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.
A quantum message is encoded into
N storage nodes (quantum systems
Q1…QN) with assistance from
NB maximally entangled bi-partite quantum systems
A1B1,…,ANBBNB, that are prepared in advance such that
B1…BNB are stored separately as entanglement assistance (EA) nodes, while
A1…ANB 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
KB of the
NB EA nodes. The capacity for this setting is the maximum size of the quantum message, given that the size of each EA node is
λB. 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,NB,KB,λB in all cases, with one exception. The capacity remains open for an intermediate range of
λB values when a strict majority of the
N storage nodes, and a strict non-zero minority of the
NB 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.
Distributed quantum information stored over nodes within a quantum network are prone to node failures, motivating the need for coded quantum networks. We encode
k qubits of quantum information over an
n-node network in a distributed fashion, with one qubit per node, using modified graph-state codes. We devise a procedure to recover the encoded quantum information in any given arbitrary network after a detected node fails. The recovery procedure is locally within four edges of the failed node. For a network with even number of nodes, we provide a class of scalable modified graph-state codes with rate $(n\ensuremath{-}2)/n$, saturating the quantum Singleton bound. This yields higher coding rates than Grassl et al.'s well-known erasure codes and has asymptotic rate 1. For odd node size
n, we design the network links to obtain an $(n\ensuremath{-}3)/n$ rate-optimal modified graph state code.
In distributed quantum storage, physical qubits of a code will be stored across the network. When qubits in one of the nodes are lost i.e. when the node is erased, the remaining nodes need to communicate with a new node to replace the lost qubits. Here, we look at the problem of how much entanglement cost is needed to perform such a distributed quantum erasure correction. We focus on distributed quantum storage based on quantum maximum distance separable (MDS) codes. We derive lower bounds on the entanglement cost when the quantum network used for the erasure correction has a star topology. We show that the simple method of downloading the non-erased qudits and performing operations at a single node is optimal when the minimal number of non-erased nodes are accessed. It remains to be seen what the entanglement cost will be when a non-minimal number of non-erased nodes are accessed. The techniques used in this work can be developed further to study the entanglement cost of quantum erasure correction in more general code families and network topologies.
This paper proposes a novel approach to distributed storage leveraging the unique properties of quantum entanglement. The core claim is that utilizing entanglement enables the construction of a distributed storage system capable of achieving long-range communication, high storage capacity, and enhanced security. The proposed mechanism involves creating entanglement between multiple quantum storage nodes, facilitating redundant information storage and inherent error correction. This approach fundamentally differs from traditional distributed storage methods relying on classical redundancy techniques. We explore the theoretical framework, outlining the key mathematical concepts and demonstrating the potential advantages of this entanglement-based system. The system's security stems from the fundamental laws of quantum mechanics, making it inherently resistant to eavesdropping and data manipulation. While no experimental validation is presented in this theoretical exploration, the presented model lays a foundation for future research and development in this promising area. The system's capacity is fundamentally limited by the number of entangled pairs, but the inherent error correction provided by entanglement significantly improves reliability. We investigate the impact of various factors on the system's performance, including entanglement fidelity and node distribution.
This paper proposes a novel approach to distributed storage leveraging the unique properties of quantum entanglement. Traditional distributed storage systems face limitations in capacity and security. This research introduces a system designed to overcome these limitations by harnessing the non-local nature of entanglement. The core concept involves encoding data into entangled quantum states distributed across multiple nodes. This enables secure data sharing, redundancy, and inherent error correction, potentially achieving significantly higher storage capacity and improved security compared to classical methods. The system utilizes the correlated nature of entangled particles to create a robust and inherently secure storage solution, addressing critical challenges in modern data management. The theoretical framework presented outlines the key principles and potential benefits of this entanglement-driven distributed storage architecture. Further research and development are needed to realize the full potential of this technology.
We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.