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Decoupling by local random unitaries without simultaneous smoothing, and applications to multi-user quantum information tasks

Pau Colomer, Andreas Winter

Abstract

We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old [Dupuis et al. Commun. Math. Phys. 328:251-284 (2014)] and new methods [Dupuis, arXiv:2105.05342], we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of Rényi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, that is for only partially known i.i.d. source or channel, which are furthermore optimal for entanglement of assistance and state merging.

Decoupling by local random unitaries without simultaneous smoothing, and applications to multi-user quantum information tasks

Abstract

We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old [Dupuis et al. Commun. Math. Phys. 328:251-284 (2014)] and new methods [Dupuis, arXiv:2105.05342], we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of Rényi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, that is for only partially known i.i.d. source or channel, which are furthermore optimal for entanglement of assistance and state merging.
Paper Structure (13 sections, 34 theorems, 156 equations, 6 figures)

This paper contains 13 sections, 34 theorems, 156 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setting and main results
  4. Lemmata
  5. Technical ingredients for the proofs
  6. Central lemmas
  7. Proving the multi-user decoupling theorems
  8. Applications
  9. Local randomness extraction
  10. Multi-party entanglement of assistance and assisted distillation
  11. Multi-party quantum Slepian-Wolf coding: state merging
  12. Quantum communication via quantum multiple access channels
  13. Discussion

Key Result

Theorem 3.2

Assume $\mathcal{T}_{A_{[k]} \rightarrow B}$ to be a CPTP map with Choi state, and consider the random channels $\mathcal{R}^{U_{[k]}}$ as above. Then, for any state $\rho_{A_{[k]}E}$, where $D_I = 2^{\abs{I}-1}\prod_{i\in I} \left(1-\frac{1}{\abs{A_i}^2}\right)^{-\frac{1}{2}}$, $\tau_B = \mathcal{T}\left({\openone_{A_{[k]}}}/{|A_{[k]}|}\right)$, the $\zeta_E^I$ are arbitrary states on $E$, $\sig

Figures (6)

  • Figure 1: generalised multipartite decoupling via local random unitary transformations $\mathcal{U}_i$ acting locally on each system $A_i$, followed by a fixed CPTP map $\mathcal{T}_{A_1\dots A_k\rightarrow B}$.
  • Figure 2: Multi-party decoupling via local random unitary transformations $\mathcal{U}_i$ followed by a fixed local CPTP map $\mathcal{T}_{A_i\rightarrow B_i}$ on each of the systems $A_i$.
  • Figure 3: Diagram of the LOCC protocol that maximally concentrates the entanglement of an initial state $\psi_{ABC_1\dots C_m}$ onto Alice's and Bob's subspaces of dimension $|A'|=|B'|=d$.
  • Figure 4: One-shot achievable rate region of a two-senders quantum Slepian-Wolf coding. Notice that the region is open towards the northeast.
  • Figure 5: One-shot achievable rate region for a MAC with three senders $A_1$, $A_2$ and $A_3$.
  • ...and 1 more figures

Theorems & Definitions (53)

  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 43 more