# Quantum Rate-Distortion Coding of Relevant Information

## TL;DR

This paper extends the classical Information Bottleneck Method to quantum data, deriving a quantum rate-distortion function using entanglement-assisted protocols and addressing unique quantum formalism challenges.

## Contribution

It introduces a quantum version of the Information Bottleneck Method for data compression, utilizing a channel communication framework and entanglement assistance.

## Key findings

- Derived quantum rate-distortion function under entanglement assistance
- Characterized optimal quantum channels for given input states
- Discussed and addressed formalism differences between quantum and classical probability

## Abstract

Rate-distortion theory provides bounds for compressing data produced by an information source to a specified encoding rate that is strictly less than the source's entropy. This necessarily entails some loss, or distortion, between the original source data and the best approximation after decompression. The so-called Information Bottleneck Method is designed to compress only `relevant' information. Which information is relevant is determined by the correlation between the data being compressed and a variable of interest, so-called side information. In this paper, an Information Bottleneck Method is introduced for the compression of quantum data. The channel communication picture is used for compression and decompression. The rate of compression is derived using an entanglement assisted protocol with classical communication, and under an unproved conjecture that the rate function is convex in the distortion parameter. The optimum channel achieving this rate for a given input state is characterised. The conceptual difficulties arising due to differences in the mathematical formalism between quantum and classical probability theory are discussed and solutions are presented.