General Mixed State Quantum Data Compression with and without Entanglement Assistance
Zahra Baghali Khanian, Andreas Winter
TL;DR
The paper addresses the general problem of compressing a finite-dimensional quantum source $\rho^{AR}$ while preserving correlations with a reference $R$, unifying Schumacher's pure-state and Koashi-Imoto (KI) mixed-state models. The authors exploit the KI decomposition to obtain a canonical KI form $\omega^{C N Q R}$, which identifies a classical part $C$, a quantum part $Q$, and a redundant part $N$ whose entropic quantities govern the limit. They derive the optimal qubit-ebit rate region for both unassisted and entanglement-assisted compression: $Q \ge S(CQ)_{\omega} - \tfrac{1}{2} S(C)_{\omega}$ with $Q+E \ge S(CQ)_{\omega}$, and characterize the corner-point achievability via Schumacher compression and dense coding, respectively. The converse uses a decoupling-based framework with auxiliary quantities $J_{\epsilon}(\omega)$ and $Z_{\epsilon}(\omega)$, proving tight asymptotic bounds and connecting to Koashi-Imoto for classical-quantum sources, while highlighting a possible gap between preserving correlations with mixed versus purifying references and outlining open questions about strong converses and source reductions.
Abstract
We consider the most general (finite-dimensional) quantum mechanical information source, which is given by a quantum system $A$ that is correlated with a reference system $R$. The task is to compress $A$ in such a way as to reproduce the joint source state $ρ^{AR}$ at the decoder with asymptotically high fidelity. This includes Schumacher's original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubit-ebit rate pairs.