Quantum channel capacities are fundamental to quantum information theory. Their definition, however, does not limit the computational resources of sender and receiver. In this work, we initiate the study of computational quantum capacities. These quantify how much information can be reliably transmitted when imposing the natural requirement that en- and decoding have to be computationally efficient. We focus on the computational two-way quantum capacity and showcase that it is closely related to the computational distillable entanglement of the Choi state of the channel. This connection allows us to show a stark computational capacity separation. Under standard cryptographic assumptions, there exists a quantum channel of polynomial complexity whose computational two-way quantum capacity vanishes while its unbounded counterpart is nearly maximal. More so, we show that there exists a sharp transition in computational quantum capacity from nearly maximal to zero when the channel complexity leaves the polynomial realm. Our results demonstrate that the natural requirement of computational efficiency can radically alter the limits of quantum communication.

翻译：量子信道容量是量子信息理论的基础。然而，其定义并未限制发送方和接收方的计算资源。在本工作中，我们开创性地研究了计算量子容量。这些容量量化了在施加编码和解码必须计算高效这一自然要求时，能够可靠传输的信息量。我们聚焦于计算双向量子容量，并证明其与信道Choi态的计算可蒸馏纠缠度密切相关。这一关联使我们能够展示一个显著的计算容量分离现象。在标准密码学假设下，存在一个具有多项式复杂度的量子信道，其计算双向量子容量为零，而其无限制对应容量却接近最大值。更进一步，我们证明当信道复杂度超出多项式范畴时，计算量子容量会出现从接近最大值到零的急剧转变。我们的结果表明，计算高效这一自然要求能够从根本上改变量子通信的极限。