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Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials
- Source :
- Linear and Multilinear Algebra (2018): 1-19
- Publication Year :
- 2017
-
Abstract
- A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain---that is, a path on $n$ vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; this has been further generalized to allow for loops (\emph{potentials} in the physics literature). We study the particularly important situation of perfect state transfer with respect to the corresponding adjacency matrix or Laplacian through the use of orthogonal polynomials. Low-dimensional examples are given in detail. Our main result is that PST with respect to the Laplacian matrix cannot occur for weighted paths on $n\geq 3$ vertices nor can it occur for certain symmetric weighted trees. The methods used lead us to a conjecture directly linking the rationality of the weights of weighted paths on $n>3$ vertices, with or without loops, with the capacity for PST between the end vertices with respect to the adjacency matrix.<br />Comment: 16 pages; some minor updates from previous version
- Subjects :
- Quantum Physics
Mathematics - Combinatorics
05C22, 05C50, 15A18, 42C05, 81P45
Subjects
Details
- Database :
- arXiv
- Journal :
- Linear and Multilinear Algebra (2018): 1-19
- Publication Type :
- Report
- Accession number :
- edsarx.1708.03283
- Document Type :
- Working Paper