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Distilling common randomness from bipartite quantum states

I. Devetak, A. Winter

TL;DR

This work addresses distilling noiseless classical common randomness from bipartite quantum states under a constraint on one-way classical communication. It derives a single-letter optimization C(R)=C^*(R)=R+D^*(R) for classical-quantum correlations, with D^*(R)=\sup_{U|X}\{ I(U;{\cal Q}) \;|\; I(U;X)−I(U;{\cal Q}) \leq R \}, and shows a regularized form for general quantum states under a tensor-product measurement restriction. The results connect the distillable common randomness to the Hayden–Jozsa–Winter compression trade-off Q^*(R) through the relation D^*(R)+Q^*(D^*(R)+R)=H({\cal Q}), providing an operational interpretation of total correlations in quantum states. For separable states, the distillable common randomness reduces to the accessible information I_{acc}(E), while for pure states it equals the entropy of entanglement, highlighting a spectrum of additivity properties. The framework lays groundwork for further exploration of multi-copy measurements, two-way communication, and broader resource inter-conversions in quantum information theory.

Abstract

The problem of converting noisy quantum correlations between two parties into noiseless classical ones using a limited amount of one-way classical communication is addressed. A single-letter formula for the optimal trade-off between the extracted common randomness and classical communication rate is obtained for the special case of classical-quantum correlations. The resulting curve is intimately related to the quantum compression with classical side information trade-off curve $Q^*(R)$ of Hayden, Jozsa and Winter. For a general initial state we obtain a similar result, with a single-letter formula, when we impose a tensor product restriction on the measurements performed by the sender; without this restriction the trade-off is given by the regularization of this function. Of particular interest is a quantity we call ``distillable common randomness'' of a state: the maximum overhead of the common randomness over the one-way classical communication if the latter is unbounded. It is an operational measure of (total) correlation in a quantum state. For classical-quantum correlations it is given by the Holevo mutual information of its associated ensemble, for pure states it is the entropy of entanglement. In general, it is given by an optimization problem over measurements and regularization; for the case of separable states we show that this can be single-letterized.

Distilling common randomness from bipartite quantum states

TL;DR

This work addresses distilling noiseless classical common randomness from bipartite quantum states under a constraint on one-way classical communication. It derives a single-letter optimization C(R)=C^*(R)=R+D^*(R) for classical-quantum correlations, with D^*(R)=\sup_{U|X}\{ I(U;{\cal Q}) \;|\; I(U;X)−I(U;{\cal Q}) \leq R \}, and shows a regularized form for general quantum states under a tensor-product measurement restriction. The results connect the distillable common randomness to the Hayden–Jozsa–Winter compression trade-off Q^*(R) through the relation D^*(R)+Q^*(D^*(R)+R)=H({\cal Q}), providing an operational interpretation of total correlations in quantum states. For separable states, the distillable common randomness reduces to the accessible information I_{acc}(E), while for pure states it equals the entropy of entanglement, highlighting a spectrum of additivity properties. The framework lays groundwork for further exploration of multi-copy measurements, two-way communication, and broader resource inter-conversions in quantum information theory.

Abstract

The problem of converting noisy quantum correlations between two parties into noiseless classical ones using a limited amount of one-way classical communication is addressed. A single-letter formula for the optimal trade-off between the extracted common randomness and classical communication rate is obtained for the special case of classical-quantum correlations. The resulting curve is intimately related to the quantum compression with classical side information trade-off curve of Hayden, Jozsa and Winter. For a general initial state we obtain a similar result, with a single-letter formula, when we impose a tensor product restriction on the measurements performed by the sender; without this restriction the trade-off is given by the regularization of this function. Of particular interest is a quantity we call ``distillable common randomness'' of a state: the maximum overhead of the common randomness over the one-way classical communication if the latter is unbounded. It is an operational measure of (total) correlation in a quantum state. For classical-quantum correlations it is given by the Holevo mutual information of its associated ensemble, for pure states it is the entropy of entanglement. In general, it is given by an optimization problem over measurements and regularization; for the case of separable states we show that this can be single-letterized.

Paper Structure

This paper contains 5 sections, 9 theorems, 107 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Classical-quantum correlations
  3. General quantum correlations
  4. Discussion
  5. Appendix

Key Result

Theorem 1

where The supremum is to be understood as one over all conditional probability distributions $p(u|x)$ for the random variable $U$ conditioned on $X$, with finite range ${\cal U}$. We may in fact restrict to the case $|{\cal U}| \leq |{\cal X}| + 1$, which in particular implies that the $\sup$ is actually

Figures (4)

  • Figure 1: $D(R)$ for the two-state ensemble ${\cal E}$.
  • Figure 2: $D(R)$ for the three-state ensemble ${\cal E}_3$.
  • Figure 3: $D(R)$ for the parametrized BB84 ensemble with $\theta = {\pi \over 8}$.
  • Figure 4: $D(R)$ for the uniform ensemble.

Theorems & Definitions (11)

  • Theorem 1: CR-rate theorem for classical-quantum correlations
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Corollary 7
  • Example 8
  • Theorem 9
  • Example 10
  • ...and 1 more