🤖 AI Summary
This work investigates the optimal quantum communication cost for computing the sum of $K$ classical messages over a finite field via a quantum erasure multiple-access channel (QEMAC). $S$ servers share pre-established entanglement; each message is replicated across exactly $alpha$ servers, and up to $eta$ response paths may be erased. The receiver recovers the sum via joint quantum measurement. Methodologically, we introduce the $N$-sum box abstraction—integrating network coding principles with the “qudit-as-dit” quantum–classical equivalence paradigm—and construct efficient encodings using absolutely maximally entangled states (AMES). Our key contributions are: (i) the first exact characterization of the sum-computation capacity both with and without arbitrary initial entanglement (quantified by $Delta_0$); and (ii) a closed-form capacity expression $C = maxig{min{2(alpha-eta)/S,,(S-2eta)/S},,(alpha-eta)/Sig}$, accompanied by an explicit capacity-achieving protocol.
📝 Abstract
The optimal quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC). <italic>K</italic> classical messages comprised of finite-field symbols are distributed across <italic>S</italic> servers, who also share quantum entanglement in advance. Each server <inline-formula> <tex-math notation="LaTeX">$sin [S]$ </tex-math></inline-formula> manipulates its quantum subsystem <inline-formula> <tex-math notation="LaTeX">${mathcal {Q}}_{s}$ </tex-math></inline-formula> according to its own available classical messages and sends <inline-formula> <tex-math notation="LaTeX">${mathcal {Q}}_{s}$ </tex-math></inline-formula> to the receiver who then computes the sum of the messages based on a joint quantum measurement. The download cost from Server <inline-formula> <tex-math notation="LaTeX">$sin [S]$ </tex-math></inline-formula> is the logarithm of the dimension of <inline-formula> <tex-math notation="LaTeX">${mathcal {Q}}_{s}$ </tex-math></inline-formula>. The rate <italic>R</italic> is 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 with <inline-formula> <tex-math notation="LaTeX">$K= {inom {S }{alpha }} $ </tex-math></inline-formula> messages where each message is replicated among a unique subset of <inline-formula> <tex-math notation="LaTeX">$alpha $ </tex-math></inline-formula> servers, and the answers from any <inline-formula> <tex-math notation="LaTeX">$eta $ </tex-math></inline-formula> servers may be erased. If no entanglement is initially available to the receiver, then we show that the capacity (maximal rate) is precisely <inline-formula> <tex-math notation="LaTeX">$C= max left {{{ min left {{{ frac {2(alpha -eta)}{S}, frac {S-2eta }{S} }}
ight }, frac {alpha -eta }{S} }}
ight }$ </tex-math></inline-formula>. The capacity with arbitrary levels of prior entanglement <inline-formula> <tex-math notation="LaTeX">$(Delta _{0})$ </tex-math></inline-formula> between the <italic>S</italic> data-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 <italic>N</italic>-sum box abstraction that rely on absolutely maximally entangled quantum states.
