Our capacity to process information depends on the computational power at our disposal. Information theory captures our ability to distinguish states or communicate messages when it is unconstrained with unrivaled beauty and elegance. For computationally bounded observers the situation is quite different -- they can, for example, be fooled to believe that distributions are more random than they actually are. In our work, we build a new foundation for a computational quantum information theory that captures the essence of complexity-constrained information theory while retaining the look and feel of the unbounded asymptotic theory. As our fundamental quantity, we define the computational relative entropy as the optimal error exponent in asymmetric hypothesis testing when restricted to polynomially many copies and quantum gates, defined in a mathematically rigorous way. Building on this foundation, we prove a computational analogue of Stein's lemma, establish computational versions of fundamental inequalities like Pinsker's bound, and demonstrate a computational smoothing property showing that computationally indistinguishable states yield equivalent information measures. We derive a computational entropy that operationally characterizes optimal compression rates for quantum states under computational limitations and show that our quantities apply to computational entanglement theory, proving a computational version of the Rains bound. Our framework reveals striking separations between computational and unbounded information measures, including quantum-classical gaps that arise from cryptographic assumptions, demonstrating that computational constraints fundamentally alter the information-theoretic landscape and open new research directions at the intersection of quantum information, complexity theory, and cryptography.

翻译：我们处理信息的能力取决于可用的计算能力。信息论在无约束条件下，以无与伦比的优美性和简洁性刻画了我们区分状态或传递信息的能力。对于计算能力有限的观测者而言，情况则大不相同——例如，他们可能被误导，认为分布比实际更加随机。在我们的工作中，我们为计算量子信息论构建了一个新的基础，该理论在保留无界渐近理论的外观和感觉的同时，捕获了复杂度约束信息论的本质。作为我们的基本量，我们以数学上严格的方式定义了计算相对熵，将其作为限制在多项式份数和量子门下的非对称假设检验中的最优误差指数。在此基础之上，我们证明了Stein引理的计算版本，建立了如Pinsker界等基本不等式的计算版本，并展示了计算平滑性质，表明计算上不可区分的态能够产生等价的信息度量。我们推导出计算熵，其在操作上刻画了计算限制下量子态的最优压缩率，并展示了我们的量适用于计算纠缠理论，证明了Rains界的计算版本。我们的框架揭示了计算信息度量与无界信息度量之间显著的分离，包括由密码学假设产生的量子-经典间隙，这表明计算约束从根本上改变了信息论的图景，并在量子信息、复杂性理论与密码学的交叉领域开辟了新的研究方向。