Quantum Quantum An Intuitive Overview An Intuitive Overview Knots ??? Knots ??? of the Theory of of the Theory of Quantum Knots Quantum Knots Samuel Lomonaco Samuel Lomonaco University of Maryland Baltimore County (UMBC) University of Maryland Baltimore County (UMBC) Email: Lomonaco@UMBC.edu Email: Lomonaco@UMBC.edu WebPage WebPage: www.csee.umbc.edu/~lomonaco : www.csee.umbc.edu/~lomonaco L- -O O- -O O- -P P • Lecture I: A Rosetta Stone for Quantum Computing • Lecture II: Quantum Knots and Mosaics, This is work in collaboration with Louis Kauffman • Lecture III: Quantum Knots & Lattices, • Lecture IV: Intuitive Overview of the Theory of Quantum Knots This talk is based on the paper: This talk is based on the paper: Lomonaco and Kauffman, Lomonaco and Kauffman, Quantum Knots and Quantum Knots and PowerPoint Lectures and Exercises can be Lattices, Lattices, to appear soon on quant to appear soon on quant- -ph ph found at: This talk was motivated by: This talk was motivated by: www.csee.umbc.edu/~lomonaco 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. 2 Processing, vol. 7, Nos. 2- -3, (2008), 85 3, (2008), 85- - 115. An earlier version can be found at: 115. An earlier version can be found at: http://arxiv.org/abs/0805.0339 http://arxiv.org/abs/0805.0339 1
This talk was also motivated by: This talk was also motivated by: Kauffman and Lomonaco, Quantum Knots, Kauffman and Lomonaco, Quantum Knots, SPIE Proc. SPIE Proc. on Quantum Information & Computation II (ed. by on Quantum Information & Computation II (ed. by Donkor Donkor, , Pirich Pirich, & Brandt), (2004), 5436 , & Brandt), (2004), 5436- -30, 268 30, 268- - Throughout this talk: 284. http://xxx.lanl.gov/abs/quant 284. http://xxx.lanl.gov/abs/quant- -ph/0403228 ph/0403228 Lomonaco, Samuel J., Jr., The modern legacies of Lomonaco, Samuel J., Jr., The modern legacies of Thomson's atomic vortex theory in classical Thomson's atomic vortex theory in classical “Knot” means either a knot or a link electrodynamics, electrodynamics, AMS PSAPM/51, Providence, RI AMS PSAPM/51, Providence, RI (1996), 145 (1996), 145 - - 166. 166. Kitaev Kitaev, Alexei Yu, , Alexei Yu, Fault Fault- -tolerant quantum computation tolerant quantum computation by anyons by anyons, , http://arxiv.org/abs/quant http://arxiv.org/abs/quant- -ph/9707021 ph/9707021 Rasetti, Mario, and Tullio Regge, Rasetti, Mario, and Tullio Regge, Vortices in He II, Vortices in He II, current algebras and quantum knots, Physica 80 A, current algebras and quantum knots, Physica 80 A, North North- -Holland, (1975), 217 Holland, (1975), 217- -2333. 2333. Thinking Outside the Box Thinking Outside the Box Quantum Mechanics Quantum Mechanics Preamble Preamble is a tool for exploring is a tool for exploring Knot Theory Knot Theory Objectives Objectives Rules of the Game Rules of the Game • We seek to create a quantum system We seek to create a quantum system Find a mathematical definition of a quantum Find a mathematical definition of a quantum that simulates a closed knotted physical that simulates a closed knotted physical knot that is knot that is piece of rope. piece of rope. • We seek to define a quantum knot in such • Physically meaningful, i.e., physically We seek to define a quantum knot in such Physically meaningful, i.e., physically a way as to represent the state of the a way as to represent the state of the implementable, and implementable, and knotted rope, i.e., the particular spatial knotted rope, i.e., the particular spatial • Simple enough to be workable and configuration of the knot tied in the rope. configuration of the knot tied in the rope. Simple enough to be workable and • We also seek to model the ways of useable. useable. We also seek to model the ways of moving the rope around (without cutting the moving the rope around (without cutting the rope, and without letting it pass through rope, and without letting it pass through itself.) itself.) 2
Aspirations Aspirations Themes Themes Formal = Knot We would hope that this definition will be We would hope that this definition will be Rewriting Theory useful in modeling and predicting the useful in modeling and predicting the System behavior of knotted vortices that actually behavior of knotted vortices that actually occur in quantum physics such as occur in quantum physics such as • In supercooled helium II Formal = Group In supercooled helium II Rewriting Representation • In the Bose System In the Bose- -Einstein Condensate Einstein Condensate • In the Electron fluid found within the In the Electron fluid found within the fractional quantum Hall effect fractional quantum Hall effect Group = Quantum Representation Mechanics Theory Overview Overview • Preamble Preamble • Mosaic Knots Mosaic Knots • Quantum Mechanics: Whirlwind Tour Quantum Mechanics: Whirlwind Tour Mosaic Knots Mosaic Knots • Quantum Knots & Quantum Knot Systems Quantum Knots & Quantum Knot Systems via Mosaics via Mosaics • Preamble to Lattice Knots Preamble to Lattice Knots • Lattice Knots Lattice Knots Transforming Knot Theory into Transforming Knot Theory into • a formal Rewriting System a formal Rewriting System Q. Knots & Q. Knot Systems Q. Knots & Q. Knot Systems via Lattices via Lattices • Future Directions & Open Questions Future Directions & Open Questions Mosaic Tiles Mosaic Tiles ( ) u T Let denote the following set of 11 Let denote the following set of 11 symbols, called symbols, called mosaic mosaic (unoriented unoriented) ) tiles tiles: Lomonaco and Kauffman, Quantum Knots and Lomonaco and Kauffman, Quantum Knots and Mosaics, Journal of Quantum Information Mosaics, Journal of Quantum Information Processing, vol. 7, Nos. 2- Processing, vol. 7, Nos. 2 -3, (2008), 85 3, (2008), 85- - 115. An earlier version can be found at: 115. An earlier version can be found at: http://arxiv.org/abs/0805.0339 http://arxiv.org/abs/0805.0339 Please note that, up to rotation, there are Please note that, up to rotation, there are exactly 5 tiles exactly tiles 3
Mosaic Knots Mosaic Knots Figure Eight Knot 5- -Mosaic Mosaic Figure Eight Knot A 4-mosaic trefoil Hopf Link 4- -Mosaic Mosaic Hopf Link Let K (n) = the set of n-mosaic knots A Cut & Paste Move A Cut & Paste Move P ← ← → 1 Sub- Sub -Mosaic Moves Mosaic Moves 4
A Cut & Paste Move A Cut & Paste Move A Cut & Paste Move A Cut & Paste Move P P ← ← → ← ← → 1 1 Planar Planar Isotopy Isotopy Moves Moves We will now re-express the as as standard moves on knot Sub Sub- -Mosaic Moves Mosaic Moves diagrams as sub-mosaic moves. 11 Planar Isotopy (PI) Moves on Mosaics 11 Planar Isotopy (PI) Moves on Mosaics Planar Isotopy (PI) Moves on Mosaics Planar Isotopy (PI) Moves on Mosaics It is understood that each of the above moves It is understood that each of the above moves P ← ← → 1 depicts all moves obtained by rotating the depicts all moves obtained by rotating the 2 2 × sub- sub -mosaics by mosaics by 0, , 90 90, , 180 180, or , or 270 270 degrees. degrees. P ← ← → P 2 ← ← 3 → For example, For example, P P P ← ← → 1 ← ← → ← ← → 4 5 represents each of the following 4 represents each of the following 4 moves: moves: P ← ← → P 6 ← ← → 7 P P ← ← → ← ← → 1 1 P P ← ← → ← ← → 8 9 P P ← ← → ← ← → 10 11 P P ← ← 1 → ← ← 1 → 5
Planar Isotopy (PI) Moves on Mosaics Planar Isotopy (PI) Moves on Mosaics Each of the PI Each of the PI 2-submosaic moves represents submosaic moves represents 2+1) 2 possible moves on an any one of the any one of the (n (n- -2+1) possible moves on an Reidemeister Reidemeister Moves Moves n-mosaic mosaic as as Sub- Sub -Mosaic Moves Mosaic Moves Reidemeister (R) Moves on Mosaics Reidemeister (R) Moves on Mosaics Reidemeister (R) Moves on Mosaics Reidemeister (R) Moves on Mosaics R ' R ← ← → 3 ← ← → 3 R ' ← ← → R 1 ← ← → 1 '' R ''' R ← ← → 3 ← ←→ 3 ' R R ← ← → ← ← → 2 2 '' ''' R R ← ← → ←→ ← 2 2 ( ) v ( iv ) R R ← ← → ← ← → 3 3 PI & R Moves on Mosaics PI & R Moves on Mosaics The Ambient Group The Ambient Group A n ( ) Each Each PI move PI move and each and each R move R move is a is a permutation permutation of the set of all of the set of all knot knot n-mosaics mosaics K (n) We define the We define the ambient ambient group group as the as the A n ( ) subgroup of the group of all permutations subgroup of the group of all permutations of the set K (n) generated by the of the set generated by the all PI all PI moves moves and and all all Reidemeister Reidemeister moves moves. In fact, each PI and R move, In fact, each PI and R move, as a as a permutation, is a product permutation, is a product of of disjoint disjoint transpositions. transpositions 6
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