How to construct spin chains with perfect state transfer

It is shown how to systematically construct the $XX$ quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST). Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A new model connected to a special case of the symmetric $q$-Racah polynomials is offered. It is also explained how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.

[1]  Joachim Stolze,et al.  Spin chains as perfect quantum state mirrors , 2005 .

[2]  Matthias Christandl,et al.  Mirror inversion of quantum states in linear registers. , 2004, Physical review letters.

[3]  Alastair Kay,et al.  Perfect, Efficent, State Transfer and its Application as a Constructive Tool , 2009, 0903.4274.

[4]  Sougato Bose,et al.  Quantum communication through spin chain dynamics: an introductory overview , 2007, 0802.1224.

[5]  G. M. L. Gladwell,et al.  Inverse Problems in Vibration , 1986 .

[6]  Duality of Orthogonal Polynomials on a Finite Set , 2001, math/0101125.

[7]  L. Vinet,et al.  A characterization of classical and semiclassical orthogonal polynomials from their dual polynomials , 2004 .

[8]  Alexei Zhedanov,et al.  Rational spectral transformations and orthogonal polynomials , 1997 .

[9]  H. Rabitz,et al.  All possible coupling schemes in XY spin chains for perfect state transfer , 2011, 1101.1156.

[10]  M. Anshelevich,et al.  Introduction to orthogonal polynomials , 2003 .

[11]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[12]  C. P. Sun,et al.  Quantum-state transfer via the ferromagnetic chain in a spatially modulated field , 2005 .

[13]  B. Shore,et al.  Coherent dynamics of N-level atoms and molecules. III. An analytically soluble periodic case , 1979 .

[14]  Gene H. Golub,et al.  The numerically stable reconstruction of a Jacobi matrix from spectral data , 1977, Milestones in Matrix Computation.

[15]  Rene F. Swarttouw,et al.  Hypergeometric Orthogonal Polynomials , 2010 .

[16]  An Exactly Solvable Spin Chain Related to Hahn Polynomials , 2011, 1101.4469.

[17]  Quantum state transfer in spin chains with q-deformed interaction terms , 2010, 1005.2912.

[18]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[19]  Edward B. Saff,et al.  Finite sequences of orthogonal polynomials connected by a Jacobi matrix , 1986 .

[20]  Pérès,et al.  Reversible logic and quantum computers. , 1985, Physical review. A, General physics.

[21]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .