Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.17781 · IEEE Transactions on Information Theory

On the Capacity of Erasure-Prone Quantum Storage With Erasure-Prone Entanglement Assistance

Hua Yan Sun, Syed A. Jafar

Published Aug 18, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.1109/tit.2026.3724917

Related papers

Entanglement-assisted quantum locally recoverable codes: bounds and constructions with availability

Rutuja Kshirsagar, Gretchen L. Matthews, Julia Shapiro

In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to $δ-1$ erased qudits can be recovered from any one of $t$ local recovery sets, each of size at most $r+δ-1$, with the recovery sets intersecting exactly in the erased coordinates, where $r$ is a (small) positive integer. We show that shared entanglement permits $t>1$, meaning that multiple local recovery sets can be available for the same set of up to $δ-1$ erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.

cs.ITCross-listedarXiv:2608.09886•Aug 11, 2026•0 citations

Capacity of Summation Over a Symmetric Quantum Erasure MAC With Partially Replicated Inputs

Yuhang Yao, Syed A. Jafar

The optimal quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC).Kclassical messages comprised of finite-field symbols are distributed acrossSservers, who also share quantum entanglement in advance. Each servers∈ [S] manipulates its quantum subsystemQsaccording to its own available classical messages and sendsQsto the receiver who then computes the sum of the messages based on a joint quantum measurement. The download cost from Servers∈ [S] is the logarithm of the dimension ofQs. The rateRis defined as the number of instances of the sum computed at the receiver, divided by the total download cost from all the servers. The main focus is on the symmetric setting withK= (Sα) messages where each message is replicated among a unique subset of α servers, and the answers from any β servers may be erased. If no entanglement is initially available to the receiver, then we show that the capacity (maximal rate) is preciselyC= max { min { 2(α−β)/S,S−2β/S}, α−β/S}. The capacity with arbitrary levels of prior entanglement (Δ0) between theSdata-servers and the receiver is also characterized, by including an auxiliary server (Server 0) that has no classical data, so that the communication cost from Server 0 is a proxy for the amount of receiver-side entanglement that is available in advance. The challenge on the converse side resides in the optimal application of the weak monotonicity property, while the achievability combines ideas from classical network coding and treating qudits as classical dits, as well as new constructions based on the N-sum box abstraction that rely on absolutely maximally entangled quantum states.

arXiv:2311.08386•IEEE Transactions on Information Theory•May 14, 2025•1 citation

Space-sharing and Singleton Bounds for Entanglement-assisted Classical Coding

Yuhang Yao, Tushita Prasad, Markus Grassl +2 more

Recent work has noted that a space-sharing argument proves the tightness of the entropic quantum Singleton bounds, which was left open in the literature for various settings involving only-quantum messages, only-classical messages, or both classical and quantum messages. Focusing on the setting of entanglement-assisted classical coding (EACC), in this letter we first elaborate upon the space-sharing argument and the tight Singleton bound for this setting, and then establish a new tight entropic Singleton bound for EACC codes with entanglement assistance distributed across a subset of encoders when only local quantum operations are allowed at each encoder.

quant-pharXiv:2603.08563•Mar 10, 2026•0 citations

Entanglement-Assisted Quantum Locally Recoverable Codes: Characterizations, Bounds, and Constructions

Yang Li, San Ling, Zhenliang Lu +2 more

Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality $r$ and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to $\ell$-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.

arXiv:2607.27091•Jul 29, 2026•0 citations

The Capacity of Classical Summation over a Quantum MAC with Arbitrarily Replicated Inputs

Yuhang Yao, Syed A. Jafar

The problem of entanglement-assisted summation over a quantum multiple access channel ($\Sigma$-QMAC) is intro-duced, involving$S$servers,$K$classical$(\mathbb{F}_{d})$data streams that are replicated arbitrarily across various subsets of servers, and a receiver who wishes to compute the sum of the$K$data streams. Independent of the data, entangled quantum systems$\mathcal{Q}_{1}, \mathcal{Q}_{2}, \cdots, \mathcal{Q}_{S}$are prepared in advance and distributed to the corresponding servers. Each server$s, s\in[S]$locally manipulates its quantum system$\mathcal{Q}_{s}$according to its classical data and sends$\mathcal{Q}_{s}$to the receiver. The total communication cost is$\log_{d}\vert \mathcal{Q}_{1}\vert +\log_{d}\vert \mathcal{Q}_{2}\vert +\cdots+\log_{d}\vert \mathcal{Q}_{S}\vert$qudits, where$\vert \mathcal{Q}_{s}$denotes the dimension of$\mathcal{Q}_{s}$. Based on a measurement of the composite system$\mathcal{Q}_{1}\mathcal{Q}_{2}\cdots \mathcal{Q}_{S}$, the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits$(\mathbf{F}_{d}$symbols) of the desired sum computed by the receiver per qudit (d-dimsional quantum system) of download. The capacity$C$is the supremum of the set of all achievable rates. As the main result of this work, the precise capacity of$\Sigma$-QMAC is obtained, from which it follows that quantum entanglements allow a factor of 2 gain in capacity (superdense coding gain) relative to capacity with no entanglements, in all cases (any$S, K, \mathbf{F}_{d}$and any data replication pattern) provided that the entanglement-assisted capacity does not exceed 1 dit/qudit (Holevo bound). Coding schemes based on a recent$N$-sum box abstraction are sufficient to achieve capacity.

Globecom•Dec 4, 2023•5 citations

Constraints on recovering quantum information after erasure

Mohammad A. Alhejji, Noah Lordi, Omkar Baraskar +2 more

Suppose that we encode quantum information into $ n $ physical carriers. With what probabilities can we perfectly recover the quantum information after sets of carriers have been erased? This question arises in the study of quantum erasure channels, for which M.B. Hastings conjectured that if each carrier is independently erased with probability $ t \geq 1/2 $, then it is impossible to recover the quantum information with asymptotically vanishing error and probability $ > 1-t $. We prove this conjecture by establishing new no-cloning constraints on realizable recovery probabilities.

quant-pharXiv:2607.17319•Jul 21, 2026•0 citations
A quantum message is encoded into NNN storage nodes (quantum systems Q1…QNQ_1\dots Q_NQ1​…QN​) with assistance from NBN_BNB​ maximally entangled bi-partite quantum systems A1B1,…,ANBBNBA_1B_1, \dots, A_{N_B}B_{N_B}A1​B1​,…,ANB​​BNB​​, that are prepared in advance such that B1…BNBB_1\dots B_{N_B}B1​…BNB​​ are stored separately as entanglement assistance (EA) nodes, while A1…ANBA_1\dots A_{N_B}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 KKK of the NNN storage nodes along with any KBK_BKB​ of the NBN_BNB​ EA nodes. The capacity for this setting is the maximum size of the quantum message, given that the size of each EA node is λBλ_BλB​. 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,λBN,K,N_B,K_B, λ_BN,K,NB​,KB​,λB​ in all cases, with one exception. The capacity remains open for an intermediate range of λBλ_BλB​ values when a strict majority of the NNN storage nodes, and a strict non-zero minority of the NBN_BNB​ 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.
In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to δ−1δ-1δ−1 erased qudits can be recovered from any one of ttt local recovery sets, each of size at most r+δ−1r+δ-1r+δ−1, with the recovery sets intersecting exactly in the erased coordinates, where rrr is a (small) positive integer. We show that shared entanglement permits t>1t>1t>1, meaning that multiple local recovery sets can be available for the same set of up to δ−1δ-1δ−1 erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.
The optimal quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC).Kclassical messages comprised of finite-field symbols are distributed acrossSservers, who also share quantum entanglement in advance. Each servers∈ [S] manipulates its quantum subsystemQsaccording to its own available classical messages and sendsQsto the receiver who then computes the sum of the messages based on a joint quantum measurement. The download cost from Servers∈ [S] is the logarithm of the dimension ofQs. The rateRis defined as the number of instances of the sum computed at the receiver, divided by the total download cost from all the servers. The main focus is on the symmetric setting withK= (Sα) messages where each message is replicated among a unique subset of α servers, and the answers from any β servers may be erased. If no entanglement is initially available to the receiver, then we show that the capacity (maximal rate) is preciselyC= max { min { 2(α−β)/S,S−2β/S}, α−β/S}. The capacity with arbitrary levels of prior entanglement (Δ0) between theSdata-servers and the receiver is also characterized, by including an auxiliary server (Server 0) that has no classical data, so that the communication cost from Server 0 is a proxy for the amount of receiver-side entanglement that is available in advance. The challenge on the converse side resides in the optimal application of the weak monotonicity property, while the achievability combines ideas from classical network coding and treating qudits as classical dits, as well as new constructions based on the N-sum box abstraction that rely on absolutely maximally entangled quantum states.
Recent work has noted that a space-sharing argument proves the tightness of the entropic quantum Singleton bounds, which was left open in the literature for various settings involving only-quantum messages, only-classical messages, or both classical and quantum messages. Focusing on the setting of entanglement-assisted classical coding (EACC), in this letter we first elaborate upon the space-sharing argument and the tight Singleton bound for this setting, and then establish a new tight entropic Singleton bound for EACC codes with entanglement assistance distributed across a subset of encoders when only local quantum operations are allowed at each encoder.
Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality rrr and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to ℓ\ellℓ-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.
The problem of entanglement-assisted summation over a quantum multiple access channel (Σ\SigmaΣ-QMAC) is intro-duced, involvingSSSservers,KKKclassical(Fd)(\mathbb{F}_{d})(Fd​)data streams that are replicated arbitrarily across various subsets of servers, and a receiver who wishes to compute the sum of theKKKdata streams. Independent of the data, entangled quantum systemsQ1,Q2,⋯ ,QS\mathcal{Q}_{1}, \mathcal{Q}_{2}, \cdots, \mathcal{Q}_{S}Q1​,Q2​,⋯,QS​are prepared in advance and distributed to the corresponding servers. Each servers,s∈[S]s, s\in[S]s,s∈[S]locally manipulates its quantum systemQs\mathcal{Q}_{s}Qs​according to its classical data and sendsQs\mathcal{Q}_{s}Qs​to the receiver. The total communication cost islog⁡d∣Q1∣+log⁡d∣Q2∣+⋯+log⁡d∣QS∣\log_{d}\vert \mathcal{Q}_{1}\vert +\log_{d}\vert \mathcal{Q}_{2}\vert +\cdots+\log_{d}\vert \mathcal{Q}_{S}\vertlogd​∣Q1​∣+logd​∣Q2​∣+⋯+logd​∣QS​∣qudits, where∣Qs\vert \mathcal{Q}_{s}∣Qs​denotes the dimension ofQs\mathcal{Q}_{s}Qs​. Based on a measurement of the composite systemQ1Q2⋯QS\mathcal{Q}_{1}\mathcal{Q}_{2}\cdots \mathcal{Q}_{S}Q1​Q2​⋯QS​, the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits(Fd(\mathbf{F}_{d}(Fd​symbols) of the desired sum computed by the receiver per qudit (d-dimsional quantum system) of download. The capacityCCCis the supremum of the set of all achievable rates. As the main result of this work, the precise capacity ofΣ\SigmaΣ-QMAC is obtained, from which it follows that quantum entanglements allow a factor of 2 gain in capacity (superdense coding gain) relative to capacity with no entanglements, in all cases (anyS,K,FdS, K, \mathbf{F}_{d}S,K,Fd​and any data replication pattern) provided that the entanglement-assisted capacity does not exceed 1 dit/qudit (Holevo bound). Coding schemes based on a recentNNN-sum box abstraction are sufficient to achieve capacity.
Suppose that we encode quantum information into $ n $ physical carriers. With what probabilities can we perfectly recover the quantum information after sets of carriers have been erased? This question arises in the study of quantum erasure channels, for which M.B. Hastings conjectured that if each carrier is independently erased with probability $ t \geq 1/2 $, then it is impossible to recover the quantum information with asymptotically vanishing error and probability $ > 1-t $. We prove this conjecture by establishing new no-cloning constraints on realizable recovery probabilities.