To compute a summation$(\Sigma)$of independent classical data streams from multiple servers over a quantum multiple access channel (QMAC), every server must directly send its quantum system containing multiple qudits to the user. Previous work$\Sigma$-QMAC in [1] considers that the servers are entangled. In this work, the maximum rate, defined as the maximum number of dits of the sum computed per qudit transmitted, is derived, and a quantum coding scheme is designed to achieve it. In practice, a server is not always directly connected to the user, necessitating a relay between them for connectivity. Motivated by this, we consider a two-layer quantum MAC (2QMAC) model, where a user is connected to a layer of mirrors, and one of the mirrors (acting as a relay) is connected to a layer of servers. Quantum systems within the same layer are entangled, whereas quantum systems across different layers are independent. In$\Sigma-2\text{QMAC}$, deriving the maximum rates of two layers simultaneously is challenging since the rates of the two layers affect each other. To address this, we first derive the maximum rate of the first layer by neglecting the second layer. Second, to relate the rates of the two layers, we derive the maximum rate of the second layer for a given feasible rate (less than the maximum rate) of the first layer. In addition, a joint coding scheme based on the$N$-sum box abstraction is designed to achieve the maximum rate of the second layer.
To compute a summation(Σ)of independent classical data streams from multiple servers over a quantum multiple access channel (QMAC), every server must directly send its quantum system containing multiple qudits to the user. Previous workΣ-QMAC in [1] considers that the servers are entangled. In this work, the maximum rate, defined as the maximum number of dits of the sum computed per qudit transmitted, is derived, and a quantum coding scheme is designed to achieve it. In practice, a server is not always directly connected to the user, necessitating a relay between them for connectivity. Motivated by this, we consider a two-layer quantum MAC (2QMAC) model, where a user is connected to a layer of mirrors, and one of the mirrors (acting as a relay) is connected to a layer of servers. Quantum systems within the same layer are entangled, whereas quantum systems across different layers are independent. InΣ−2QMAC, deriving the maximum rates of two layers simultaneously is challenging since the rates of the two layers affect each other. To address this, we first derive the maximum rate of the first layer by neglecting the second layer. Second, to relate the rates of the two layers, we derive the maximum rate of the second layer for a given feasible rate (less than the maximum rate) of the first layer. In addition, a joint coding scheme based on theN-sum box abstraction is designed to achieve the maximum rate of the second layer.
The problem of entanglement-assisted summation over a quantum multiple access channel (Σ-QMAC) is intro-duced, involvingSservers,Kclassical(Fd)data streams that are replicated arbitrarily across various subsets of servers, and a receiver who wishes to compute the sum of theKdata streams. Independent of the data, entangled quantum systemsQ1,Q2,⋯,QSare prepared in advance and distributed to the corresponding servers. Each servers,s∈[S]locally manipulates its quantum systemQsaccording to its classical data and sendsQsto the receiver. The total communication cost islogd∣Q1∣+logd∣Q2∣+⋯+logd∣QS∣qudits, where∣Qsdenotes the dimension ofQs. Based on a measurement of the composite systemQ1Q2⋯QS, the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits(Fdsymbols) of the desired sum computed by the receiver per qudit (d-dimsional quantum system) of download. The capacityCis the supremum of the set of all achievable rates. As the main result of this work, the precise capacity ofΣ-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,Fdand any data replication pattern) provided that the entanglement-assisted capacity does not exceed 1 dit/qudit (Holevo bound). Coding schemes based on a recentN-sum box abstraction are sufficient to achieve capacity.
The Σ-QMAC problem is introduced, involvingSservers,Kclassical (Fd) data streams, andTindependent quantum systems. Data stream Wk,k∈ [K] is replicated at a subset of servers W(k) ⊂ [S], and quantum system Qt,t∈ [T] is distributed among a subset of servers ε(t) ⊂ [S] such that Servers∈ ε(t) receives subsystem Qt,sof Qt= (Qt,s)s∈ε(t). Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is Σt∈[T]Σs∈ε(t)logd|Qt,s| qudits, where |Q| is the dimension of Q. The states and measurements of (Qt)t∈[T]are required to be separable acrosst∈ [T] throughout, but for eacht∈ [T], thesubsystemsof Qtcan be prepared initially in an arbitrary (independent of data) entangled state, manipulated arbitrarily by the respective servers, and measured jointly by the receiver. From the measurements, the receiver must recover the sum of all data streams. Rate is defined as the number of dits (Fdsymbols) of the desired sum computed per qudit of download. The capacity of Σ-QMAC, i.e., the supremum of achievable rates is characterized for arbitrary data and entanglement distributions W, ε. For example, in the symmetric setting withK= (Sα) data-streams, each replicated among a distinct α-subset of [S], andT= (Sβ) quantum systems, each distributed among a distinct β-subset of [S], the capacity of the Σ-QMAC is 1/βTΣmin(α,β)γ=(α+β−S)+ min(β, 2γ) · (αγ) · (S−αβ−γ). Coding based on the N-sum box abstraction is optimal in every case. Notably, for everyS≠ 3 there exists an instance of the Σ-QMAC whereS-party entanglement is necessary to achieve the fully entangled capacity.
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.
Network function computation is an active topic in network coding, with much recent progress for linear (over a finite field) computations over broadcast (LCBC) and multiple access (LCMAC) channels. Over a quantum multiple access channel (QMAC) with quantum-entanglement shared among transmitters, the linear computation problem (LC-QMAC) is non-trivial even when the channel is noiseless, because of the challenge of optimally exploiting transmit-side entanglement through distributed coding. Given an arbitrary linear function of data streams defined in a finite fieldFd, the LC-QMAC problem seeks the optimal communication cost (minimum number of qudits that need to be sent by the transmitters to the receiver, per computation instance) over a noise-free QMAC, when the independent input data streams originate at the corresponding transmitters, who share quantum entanglement in advance. As our main result, we fully solve this problem forK=3transmitters (K≥4settings remain open). Coding schemes based on theN-sum box protocol (along with time-sharing and batch-processing) are shown to be information theoretically optimal in all cases.
The quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC).K- messages are distributed acrossSservers so that each server knows a subset of the messages. Each servers∈[S]sends a quantum subsystemQsto the receiver who computes the sum of the messages. The download cost from Servers∈[S]is the logarithm of the dimension ofQs. The rateR: is defined as the number of instances of the sum computed at the receiver, divided by the total download cost from all the servers. In the symmetric setting withK=(αS)messages where each message is replicated among a unique subset ofαservers, and the answers from anyβservers may be erased, the rate achieved isR=max{min{S2(α−β),1−S2β},Sα−β}.
A multiple access channel (MAC) consists of multiple senders simultaneously transmitting their messages to a single receiver. For the classical-quantum case (CQ MAC), achievable rates are known assuming that all the messages are decoded, a common assumption in quantum network design. However, such a conventional design approach ignores the global network structure, i.e., the network topology. When a CQ MAC is given as a part of quantum network communication, this work shows that computation properties can be used to boost communication speeds with code design dependent on the network topology. We quantify achievable quantum communication rates of codes with the computation property for a two-sender CQ MAC. When the two-sender CQ MAC is a boson coherent channel with binary discrete modulation, we show that it achieves the maximum possible communication rate (the single-user capacity), which cannot be achieved with conventional design. Further, such a rate can be achieved by different detection methods: quantum (with and without quantum memory), on-off photon counting, and homodyne (each at different photon power). Finally, we describe two practical applications, one of which cryptographic.