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

Abstract

The Σ-QMAC problem is introduced, involving S servers, K classical (F d ) data streams, and T independent quantum systems. Data stream W k , k ∈ [K] is replicated at a subset of servers W(k) ⊂ [S], and quantum system Q t , t ∈ [T ] is distributed among a subset of servers E(t) ⊂ [S] such that Server s ∈ E(t) receives subsystem Q t,s of Q t = (Q t,s ) s∈E(t) . Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is t∈[T ] s∈E(t) log d |Q t,s | qudits, where |Q| is the dimension of Q. The states and measurements of (Q t ) t∈[T ] are required to be separable across t ∈ [T ] throughout, but for each t ∈ [T ], the subsystems of Q t can 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 (F d symbols) 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, E. For example, in the symmetric setting with K = S α data-streams, each replicated among a distinct α-subset of [S], and T = S β quantum systems, each distributed among a distinct β-subset of [S], the capacity of the Σ-QMAC is * Presented in part at IEEE GLOBECOM 2023. Connection to Simultaneous Message Passing (SMP):The underlying quantum multiple access (QMAC) communication model in the Σ-QMAC is similar to what is known in the literature as simultaneous message passing (SMP) model with quantum messages [21,24,47] . A noteworthy distinction is that the SMP model is typically studied from a communication complexity perspective which does not allow batch processing, whereas since our perspective is information theoretic, batch processing is not only allowed, it is essential to our problem formulation. For brevity, and to underscore the information theoretic perspective, we say QMAC when we mean an SMP model with quantum messages and batch processing. Connection to Quantum Metrology: The Σ-QMAC is conceptually related to various physically motivated and commonly studied models in the active area of distributed quantum sensing and quantum metrology. A general theme in this area is how the entanglement across quantum sensors allows higher precision (approaching the Heisenberg limit) in the computation of a function of distributed classical parameters, than what is possible without entanglement (the standard Quantum limit) [48]. For example, note the similarity of Fig. 1 and the quantum metrology protocol illustrated in [49]. In the quantum metrology protocol, entangled quantum systems are distributed to sensors (which take the role of servers in Fig. 1), classical parameters are encoded into them through quantum transducers, and the quantum systems are sent to a central receiver where the desired function is estimated by a joint measurement....