On the Capacity of Distributed Quantum Storage

TL;DR

This paper studies the fundamental capacity of distributed quantum storage with heterogeneous node sizes and prescribed erasure patterns by modeling storage as graphs with hyperedge decoding sets. It develops a Shannon-theoretic framework where the capacity C(\\mathcal{G}) is characterized via quantum information inequalities (Intersection, Wheel bounds) and an achievable scheme based on CSS codes that reduces to a classical secure storage problem; the achievability leverages nontrivial interference alignment. The authors derive exact or tight bounds for several graph families (MDS, Wheel, Fano, and intersection graphs) and provide concrete constructions (including space-sharing and aligned classical codes) to show positive capacity where possible. They discuss maximal graphs, extend results to small graphs, and illustrate the classical-to-quantum translation in depth, highlighting the key role of alignment in achieving secure, decodable storage in heterogeneous quantum networks. Finally, the work outlines open problems such as the general wheel/intersection capacities, additivity questions, and the broader implications for scalable quantum storage design.

Abstract

A distributed quantum storage code maps a quantum message to N storage nodes, of arbitrary specified sizes, such that the stored message is robust to an arbitrary specified set of erasure patterns. The sizes of the storage nodes, and erasure patterns may not be homogeneous. The capacity of distributed quantum storage is the maximum feasible size of the quantum message (relative to the sizes of the storage nodes), when the scaling of the size of the message and all storage nodes by the same scaling factor is allowed. Representing the decoding sets as hyperedges in a storage graph, the capacity is characterized for various graphs, including MDS graph, wheel graph, Fano graph, and intersection graph. The achievability is related via quantum CSS codes to a classical secure storage problem. Remarkably, our coding schemes utilize non-trivial alignment structures to ensure recovery and security in the corresponding classical secure storage problem, which leads to similarly non-trivial quantum codes. The converse is based on quantum information inequalities, e.g., strong sub-additivity and weak monotonicity of quantum entropy, tailored to the topology of the storage graphs.

Figures (6)

• Figure 1: (Left) The 'wheel graph $\mathcal{W}_4$' representing a $4$-partite quantum storage system $Q_1Q_2Q_3Q_4$, where each storage node $Q_n$ is represented by a vertex $n$, $n\in[4]$, and decoding sets $\{Q_1,Q_2\}$, $\{Q_1,Q_3\}$, $\{Q_1,Q_4\}$, $\{Q_2,Q_3,Q_4\}$ are represented by the red, blue, green, and black hyperedges, respectively. (Right) The storage capacity $C(\mathcal{W}_4)=\min(\lambda_1,\lambda_2,(\lambda_1+\lambda_2)/3)$ is shown as a function of $\max(\lambda_1,\lambda_2)$ assuming without loss of generality that $\min(\lambda_1,\lambda_2)=1$ and $\lambda_2\leq \lambda_3\leq \lambda_4$.
• Figure 2: Sketch of achievability argument $(\lambda_0=1)$ for a particular wheel graph $\mathcal{W}_4$ setting, with size constraints $(\lambda_1,\lambda_2,\lambda_3,\lambda_4)=(1,2,2,2)$. Quantum nodes are mapped to classical nodes of corresponding sizes over $\mathbb{F}_q$, decodability and security constraints (stated here in entropic terms) define the classical secure storage problem, and an $\mathbb{F}_q$-linear solution to the classical problem is mapped to a quantum distributed storage code via a CSS code construction.
• Figure 3: Erasure channel corresponding to the storage capacity problem. Given any $e\in\mathcal{E}$, the channel $\mathcal{N}^e$ erases the storage nodes $Q_i$ for all $i\in[N]\setminus e$, leaving only storage nodes $Q_j, j\in e$ for the decoder $\hbox{DEC}^e$. Note that the encoder does not depend on $e$, because the positions of erasures are not known at the time of encoding, but the decoder is chosen based on the realization of $e$ because the decoding is done with the knowledge of which storage nodes are erased.
• Figure 4: The Fano graph is shown. There are $7$ storage nodes and $7$ decoding sets. The decoding sets are the hyperedges corresponding to the $6$ straight lines and the circle. The solution for the corresponding classical secure storage problem is shown with secret '$a$' and noise '$b_1,b_2,b_3$,' all i.i.d. uniform in $\mathbb{F}_q$ where $\mathbb{F}_q$ is any finite field with an even characteristic.
• Figure 5: The intersection graph $\hbox{[}1.25]{$⊓$}_{4,2}$ is shown. Storage nodes $\{Q_{\overline{\mathcal{S}}}\}$ appear as empty circles, decoding sets $\{\mathcal{D}(e_i)\}$ as filled rectangles. Arrows from storage nodes to decoding sets indicate membership of the decoding set. Decoding set $\mathcal{D}(e_1)$ and its participating nodes are shown in blue, and the nodes in the complement $\mathcal{D}^c(e_1)$ are shown in red. Also shown is the optimal classical secure storage code over any finite field $\mathbb{F}_q$, with secret $(a\in\mathbb{F}_q)$ and noise $(b_1,b_2\in\mathbb{F}_q)$, respectively. The optimal classical code translates into an optimal quantum code via Theorem \ref{['thm:css']}.

Theorems & Definitions (17)

• Remark 1
• definition 1: Maximal Storage Graph
• Theorem 1
• Theorem 2
• Corollary 1
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6