Anzeige
Erschienen in:
01.03.2015
Quantum knots and the number of knot mosaics
Erschienen in: Quantum Information Processing | Ausgabe 3/2015
EinloggenAktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot \((m,n)\)-mosaic is an \(m \times n\) matrix of mosaic tiles (\(T_0\) through \(T_{10}\) depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. \(D^{(m,n)}\) is the total number of all knot \((m,n)\)-mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. \(D^{(m,n)}\) is already found for \(m,n \le 6\) by the authors. In this paper, we construct an algorithm producing the precise value of \(D^{(m,n)}\) for \(m,n \ge 2\) that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics. where \(2^{m-2} \times 2^{m-2}\) matrices \(X_{m-2}\) and \(O_{m-2}\) are defined by for \(k=0,1, \cdots , m-3\), with \(1 \times 1\) matrices \(X_0 = \begin{bmatrix} 1 \end{bmatrix}\) and \(O_0 = \begin{bmatrix} 1 \end{bmatrix}\). Here \(\Vert N\Vert \) denotes the sum of all entries of a matrix \(N\). For \(n=2\), \((X_{m-2}+O_{m-2})^0\) means the identity matrix of size \(2^{m-2} \times 2^{m-2}\).
$$\begin{aligned} D^{(m,n)} = 2 \, \Vert (X_{m-2}+O_{m-2})^{n-2} \Vert \end{aligned}$$
$$\begin{aligned} X_{k+1} = \begin{bmatrix} X_k&O_k \\ O_k&X_k \end{bmatrix} \ \hbox {and } \ O_{k+1} = \begin{bmatrix} O_k&X_k \\ X_k&4 \, O_k \end{bmatrix} \end{aligned}$$