverse the order of the crossigs. There is a atural moomorphism ' Geerators of the raid Group Relatios Amog the Geerators of The braid group is geerated by b b b b b i, i i i i i i, fr o i j i j j i 3

4 0/6/0 b b i, i i i i i i, fr o ij i j j i Reidemeister 3 Move Plaar Isotopy Move = = A Presetatio of the raid Group A raid Is Almost a Permutatio b, b,, b : b b,i i i i i i i, i j, i, j i j j i i ibibi i i,i b, b,, b : i j j i, i j, i, j Natural Epimorphism b b, i i i i i i i S b, b,, b : bi, i i j j i, i j, i, j Why is the braid group importat for Q Comp? Why is the raid Group Importat??? The represetatios of the Symmetric S are the basic buildig blocks for the represetatios of the uitary group U used i quatum mechaics, The braid group sits above the symmetric group S, i.e., there is a atural epimorphism S Thus, ew represetatios of the braid group will give us ew represetatios of the uitary group U, i.e., quatum gates Claim: These quatum gates ca be implemeted i quatum systems that are resistat to decoherece because of topological obstructios, e.g., i terms of the fractioal quatum Hall effect, ayoic systems 4

5 0/6/0 Ayos: A Very rief Overview Ayos are quatum systems that are cofied to two dimesios. They were first proposed by Nobel Laureate F. Wilczek. See for example, Wilczek, F., Fractioal statistics ad ayo supercoductivity, World Scietific Press, (990). Ayos ca used to explai the fractioal quatum Hall effect A raid Represets the Movemet of Holes i a Disc This braidig ca be used to represet Ayo exchages A Ayoic braidig correspods to a Uitary trasformatio Recall: Q.M.= Qroup Rep. Theory Ayos Ca Also Fuse or Split Ayos: A Very rief Overview (Cot.) A C Quatum Topology gives us the tools eeded to fid ew uitary represetatios based o fusig ad braidig These ew uitary trasformatios are created with a object called a uitary topological modular fuctor which we call simply a ayo model. Recall: Q.M.= Qroup Rep. Theory ( ) For each iteger 0, let T be the set of symbols b b b raid Mosaics b0 b b b called braid -straded tiles, or simply tiles, ad also respectively deoted by b0, b, b,, b 5

6 0/6/0 The Set (, ) of raid (,l)-mosaics The Set (, ) of raid (,l)-mosaics Def. A braid (,l)- mosaic is a sequece of legth l bj(), bj(),, bj( ) of braid -tiles. Let all braid (,l)-mosaics. (, ) be the set of Example: The braid (3,8)-mosaic b (3,8) is a elemet of. b b b b b b b Observatio: The cardiality of the set of braid (,l)-mosaics is (, ) raid Mosaic Moves raid Moves for the set of raid (,l)-mosaics (, ) Def. A braid move o a braid mosaic is a (cut & paste) operatio that trasforms ito aother braid by replacig a submosaic of by aother. The locatio of the braid move is the locatio of the leftmost symbol i effected by the move. Example: = The Plaar Isotopy Moves Move P b b i Observatio: The umber of Example: i for P 0 i moves is 6

7 0/6/0 The Plaar Isotopy Moves The Reidemeister Moves Move P i j j i for 0 i, j & i j Move R i i for 0 i Observatio: The umber of 6 P moves is Observatio: The umber of R moves is Example: Example: Move R3 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 The Reidemeister Moves 0 i or i 6 bi i ibib b i i 4 bi i ibib bi i b i i ibi i i b b i i i i i i bibi i i i i 4 bi i i i i b i 3 The Reidemeister Moves Observatio: The umber of 6 if if 5 # RMoves 3 8 if 4 R 3 moves is if 3 0 if 3 The Reidemeister R 3 Moves Examples: The Ambiet Group 7

8 0/6/0 raid Mosaic Moves Are Permutatios Each braid mosaic move acts as a local trasf o a braid (, l )-mosaic wheever its coditios are met. If its coditios are ot met, it acts as the idetity trasformatio. Ergo, each braid mosaic move is a permutatio (, ) o the set of all braid (, l)-mosaics The Ambiet Group A(,l) We defie the ambiet group A(,l) as the subgroup of the group of all (, ) permutatios of the set geerated by the all braid (,l)-moves. I fact,each braid mosaic move, as a permutatio, is a product of disjoit traspositios. The raid Mosaic Ijectio : (, ) (, ) raid Type We defie the braid mosaic ijectio (, ) (, ) : as (, ) (, ) b b b ' b b b j() j() j( ) j() j() j( ) Mosaic raid Type Def. Two braid (,l)-mosaics ad are of the same braid (,l)-mosaic type, writte ~ ' provided there exists a elemet of the ambiet group A(,l) that trasforms ito. Two (,l)-mosaics ad are of the same braid type if there exists a o-egative egative iteger k such that k k k i ' Part Quatum raids & Quatum raid Systems 8

9 0/6/0 The Hilbert Space ( ) Let H be the - dimesioal Hilbert space with orthoormal basis labeled by the tiles We defie the Hilbert space H (,l)- mosaics as H (, ) ( ) k This is the Hilbert space with iduced orthoormal basis (, ) b, b, b,, b 0 ( ) of (,l)-mosaics (, ) b ( ) : j( k) k jk of braid The Hilbert Space (, ) We idetify each basis ket b k j( k ) with a ket labeled by a braid (,l)-mosaic. For example, i the braid (3,4)-mosaic Hilbert space (3,4), the basis ket b b b b 0 of raid (,l)-mosaics is idetified with braid (3,4)-mosaic labeled ket A quatum braid is a elemet of (3,4) A Example of a Quatum raid The Ambiet Group A(,l) as a Uitary Group A quatum braid (3,)-mosaic We idetify each elemet g A (, ) with the liear trasformatio defied by (, ) (, ) g Sice each elemet g A (, ) is a permutatio, it is a liear trasformatio that simply permutes basis elemets. Hece, uder this idetificatio, the ambiet group A (, ) becomes a discrete group of uitary trasfs o the Hilbert space (,. ) A Example of the R R A (, ) Group Actio A (3,)-move The Quatum raid System Def. A quatum braid system is a quatum system havig (, ) as its state space, ad havig the Ambiet group A (, ) as its set of accessible uitary trasformatios. (, ), (, ) The states of quatum system A are quatum braids. The elemets of the ambiet group A (, ) are quatum moves. A (, ), (, ) A (, ), (, ) 9

10 0/6/0 Quatum raid Type Def. Two quatum braid (,l)-mosaics ad are of the same braid (,l) -type, writte, provided there is a elemet g A (, ) s.t. g They are of the same braid type, writte, provided there is a iteger m 0 such that m m m Two Quatum raids of the Same raid Type R R A (3,) move Hamiltoias for A ( ) Hamiltoias of the Geerators of the Ambiet Group Each geerator g A (, ) is the product of disjoit traspositios, i.e.,,,, g K K K K K K Choose a permutatio so that,,, g K K K K K K Hece, 3 3 g 0 0 0, where 0 I 0 Also, let 0 0, ad ote that i l s 0, s For simplicity, we always choose the brach s. 0 H il g g Hamiltoias for A (, ) Observables I which are Quatum raid Ivariats 0

11 0/6/0 Observable Q. raid Ivariats Observable Q. raid Ivariats Questio. What do we mea by a physically observable braid ivariat? Questio. ut which observables are actually braid ivariats? (, ) Let, A (, ) be a quatum braid system. The a quatum observable is a Hermitia operator o the Hilbert space (, ). Def. A observable is a ivariat of quatum braids provided UU for all U A(, ) Observable Q. Kot Ivariats Questio. ut how do we fid quatum braid ivariat observables? (, ) Theorem. Let, A (, ) braid system, ad let (, ) W j j be a quatum be a decompositio of the represetatio (, ) (, ) A(, ) ito irreducible represetatios. The, for each j, the projectio operator for the subspace W j is a quatum braid observable. P j Observable Q. raid Ivariats (, ) Theorem. Let, A (, ) be a quatum braid system, ad let be a observable o (, ). Let St be the stabilizer subgroup for, i.e., St U A U U (, ) : The the observable UU UA( )/ St is a quatum braid ivariat, where the above sum is over a complete set of coset represetatives of St i A (, ). Future Directios & Ope Questios Future Directios & Ope Questios Presetatio of the ambiet group A(,l) How is the ambiet group A(,l) related to the homology group of the braid group? Ca quatum braids be used to simplify the search for uitary represetatios of the braid group?

12 0/6/0 Future Directios & Ope Questios The Yag-axter relatio lives i the ambiet group A(,l). Ca it be lifted to the Lie algebra of the uitary group (, U? If so, the search for uitary reps of the braid group reduces to the task of associatig Hamiltoias with the geerators of the braid group. Future Directios & Ope Questios If so, we could choose a assigmet of Hamiltoias H j g j which is cosistet with the Yag-axter relatio. These Hamiltoias determie a uitary evolutio of Schroediger s equatio, which is a uitary represetatio of the braid group. Future Directios & Ope Questios As a example, we have foud Hamiltoias that produce the Fiboacci represetatio.??? Questio: Ca we fid a geeral way to lift the Yag-axter relatio?