3. Results Beyond the State of the Art
The project advances the state of the art in several directions, including the following:
1. This work offers novel insights into the task of “quantum state exchange”, a generalisation of the well-understood quantum state merging problem. In quantum state merging, a sender seeks to merge its quantum state with that of a receiver. In contrast, the quantum state exchange task requires both parties to exchange their quantum systems with one another. This study provides a new perspective on quantum data compression and symmetry.
2. The network generalisation of the state merging problem has remained an open problem since the inception of state merging, involving a scenario where multiple senders aim to merge their quantum systems with a single receiver. This project addresses this longstanding question by providing various examples and structures within the network, enabling the determination of minimal resource requirements for successful distributed merging.
3. This project provides mathematical proofs characterising the minimal imperfect resources required for simulating noisy quantum channels. It demonstrates how entanglement offers an “advantage” even in simulating noisy classical channels, surpassing the capabilities of the classical counterpart resource, namely shared randomness.
4. Establishing the capacity of quantum channels is a fundamental question in information theory of central operational relevance. This project mathematically determines the capacity of quantum channels for transmitting imperfect and noisy entanglement, introducing a new quantity called the generalised capacity of a quantum channel.
5. For the quantum capacity of anti-degradable channels, this project provides a rigorous mathematical proof demonstrating that the pretty strong converse property holds: no information can be transmitted above the capacity with an error smaller than 1.
6. For imperfect quantum data, this project mathematically establishes the optimal trade-off between the error (loss in fidelity) in recovering such data and the minimal number of qubits required for its compression.
