The role of the ℓ1-norm in quantum information theory and two types of the Yang–Baxter equation

The role of the l1-norm in the Yang–Baxter system has been studied through Wigner's D-functions, where l1-norm means ∑i|Ci| for |Ψ = ∑iCi|ψi with |ψi being the orthonormal basis. It is shown that the existing two types of braiding matrices, which can be viewed as particular solutions of the Yang–Baxter equation (YBE) with different spectral parameters can be unified in the 2D YBE. We prove that the maximum of the l1-norm is connected with the maximally entangled states and topological quantum field theory with two-component anyons, while the minimum leads to the deformed permutation related to the familiar integrable models.

[1]  John Preskill,et al.  Lecture Notes for Physics 219: Quantum Computation , 2004 .

[2]  Evgeny Sklyanin,et al.  QUANTUM SPECTRAL TRANSFORM METHOD. RECENT DEVELOPMENTS , 1982 .

[3]  M. Freedman,et al.  Topologically protected qubits from a possible non-Abelian fractional quantum Hall state. , 2004, Physical review letters.

[4]  M. E. Rose.  Elementary Theory of Angular Momentum , 1957 .

[5]  L. A. Takhtadzhan,et al.  THE QUANTUM METHOD OF THE INVERSE PROBLEM AND THE HEISENBERG XYZ MODEL , 1979 .

[6]  Zhenghan Wang.  Topologization of electron liquids with Chern-Simons theory and quantum computation , 2006 .

[7]  R. Baxter.  Partition function of the eight vertex lattice model , 1972 .

[8]  J. B. McGuire,et al.  Study of Exactly Soluble One-Dimensional N-Body Problems , 1964 .

[9]  Louis H. Kauffman,et al.  Braiding operators are universal quantum gates , 2004, quant-ph/0401090.

[10]  D. Varshalovich,et al.  Quantum Theory of Angular Momentum , 1988 .

[11]  C. Yang,et al.  S MATRIX FOR THE ONE-DIMENSIONAL N-BODY PROBLEM WITH REPULSIVE OR ATTRACTIVE delta-FUNCTION INTERACTION. , 1968 .

[12]  Jie Tang,et al.  Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets. , 2008, Medical physics.

[13]  Mauro Spera,et al.  On Uncertainty, Braiding and Entanglement in Geometric Quantum Mechanics , 2006 .

[14]  Kang Xue,et al.  Optical simulation of the Yang-Baxter equation , 2007, 0711.4703.

[15]  Michael Baake,et al.  Quantum mechanics versus classical probability in biological evolution , 1998 .

[16]  Michael Larsen,et al.  A Modular Functor Which is Universal¶for Quantum Computation , 2000, quant-ph/0001108.

[17]  Xin Wan,et al.  Constructing functional braids for low-leakage topological quantum computing , 2008, 0802.4213.

[18]  Yong Zhang,et al.  Yang–Baxterizations, Universal Quantum Gates and Hamiltonians , 2005, Quantum Inf. Process..

[19]  C. Yang.  Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction , 1967 .

[20]  F. A. Bais,et al.  Quantum groups and non-Abelian braiding in quantum Hall systems , 2001 .

[21]  Elliott H Lieb,et al.  Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[22]  S. Simon,et al.  Non-Abelian Anyons and Topological Quantum Computation , 2007, 0707.1889.

[23]  Kang Xue,et al.  Berry phase and quantum criticality in Yang-Baxter systems , 2008, 0806.1369.

[24]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[25]  Frank Wilczek,et al.  2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states , 1996 .

[26]  Kang Xue,et al.  Braiding transformation, entanglement swapping, and Berry phase in entanglement space , 2007, 0704.0709.

[27]  Jie Tang,et al.  Performance comparison between total variation (TV)-based compressed sensing and statistical iterative reconstruction algorithms , 2009, Physics in medicine and biology.