Distributed Problems in Quantum Communication

Locality might prohibit Alice and Bob from using quantum entanglement to transmit information superluminally, but it does not prevent them from either reducing the communication required for some problems, or playing impossible games requiring no communication at all. Accordingly, we will review the polynomial method for determining quantum communication complexity lower bounds. We will also review quantum psuedo-telepathy, which permits certain distributed tasks amongst entangled parties to be completed without communication. Communication complexity quantifies the communication requirements of two separated parties, Alice and Bob, living far-apart in a country where telephone calls and email are extremely expensive. Fortunately, Alice and Bob each have quantum supercomputers, an infinite supply of nuclear energy, and can communicate without fear of eavesdroppers, since Eve has recently been executed. They want to compute some function f : R→ {0, 1} such that R ⊆ X × Y . The function f is a total function if R = X × Y , and a promise function if otherwise. Alice and Bob have two n-bit string inputs x and y from X = Y = {0, 1}n, and want to compute the value of the function f (x, y) on their inputs. However, because communication is so expensive, they want to minimize information transmitted between them, and accordingly define the communicational complexity of a total Boolean function f : X × Y → {0, 1} as the minimum number of (qu)bits required for either of them to evaluate f (x, y). Our goal here is to quantify the limits of quantum communication between Alice and Bob. We will first provide a method for proving lower bounds for the quantum communicational complexity of a given Boolean function f and relate it to some other important results and conjectures in classical information theory. We will discuss Mermin-Peres magic square