Computational Two-Way Quantum Capacity Vanishes for Polynomial Complexity Channels

Researchers are increasingly focused on understanding the limits of reliable communication under realistic constraints. Johannes Jakob Meyer, Jacopo Rizzo, and Asad Raza, from the Dahlem Center for Complex Quantum Systems at Freie Universität Berlin, alongside Lorenzo Leone of the Università degli Studi di Salerno, Sofiene Jerbi and Jens Eisert et al, have now investigated computational capacities, quantifying information transmission when encoding and decoding must be computationally efficient. This work introduces a study of the computational two-way capacity, revealing a surprising link between this capacity and the computational distillable entanglement of a quantum channel’s Choi state. Crucially, the team demonstrate that even channels with near-maximal unbounded capacity can have vanishing computational capacity if their complexity exceeds polynomial limits, highlighting a dramatic shift in communication bounds when computational efficiency is considered. These findings fundamentally alter our understanding of how much information can truly be transmitted, given the practical limitations of computation.

Their research, published recently, moves beyond traditional quantum channel capacity definitions, which disregard the computational resources of sender and receiver, to quantify information transmission rates under the constraint of efficient encoding and decoding. The team focused specifically on the computational two-way quantum capacity, revealing a surprising connection to the computational distillable entanglement within the channel’s Choi state. This innovative approach allowed researchers to demonstrate a stark separation in computational capacity, proving that a quantum channel of polynomial complexity can possess a vanishing computational two-way capacity despite exhibiting a nearly maximal unbounded capacity.

This means that even with a channel that isn’t overwhelmingly complex to operate, the amount of reliably transmitted information is severely limited when computational efficiency is prioritised. Furthermore, the study unveils a sharp transition, from nearly maximal to zero, in computational capacity as channel complexity moves beyond the polynomial realm, highlighting the critical role of computational constraints in quantum communication. Experiments show that the computational two-way quantum capacity is intimately linked to the computational distillable entanglement of the Choi state, mirroring the relationship observed in traditional quantum information theory. This connection provides a powerful new tool for quantifying and assessing the limits of quantum communication under polynomial constraints, ensuring that computed capacities are operationally achievable.
The team established a new direction within computational quantum information theory, building upon the foundational premise that tasks are feasible only if computational steps grow polynomially with input size. Most notably, the research establishes that under standard cryptographic assumptions, a physically meaningful quantum channel with polynomial complexity can have a computational two-way quantum capacity that effectively disappears, while its traditional capacity remains high. This phenomenon, termed computationally bound capacity, suggests that the inherent limits of quantum communication are dramatically altered when computational efficiency is a primary concern. The study also demonstrates a sharp transition in capacity for generalized dephasing channels, dependent on their Choi-rank, with nearly optimal capacity achieved in the polynomial regime and vanishing capacity beyond it.

This work underscores the necessity of reassessing the foundations of quantum communication theory in light of the potentially unreasonable resources implicitly required by traditional models. . Experiments confirm this phenomenon under standard cryptographic assumptions, highlighting a sharp transition in computational quantum capacity from nearly maximal to zero as channel complexity moves beyond the polynomial realm. These results demonstrate that imposing computational efficiency as a requirement radically alters the limits of quantum communication. The team measured the computational two-way quantum capacity, a novel concept assessing limits with polynomial constraints on resources, specifically, polynomially many copies used and quantum gates implemented in encoding and decoding.

Data shows a close relationship between this computational capacity and the computational distillable entanglement of the channel’s Choi state, mirroring the connection found in traditional quantum information theory. Researchers established that the computational capacity is intricately linked to the resource of entanglement in a bipartite scenario, providing a quantitative tool for assessing its limits. Measurements confirm that the computational two-way quantum capacity captures the limits of quantum information transmission with bidirectional classical communication, but with operations limited to polynomial complexity. Tests prove the existence of physically meaningful quantum channels with polynomial complexity exhibiting vanishing computational two-way quantum capacity, while their traditional counterparts maintain substantial capacity.

Specifically, the study reveals a sharp transition in computational capacity dependent on the Choi-rank of a generalized dephasing channel. In the polynomial Choi-rank regime, a computationally efficient protocol achieves nearly optimal quantum capacity, but this capacity can vanish when the Choi-rank becomes super-polynomial. The breakthrough delivers a new understanding of quantum communication theory, suggesting a reassessment of foundational premises in light of potentially super-polynomial resource requirements. Further analysis quantified the complexity of quantum channels, defining it as the number of 2-local gates required for operations, alongside measurements projecting onto |0⟩ or ignoring the system. The work adopts a computational model where tracing out is a free operation, allowing for gate-constrained versions of quantum information processing tasks like entanglement distillation.