# On the Entanglement Cost of One-Shot Compression

## TL;DR

This paper investigates the fundamental limits of quantum state compression with entanglement assistance in a one-shot setting, revealing cases where entanglement is essential and can drastically reduce communication costs.

## Contribution

It demonstrates that shared entanglement can significantly lower communication costs in quantum compression, and constructs ensembles that saturate known bounds even with relaxed protocols.

## Key findings

- Shared entanglement reduces communication cost arbitrarily.
- Certain ensembles are incompressible without entanglement.
- Entanglement can be more resource-efficient than classical communication.

## Abstract

We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.

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Source: https://tomesphere.com/paper/1905.02110