# Compression for quantum population coding

## TL;DR

This paper introduces an asymptotically faithful quantum data compression protocol for parametric families of quantum states, balancing quantum and classical memory, and establishing fundamental limits on quantum memory requirements.

## Contribution

It develops a quantum version of local asymptotic normality for efficient compression, and determines the minimal memory size and quantum-classical ratio needed for faithful encoding.

## Key findings

- Minimum memory size for asymptotic faithfulness is (f/2)log(n).
- Quantum and classical memory ratio can be made arbitrarily small but not zero.
- Classical-only protocols cannot be faithful unless all states commute.

## Abstract

We study the compression of n quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with f independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size (f/2)log(n), including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of n identically prepared modes. In both cases, we show that (f/2)log(n) is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.