量子信道容量

Quantum channel is the abstract model for information processing such as quantum states stored in cold atoms、photons transmitted via optical fibre and so on. Quantum channel can be used to transmit classsical information, quantum information, and private information. The ubiquitous noise introduces errors in the transmission that must be corrected via quantum error correction. The fundamental bounds on quantum error correction are the capacities, which quantify the amount of information that can be protected. Distinct from classical channel, the capacites on quantum channel can be non-additive, that is two channels used in paralell can transmit more information than the same two used separately, somewhat like 1+1>2. The reason is that the entangled states that have no classical counterpart could be chosen as inputs and might help transmission. Even there exists a startling effect called superactvation, 0+0>0, where two channels that cannot transmit information in separate use could do in parallel. It's a good news that entanglememt can help transmission. However, a bad news is that the capacites cannot be effectively evaluated without additivity. Therefore, the quesiton on the capacities remains surprisingly wide open and the computable bounds are extremely desirable. When entangled states are shared between the input and output as an assisting resource, additivity survives and the capacities are computable. In this proposal, the topic is on quantum channel capacities, especially the computable bounds, the superactivation, the criteria for additivity and non-additivity, the assisted capacities, and finally the concept of potential capcities. The topic is one of the most fundamental questions in quantum information theory and the solution could serve as a benchmark of quantum information processing in physical realization. As pointed out, the question is wide open, so the proposal is timely.

量子信道是冷原子存储量子态、光纤传递光子等信息过程的抽象模型。噪声无处不在，导致错误，错误通过纠错码纠正，不同的纠错码保护不同的信息量，容量是其中的极大值。量子信道能传递经典信息、量子信息和秘密信息。与经典信道不同，量子信道容量一般不具有可加性，也即有可能出现1+1>2的情况，两个信道一起用比分开用能传递更多信息。原因是：纠缠的存在扩大了输入的选择范围，因而能提高信息传递，甚至出现奇异的超级激发效应，两个不能传递信息的信道一起用就能传递信息，0+0>0！但由于可选范围太大，容量作为极值变得不可计算。因此一个急需解决的问题是容量的可计算上界。另外，研究在什么样的辅助资源下，可加性成立，以及可加性判据，也是一个关键。本项目研究量子信道容量，内容包括：一、可计算上界；二、超级激发；三、可加与不可加性判据；四、在什么辅助资源下可加性成立；五、潜在容量的概念。本项目对量子信息论具有重要意义。

量子信道容量是量子香农理论研究的主要课题。该项目研究包括：（1）强反定理：证明强反定理在entanglement-breaking和Hadamard信道传输经典信息时成立；和正定理一起表明容量是传输信息是否可靠的相变点。发现了三明治式的量子瑞利相对熵函数(此函数也为苏黎世 ETH 研究组独立发现)，证明其满足数据处理不等式。（2）潜在容量：引入量子信道潜在容量的概念，给出可计算上界，证明有燥信道在任何 contextual channel 下都不能被激发为理想信道。（3）传输模型：发展了新的信息传输模型，研究传输量子信息和经典信息，发现存在超级激发(量子信息)和 superadditivity(经典信息)。（4）量子相干：建立了量子相干的资源理论。引入量子相干转化的两个基本过程：coherence distillation 和 coherence cost，并给出 distillable coherence 和 coherence cost 的计算公式；证明该理论是非可逆性理论，但不存在 bound coherence；进一步给出是否可逆的判据。