Quantum dense coding properties between two identical and spatially separated atoms in free space with different initial states is investigated. It shows that dense coding capacity χ experienced a sharp decline firstly and then gradually increased to be one steady value 1 with increasing Γt. The realization of dense coding capacity χ is found to be strongly dependent on the initial states. It is worth noting that the initial pure state |ee〉 is not useful for dense coding in this system, due to the dense coding capacity χ is always less than 1(a valid dense coding capacity satisfies χ > 1). Otherwise, for the initial entangled state and mixed state, one can obtain the valid dense coding capacity, the results show that one threshold value of tc is exists, and when t < tc the dense coding capacity is valid. Tuning the atomic distance between the two atoms slightly broaden the valid dense coding region and improve the value of tc. Decreasing the purity a of initial states not only broaden the region but also prolong the effective time where one can carry out the valid dense coding successfully.

中文翻译：

---

自由空间中两个空间分离的原子之间的量子密集编码特性

研究了自由空间中具有不同初始状态的两个相同且空间分离的原子之间的量子密集编码特性。可以看出，随着 Γt 的增加，密集编码容量 χ 先急剧下降，然后逐渐增加到一个稳定值 1。发现密集编码容量 χ 的实现强烈依赖于初始状态。值得注意的是，初始纯状态 |ee> 在该系统中对于密集编码没有用，因为密集编码容量 χ 总是小于 1（有效密集编码容量满足 χ > 1）。否则，对于初始纠缠态和混合态，可以得到有效的密集编码容量，结果表明存在一个阈值tc，当t<tc时密集编码容量有效。调整两个原子之间的原子距离会稍微加宽有效的密集编码区域并提高 tc 的值。降低初始状态的纯度 a 不仅扩大了区域，而且延长了成功进行有效密集编码的有效时间。