Perfect state transfer on weighted graphs of the Johnson scheme

Letters in Mathematical Physics - We characterize quantum perfect state transfer on real-weighted graphs of the Johnson scheme $${\mathcal {J}}(n,k)$$ , which represent spin networks with...     

Perfect state transfer in Laplacian quantum walk

For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix \(U(t) = \exp (-itM)\), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time \(\tau \) if the (u, v)-entry of \(U(\tau )\) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an n-vertex graph has perfect state transfer at time \(\tau \) relative to the Laplacian, then so does its complement if \(n\tau \in 2\pi {\mathbb {Z}}\). As a corollary, the join of \(\overline{K}_{2}\) with any m-vertex graph has perfect state transfer relative to the Laplacian if and only if \(m \equiv 2\pmod {4}\). This was previously known for the join of \(\overline{K}_{2}\) with a clique (Bose et al. in Int J Quant Inf 7:713–723, 2009). If a graph G has perfect state transfer at time \(\tau \) relative to the normalized Laplacian, then so does the weak product \(G \times H\) if for any normalized Laplacian eigenvalues \(\lambda \) of G and \(\mu \) of H, we have \(\mu (\lambda -1)\tau \in 2\pi {\mathbb {Z}}\). As a corollary, a weak product of \(P_{3}\) with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of \(P_{3}\) has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (Godsil in Discret Math 312(1):129–147, 2011).     

Quantum walks on embeddings

Journal of Algebraic Combinatorics - We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence...     

Rhodamine B-based “turn-on” fluorescent and colorimetric chemosensors for highly sensitive and selective detection of mercury (II) ions

Two novel macromolecules based on 2-[3-(2-aminoethylsulfanyl)propylsulfanyl]ethanamine covalently bound to one and two units of rhodamine-B moieties, 1 and 2, were prepared and utilized as fluoroionophores and chromophores for the optical detection of Hg²⁺ ions. The sensors were readily prepared by a conventional two-step synthesis. Especially, sensor 1 exhibits high sensitivity and selective OFF–ON fluorescence enhancement and chromogenic change upon binding to Hg²⁺, which served as a “naked-eye” indicator by a noticeable color change of the solution (from colorless to pink–red color). 1 is shown to discriminate various competing metal ions, particularly Ag⁺ and Cu²⁺, as well as Cd²⁺, Na⁺, Li⁺, K⁺, Ba²⁺, Co²⁺, Ni²⁺, Mg²⁺, Mn²⁺ and Al³⁺, with a detection limit of 10 ppb.