Incompressibility of classical distributions

TL;DR

This paper establishes a fundamental lower bound on the data transmission rate for classical state ensembles in quantum compression, demonstrating that certain classical ensembles are inherently incompressible even with shared entanglement and constant error tolerance.

Contribution

It provides a general, robust lower bound on the compression rate for classical states, showing that some ensembles cannot be significantly compressed beyond the Holevo information.

Findings

Proves a near-maximal separation between achievable rate and Holevo information for a two-state ensemble.

Demonstrates that classical ensembles can be inherently incompressible, with minimal reduction in communication cost.

Links incompressibility to limitations in cloning and distinguishing classical states.

Abstract

In blind compression of quantum states, a sender Alice is given a specimen of a quantum state $\rho$ drawn from a known ensemble (but without knowing what $\rho$ is), and she transmits sufficient quantum data to a receiver Bob so that he can decode a near perfect specimen of $\rho$. For many such states drawn iid from the ensemble, the asymptotically achievable rate is the number of qubits required to be transmitted per state. The Holevo information is a lower bound for the achievable rate, and is attained for pure state ensembles, or in the related scenario of entanglement-assisted visible compression of mixed states wherein Alice knows what state is drawn. In this paper, we prove a general and robust lower bound on the achievable rate for ensembles of classical states, which holds even in the least demanding setting when Alice and Bob share free entanglement and a constant per-copy error is allowed. We apply the bound to a specific ensemble of only two states and prove a near-maximal separation (saturating the dimension bound in leading order) between the best achievable rate and the Holevo information for constant error. This also implies that the ensemble is incompressible -- compression does not reduce the communication cost by much. Since the states are classical, the observed incompressibility is not fundamentally quantum mechanical. We lower bound the difference between the achievable rate and the Holevo information in terms of quantitative limitations to clone the specimen or to distinguish the two classical states.