The Capacity of Classical Summation Over a Quantum MAC With Arbitrarily Distributed Inputs and Entanglements

The Sigma-QMAC problem is introduced, involving S servers, K classical (F-d) data streams, and T independent quantum systems. Data stream W-k, k is an element of[K] is replicated at a subset of servers W(k) subset of [S] , and quantum system Q(t), t is an element of[T] is distributed among a subset of servers E(t) subset of [S] such that Server s is an element of epsilon(t) receives subsystem Q(t,s) of Q(t) = (Q(t,s))(s is an element of epsilon(t)). Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is Sigma(t is an element of[T]) Sigma(s is an element of E(t)) log(d vertical bar)Q(t,s vertical bar) qudits, where vertical bar Q vertical bar is the dimension of Q . The states and measurements of (Q(t))(t is an element of[T]) are required to be separable across t is an element of[T] throughout, but for each t is an element of[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 Sigma-QMAC, i.e., the supremum of achievable rates is characterized for arbitrary data and entanglement distributions W, epsilon. For example, in the symmetric setting with K = ((S)(alpha)) data-streams, each replicated among a distinct alpha-subset of [S], and T = ((S)(beta)) quantum systems, each distributed among a distinct beta-subset of [S], the capacity of the Sigma-QMAC is 1/beta T Sigma(min(alpha,beta))(gamma=(alpha+beta-S)+) min(beta, 2 gamma)center dot ((alpha)(gamma))center dot ((S-alpha)(beta-gamma)). Coding based on the N-sum box abstraction is optimal in everycase. Notably, for every S not equal 3there exists an instance of the Sigma-QMAC where S-party entanglement is necessary to achieve the fully entangled capacity