The computational two-way quantum capacity

TL;DR

This paper introduces the concept of computational quantum capacities, focusing on the computational two-way quantum capacity, and reveals significant capacity separations under cryptographic assumptions, highlighting the impact of computational efficiency on quantum communication limits.

Contribution

It defines and analyzes the computational two-way quantum capacity, establishing its relation to computational distillable entanglement and demonstrating capacity separation under cryptographic assumptions.

Findings

Existence of a polynomial complexity channel with zero computational capacity but near-maximal unbounded capacity.

Sharp transition in computational capacity when channel complexity exceeds polynomial bounds.

Computational efficiency constraints can drastically reduce quantum communication capabilities.

Abstract

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.