A Study of Information Transmission over Quantum Channels in the One-Shot Setting

Sayantan Chakraborty
Pranab Sen
Thursday, 1 Sep 2022, 17:30 to 19:30
In this thesis we tackle two main problems :
1. Coding techniques for sending quantum information across a multi-terminal quantum channel in the one-shot regime.
2. Coding techniques for sending private information across a classical-quantum multiple access channel, while ensuring that the transmitted messages remain hidden from an eavesdropper.
We first consider the Quantum Multiple Access Channel or QMAC and show the existence of coding theorems for the task of sending EPR pairs from two independent senders Alice and Bob to a receiver Charlie using this channel. For this purpose we develop two main tools:
1. A multi-terminal decoupling theorem.
2. One-shot quantum rate splitting.
While the multi-terminal decoupling theorem allows us to recover an ideal pentagonal achievable rate region, for entanglement transmission across the QMAC, these bounds are not easily generalised to recover the best inner bounds known for this problem in the asymptotic iid setting, due to the issue of Simultaneous Smoothing, which remains a major open problem in quantum information theory.
To get around this problem, we adapt the classical iid technique of rate splitting to the one-shot quantum setting. This new technique allows us to recover the best known achievable rate region known for this problem in the asymptotic iid regime. Our techniques also allow us to show the existence of coding schemes which achieve the best known non-trivial rate region for entanglement transmission across the Quantum Interference Channel or QIC, both in the one-shot and in the asymptotic iid regime.
We next consider the problem of sending private classical information from two independent senders Alice and Bob to a legitimate receiver Charlie, in the presence of an eavesdropper Eve, via a wiretap classical-quantum MAC. This problem is considerably hard to solve since we require that Alice and Bob's messages should be jointly secret from Eve, which requires a joint quantum covering lemma.
Unfortunately, such a lemma remains out of reach of current techniques, since it reduces to the Simultaneous Smoothing conjecture.
We get around this issue by developing a successive cancellation style covering lemma which, along with some other tools, allows us to recover a non-trivial achievable rate region in the one-shot regime and also recover the natural pentagonal rate region that one would expect for this problem, in the asymptotic iid regime.