On the undecidability of quantum channel capacities
Archishna Bhattacharyya, Arthur Mehta, Yuming Zhao
TL;DR
The authors address whether quantum channel capacities admit efficient computation, showing that computing the quantum capacity $Q(\Phi)$ is $\mathsf{QMA}$-hard and that the maximal-entanglement-assisted one-shot zero-error classical capacity $C^{(1)}_{0,M,E}(\Phi)$ is uncomputable. Their approach combines a circuit-based description of channels, a novel projective direct sum construction to create a controlled capacity gap, and properties of degradable/anti-degradable channels to connect capacity evaluations to entropic quantities. They further relate a quantum graph parameter, the quantum independence number $\alpha_q(G)$, to $C^{(1)}_{0,M,E}(\Φ_G)$, leveraging the $\textsf{MIP}^{\ast}=\textsf{RE}$ undecidability result to establish uncomputability. While not resolving the full uncomputability of $Q(\Phi)$, the work provides the first concrete undecidability result for a capacity-driven quantum-information task and demonstrates deep ties between quantum information, circuit complexity, and nonlocal games. These findings imply that, in general, no efficient algorithm can compute quantum channel capacities under current complexity-theoretic assumptions, highlighting fundamental limits in quantum information theory.
Abstract
An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the maximal-entanglement-assisted zero-error one-shot classical capacity is uncomputable.