# A hypothesis testing approach for communication over entanglement assisted compound quantum channel

## TL;DR

This paper develops a hypothesis testing framework for communication over entanglement-assisted compound quantum channels, providing near-optimal bounds and new techniques for quantum hypothesis testing and decoding.

## Contribution

It introduces a one-shot achievability and converse bounds for entanglement-assisted compound quantum channels using quantum hypothesis testing divergence.

## Key findings

- Near-optimal bounds in one-shot quantum setting.
- A new approach for constructing unions of projectors.
- Applications to testing composite quantum hypotheses.

## Abstract

We study the problem of communication over a compound quantum channel in the presence of entanglement. Classically such channels are modeled as a collection of conditional probability distributions wherein neither the sender nor the receiver is aware of the channel being used for transmission, except for the fact that it belongs to this collection. We provide near optimal achievability and converse bounds for this problem in the one-shot quantum setting in terms of quantum hypothesis testing divergence. We also consider the case of informed sender, showing a one-shot achievability result that converges appropriately in the asymptotic and i.i.d. setting. Our achievability proof is similar in spirit to its classical counterpart. To arrive at our result, we use the technique of position-based decoding along with a new approach for constructing a union of two projectors, which can be of independent interest. We give another application of the union of projectors to the problem of testing composite quantum hypotheses.