量子自旋系统中通信的数论性质
Number-Theoretic Nature of Communication in Quantum Spin Systems
摘要：
在过去的十年里，人们对在量子力学对象网络上传输信息的协议产生了浓厚的兴趣。已经提出了各种控制方法和网络拓扑结构，其基础是具有完美保真度的传输-即确定性和无信息丢失-通过具有多于几个粒子的未调制自旋链是不可能的。解决由Bose [Phys. Rev. Lett. 91，207901（2003）]，我们确定了未调制链（具有XY哈密顿量）中允许保真度任意接近1的传输的量子位的确切数量，这种现象称为相当好的状态传输。我们证明了当且仅当节点数为n=p-1，2 p-1，其中p是素数，或n=2^{m}-1时，这种情况发生。这一结果突出了量子自旋系统动力学在重新解释整数算术结构问题上的潜力，在这种情况下，是素数。
英文摘要：
The last decade has witnessed substantial interest in protocols for transferring information on networks of quantum mechanical objects. A variety of control methods and network topologies have been proposed, on the basis that transfer with perfect fidelity --- i.e. deterministic and without information loss --- is impossible through unmodulated spin chains with more than a few particles. Solving the original problem formulated by Bose [Phys. Rev. Lett. 91, 207901 (2003)], we determine the exact number of qubits in unmodulated chains (with XY Hamiltonian) that permit the transfer with fidelity arbitrarily close to 1, a phenomenon called pretty good state transfer. We prove that this happens if and only if the number of nodes is n=p-1, 2p-1, where p is a prime, or n=2^{m}-1. The result highlights the potential of quantum spin system dynamics for reinterpreting questions about the arithmetic structure of integers, and, in this case, primality.
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