A Note on the Speed of Perfect State Transfer

TL;DR

This paper simplifies the proof of the speed limit for perfect quantum state transfer in one-dimensional spin chains, specifically addressing the case of odd chain lengths, and aligns it with the even case proof.

Contribution

It provides a more straightforward and unified proof for the speed bound in odd-length spin chains, enhancing theoretical understanding.

Findings

Simplified proof for odd N case of state transfer speed bound

Unified proof approach for even and odd N cases

Clarified the theoretical limits of quantum state transfer speed

Abstract

In Phys. Rev. A 74, 030303 (2006), Yung showed that for a one-dimensional spin chain of length $N$ and maximum coupling strength $J_{\max}$, the time $t_0$ for a quantum state to transfer from one end of the chain to another is bounded by $J_{\max} t_0\geq\pi N/4$ (even $N$) and $J_{\max} t_0\geq\pi\sqrt{N^2-1}/4$ (odd $N$). The proof for even $N$ was elegant, but the proof for odd $N$ was less so. This note provides a proof for the odd $N$ case that is simpler, and more in keeping with the proof for the even case.