[Paper Review] Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting

Sayantan Chakraborty, Aditya Nema+1 more|arXiv (Cornell University)|Feb 2, 2021

Quantum Information and Cryptography23 references4 citations

TL;DR

This paper presents the first non-trivial one-shot inner bounds for unassisted fully quantum multiple access and interference channels using rate splitting and successive cancellation techniques. It extends classical rate splitting to the quantum one-shot setting, achieving novel rate regions that approach known asymptotic limits and providing new results for limited entanglement assistance in both settings.

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.

Motivation & Objective

• To establish the first non-trivial one-shot inner bounds for unassisted two-sender quantum multiple access channels (QMAC) and quantum interference channels (QIC).
• To extend the classical rate splitting technique to the quantum one-shot setting, avoiding reliance on time sharing.
• To derive achievable rate regions for quantum information transmission in both one-shot and asymptotic i.i.d. settings, including cases with limited entanglement assistance.
• To generalize prior work by Sen (2018a) on classical information to quantum information in unassisted settings.
• To provide novel inner bounds for QIC in the asymptotic i.i.d. regime, which were previously unknown.

Proposed method

• Employ rate splitting in the quantum one-shot setting to decompose sender systems into parts for successive decoding.
• Use successive cancellation decoding strategies where one sender's message is decoded before the other, leveraging classical decoding order principles.
• Define achievable rate tuples via regularized quantum information measures such as coherent information and conditional entropy.
• Introduce control states |σₖ⟩ involving entangled pure states |Ω⟩ and |Δ⟩ to model shared resources and splitting operations.
• Apply isometries Uθ that split the system A"ᵏ into A"ᵏ₀ and A"ᵏ₁ to model rate splitting, with θ ∈ [0,1] controlling the split.
• Construct two sets of achievable rates, 𝒜θᵏ and 𝒷θᵏ, corresponding to splitting on Alice’s or Bob’s system, and take their union to form the full rate region.

Experimental results

Research questions

• RQ1 Can rate splitting be extended from classical asymptotic i.i.d. channels to the quantum one-shot setting for fully quantum channels?
• RQ2 What are the achievable one-shot rate regions for unassisted quantum multiple access and interference channels?
• RQ3 How do the one-shot inner bounds compare to known asymptotic i.i.d. results, particularly for QIC?
• RQ4 What is the impact of limited entanglement assistance on one-shot and asymptotic rate regions for QMAC and QIC?
• RQ5 Can successive cancellation decoding strategies be adapted to the quantum one-shot setting without relying on time sharing?

Key findings

• The one-shot inner bound for QMAC asymptotically approaches the rate region of Yard et al. (2005) in the i.i.d. limit.
• For QIC, the paper derives novel non-trivial inner bounds in the asymptotic i.i.d. setting, which were previously unknown.
• The one-shot inner bounds for both QMAC and QIC are valid under limited entanglement assistance, with new results even in the asymptotic i.i.d. case for QIC.
• The achievable rate regions are expressed through regularized quantum information quantities such as H(A"ᵏ)σₖ and I(A"ᵏ > BCᵏ)𝒰𝒩·σₖ.
• The rate splitting construction via isometries Uθ on either Alice’s or Bob’s system yields two distinct rate sets, whose union forms the complete achievable region.
• The method avoids time sharing and instead uses structured rate splitting, enabling a direct extension of classical techniques to the quantum one-shot domain.