Quantum networks: Anti-core of spin chains

The purpose of this paper is to exhibit a quantum network phenomenon - the anti-core---that goes against the classical network concept of congestion core. Classical networks idealized as infinite, Gromov hyperbolic spaces with least-cost path routing (and subject to a technical condition on the Gromov boundary) have a congestion core, defined as a subnetwork that routing paths have a high probability of visiting. Here, we consider quantum networks, more specifically spin chains, define the so-called maximum excitation transfer probability $p_{\max}(i,j)$ between spin $i$ and spin $j$, and show that the central spin has among all other spins the lowest probability of being excited or transmitting its excitation. The anti-core is singled out by analytical formulas for $p_{\mathrm{max}}(i,j)$, revealing the number theoretic properties of quantum chains. By engineering the chain, we further show that this probability can be made vanishingly small.