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Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting

Sayantan Chakraborty, Aditya Nema, Pranab Sen

TL;DR

It is proved the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassistedTwo-Sender two-receiver quantum interference channel ( QIC).

Abstract

We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation}, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard, Devetak and Hayden. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting

TL;DR

It is proved the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassistedTwo-Sender two-receiver quantum interference channel ( QIC).

Abstract

We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation}, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard, Devetak and Hayden. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

Paper Structure

This paper contains 29 sections, 20 theorems, 232 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Subsequent Works:
  3. Organisation of the Paper
  4. Preliminaries
  5. Notation
  6. Smooth Entropies
  7. The op Operator
  8. The Smooth Single Sender Decoupling Theorem
  9. Miscellaneous Useful Facts
  10. Quantum Channel Capacities: Definitions and Previous Work
  11. Entanglement Transmission over the QMAC
  12. Unassisted vs. Rate Limited Assistance
  13. Quantum Rate Splitting I
  14. Rate Splitting for point-to-point Channels
  15. Rate Splitting in the Classical Regime
  16. ...and 14 more sections

Key Result

Theorem 4.7

Given the QMAC $\mathcal{N}^{A'B'\to C}$ its capacity region is given by the closure of where the region $\mathcal{Q}(\mathcal{M})$ equals the pairs of non-negative rates $(R_A,R_B)$ satisfying where all the entropic quantities are computed with respect to the control state for a pair of pure states $\ket{\Omega}^{AA'}$ and $\ket{\Delta}^{BB'}$.

Figures (9)

  • Figure 1: Achievable rate region per channel use for the classical MAC in the asymptotic iid setting.
  • Figure 2: Achievable rate region for the unassisted quantum MAC per channel use in the asymptotic iid setting.
  • Figure 3: One-shot achievable rate region for the unassisted QMAC (for single channel use only), contained inside the 'ideal' pentagonal region demarcated by the dashed line, and approaching it in the asymptotic iid limit. $O(\log \varepsilon)$ additive factors have been ignored in the figure.
  • Figure 4: The 'corner' point $P'$ can be obtained by successive cancellation following the order Alice0$\to$ Bob $\to$ Alice1 with splitting of Alice followed by one use of the unassisted QMAC. Point $P'$ projects down to point $P$ in Figure \ref{['fig:QMAConeshot1Intro']}. Only the 'dominant face' of the rate region is shown. Successive cancellation can only obtain the corner points of the dominant face and all 'sub-points' by 'resource wasting'. It cannot obtain 'middle' points of the 'dominant' face. $O(\log \varepsilon)$ additive factors have been ignored in the figure.
  • Figure 5: One-shot achievable rate region (for single channel use only) for the unassisted QIC. The trivial region is shown dotted. Alice can sacrifice her rate in order to boost Bob's rate with respect to the trivial region, as shown by the solid rectangle. The dashed rectangle can be similarly obtained by Bob sacrificing his rate in order to boost Alice's. $O(\log \varepsilon)$ additive factors have been ignored in the figure.
  • ...and 4 more figures

Theorems & Definitions (50)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.8
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.5
  • Definition 4.6
  • Theorem 4.7
  • ...and 40 more