Pretty Good State Transfer in Qubit Chains - The Heisenberg Hamiltonian

Abstract

Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with n qubits there is pretty good state transfer between the nodes at the j-th and (n-j+1)-th position if n is a power of 2. Moreover, this condition is also necessary for j=1. We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory.

The first version of this paper made available on Arxiv contained the false claim that pretty good state transfer occurs between end vertices in a path of length $n$ where $n$ is a prime congruent to $1$ modulo $4$. This claim has been corrected in this version

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Cited by in corpus (13)

• Perfect quantum state transfer using Hadamard diagonalizable graphs
• Designing pretty good state transfer via isospectral reductions
• Quantum routing of information using chiral quantum walks
• Pretty good quantum state transfer on isotropic and anisotropic Heisenberg spin chains with tailored site dependent exchange couplings
• Understanding the propagation of excitations in quantum spin chains with different kind of interactions
• Quantum state transfer between twins in weighted graphs
• Pretty good state transfer via adaptive quantum error correction
• Perfect state transfer on hypercubes and its implementation using superconducting qubits
• Quantum state transfer using 1D Heisenberg Hamiltonian on quasi-1D lattices
• State Transfer in Latent-Symmetric Networks
• Laplacian pretty good fractional revival
• Memory effects in repeated uses of quantum channels
• Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer