Much work has been done in the last two decades on the topic of quantum state transfer in a quantum spin network. One can model such a system of interacting qubits using an undirected graph, and studying vertex-to-vertex dynamics. This setup has recently been relaxed to allow for dynamics between linear combinations of two vertex states, i.e.\ from u=ea+seb to μ=eα+reβ, where r=s is either −1 (which corresponds to pair state transfer) or +1 (which corresponds to plus state transfer), or more recently r=s is taken to be any real number (which corresponds to s-pair state transfer).
Here, we broaden the investigation of s-pair state transfer to \textit{perfect (s,r)-state transfer}, which is perfect state transfer from u=ea+seb to μ=eα+reβ (up to some dilation) where r,s∈C. We identify infinite families of graphs with perfect (s,r)-state transfer and provide characterizations of cases when ∣r∣=∣s∣ and when ∣r∣=∣s∣, showing situations when the degree of entanglement between vertex states is preserved, and when it is not preserved. The latter is particularly important as it represents perfect state transfer from an entangled pair of qubits to another one where the degree of entanglement need not be the same\mdash in fact, it can be set up so as to ``boost'' (increase) entanglement. We provide an algorithm that finds the vector with two nonzero entries that maximizes the fidelity of transfer for a fixed time t starting from a given s-pair state u. Finally, we provide a sensitivity analysis, with respect to readout time errors, of perfect (s,r)-state transfer.
A weighted graph G with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in G. In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each k≥3, (i) there are infinitely many connected unweighted graphs with maximum degree k admitting pair state transfer at τ∈{2π,2π}, and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree k admitting plus state transfer at τ∈{2π,2π}. Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at τ∈{2π,2π}, and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at τ∈{2π,2π}. Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.