On the Capacity of Erasure-prone Quantum Storage with Erasure-prone Entanglement Assistance

TL;DR

The paper studies the capacity of erasure-prone quantum storage with erasure-prone entanglement assistance (QS̃EÃ) by linking it to an analogous classical problem with shared randomness (CS̃SRÃ). It derives exact capacity regions in three regimes and shows that quantum and classical capacities coincide where settled, leaving one open regime for which a conjectured inner bound is provided. A key methodological step is translating optimal classical linear SR-assisted codes to quantum CSS codes, yielding a unified framework (Theorem ab) that explains the capacity parity. The work advances the theory of quantum storage under practical erasure models and illuminates the deep connections between quantum and classical erasure coding through CSS constructions and linear coding schemes.

Abstract

A quantum message is encoded into $N$ storage nodes (quantum systems $Q_1\dots Q_N$) with assistance from $N_B$ maximally entangled bi-partite quantum systems $A_1B_1, \dots, A_{N_B}B_{N_B}$, that are prepared in advance such that $B_1\dots B_{N_B}$ are stored separately as entanglement assistance (EA) nodes, while $A_1\dots A_{N_B}$ are made available to the encoder. Both the storage nodes and EA nodes are erasure-prone. The quantum message must be recoverable given any $K$ of the $N$ storage nodes along with any $K_B$ of the $N_B$ EA nodes. The capacity for this setting is the maximum size of the quantum message, given that the size of each EA node is $λ_B$. All node sizes are relative to the size of a storage node, which is normalized to unity. The exact capacity is characterized as a function of $N,K,N_B,K_B, λ_B$ in all cases, with one exception. The capacity remains open for an intermediate range of $λ_B$ values when a strict majority of the $N$ storage nodes, and a strict non-zero minority of the $N_B$ EA nodes, are erased. As a key stepping stone, an analogous classical storage (with shared-randomness assistance) problem is introduced. A set of constraints is identified for the classical problem, such that classical linear code constructions translate to quantum storage codes, and the converse bounds for the two settings utilize similar insights. In particular, the capacity characterizations for the classical and quantum settings are shown to be identical in all cases where the capacity is settled.

Figures (4)

• Figure 1: A $\widetilde{\hbox{QS}}\hbox{EA}$ setting studied in Grassl_Huber_Winter is illustrated. A quantum message $Q_0$ (possibly entangled with a purifying reference system $R$) is encoded into $N$ quantum storage systems $(Q_1\dots Q_N)$ by an encoder with entanglement assistance $A$. Any $N-K$ of these $N$ systems are erased by a channel. A decoder must recover the quantum information from the remaining (unerased) systems and the entanglement assistance $B$. Note that the entanglement assistance is perfect (not prone to erasures). A feasible coding scheme specifies an encoder ENC, and an array of decoders $\hbox{DEC}^{\mathcal{K}}$, guaranteeing recovery for every $\mathcal{K}\in\binom{[N]}{K}$.
• Figure 2: The $\widetilde{\hbox{QS}}\widetilde{\hbox{EA}}$ setting studied in this work. Note that the entanglement assistance systems $B$ are also subject to erasures.
• Figure 3: Constructing a $\widetilde{\hbox{QS}}\widetilde{\hbox{EA}}$ code from a $\widetilde{\hbox{CS}}\widetilde{\hbox{SRA}}$ code for $(N,K,N_B,K_B)=(3,1,3,2)$ and $(\lambda_0,\lambda_B)=(1/2,5/6)$. First, quantum systems are mapped to classical systems. In addition to the decodability and security constraints identified in prior works Sun_Jafar_QuStorageSmithHayashi_Song, here the constraints for the classical setting also include (shown in blue) recovery of the shared-randomness from the storage systems. Next, a classical code is constructed by optimally aligning shared-randomness, local randomness (not needed in this example) and message terms to satisfy all constraints. Finally, the classical code is mapped to a quantum code via the CSS construction Calderbank_ShorSteane.
• Figure 4: Left: Capacity region for Case $2$, i.e., when $K_B/N_B>1/2,K/N\geq 1/2$. Note that the upper boundary of the capacity region represents the capacity $C_Q(\lambda_B)=C(\lambda_B)$. Right: Inner and outer bounds for capacity region in Case $3$, i.e., when $K/N < 1/2, K_B/N_B > 1/2$. Note that the bounds match if $\lambda_B\leq K/(2K_B)$ or if $\lambda_B\geq (N-2K)/N_B+K/K_B$, providing an exact capacity characterization in both cases.

Theorems & Definitions (8)

• Theorem 1
• definition 1: Linear Scheme
• Theorem 2
• Remark 1
• Remark 2
• Lemma 1
• Lemma 2
• Lemma 3