On the distributed compression of quantum information

TL;DR

The paper investigates distributed compression of correlated quantum sources and contrasts quantum behavior with the classical Slepian–Wolf result, showing that correlations do not universally reduce total rates for irreducible product-state sources.It establishes a tight lower bound on the sum rate for irreducible product-state ensembles and presents an optimal, fully exploiting scheme for Bell-state sources, highlighting fundamental differences between classical and quantum correlations in distributed compression.The work connects distributed compression to information-disturbance, entanglement distillation, and quantum error correction, and explores intermediate regimes via additional examples and hybrid strategies.Overall, the paper maps out a nuanced landscape of achievable rate regions in distributed quantum compression and points to open problems, including a unified theory for general ensembles and the implications of different informational models.

Abstract

We consider the problem of distributed compression for correlated quantum sources. The classical version of this problem was solved by Slepian and Wolf, who showed that distributed compression could take full advantage of redundancy in the local sources created by the presence of correlations. We show that, in general, this is not the case for quantum sources by proving a lower bound on the rate sum for irreducible sources of product states which is stronger than the one given by a naive application of Slepian-Wolf. Nonetheless, strategies taking advantage of correlation do exist for some special classes of quantum sources. For example, Devetak and Winter demonstrated the existence of such a strategy when one of the sources is classical. Here we find optimal non-trivial strategies for a different extreme, sources of Bell states. In addition, we illustrate how distributed compression is connected to other problems in quantum information theory, including information-disturbance questions, entanglement distillation and quantum error correction.

Figures (4)

• Figure 1: Achievable rate region for Slepian-Wolf encoding.
• Figure 2: Cloning as distributed compression. Solid lines represent noiseless quantum channels and dashed lines correlation in the ensemble ${\cal E}_{AB}$. The encoders are each given a copy of $|\varphi\rangle$ while the decoder tries to produce the state $|\varphi\rangle|\varphi\rangle$.
• Figure 3: Measurement without disturbance as distributed compression. This time the encoders are given the states $|\varphi_i\rangle$ and $|\psi_i\rangle$, and the decoder attempts to produce $|\varphi_i\rangle|\psi_i\rangle$.
• Figure 4: Hidden orthogonality: (a) depicts the ensemble states $\{|\varphi_i\rangle|\psi_i\rangle\}$ while (b) shows the ensemble after Bob has performed the first half of his compression operation.

Theorems & Definitions (8)

• Theorem 2.1: Slepian-Wolf SlepianW73. See also CoverT, p. 407
• Lemma 3.1: Barnum et al. BarnumHJW01, Lemma 6.1
• Proposition 3.2: Barnum et al. BarnumHJW01, Lemma 6.4
• Theorem 3.3
• Theorem 4.1
• Theorem 5.1: Holevo-Schumacher-Westmoreland Holevo98bSchumacherW97
• Theorem 5.2
• Lemma 5.3