Quantum Advantage in Non-Interactive Source Simulation

TL;DR

The paper studies non-interactive source simulation using quantum resources (EA-NISS) versus classical common randomness (CR-NISS) to reproduce a target distribution $Q_{U,V}$ from IID inputs $(X^d,Y^d)$ drawn from $P_{X,Y}$. It develops a Boolean Fourier-analytic framework to analyze feasibility and proves that for binary-output targets, EA-NISS and CR-NISS yield identical feasible sets, implying no quantum advantage in this regime. For non-binary outputs, it constructs a concrete EA-NISS example that cannot be simulated by CR-NISS and proves that the CR-feasible set has measure zero inside the EA-feasible set, demonstrating a quantum advantage. The results rely on a combination of Fourier analysis, rank considerations, and careful construction of classical simulations and show that quantum effects matter only beyond binary outputs in NISS contexts.

Abstract

This work considers the non-interactive source simulation problem (NISS). In the standard NISS scenario, a pair of distributed agents, Alice and Bob, observe a distributed binary memoryless source $(X^d,Y^d)$ generated based on joint distribution $P_{X,Y}$. The agents wish to produce a pair of discrete random variables $(U_d,V_d)$ with joint distribution $P_{U_d,V_d}$, such that $P_{U_d,V_d}$ converges in total variation distance to a target distribution $Q_{U,V}$. Two variations of the standard NISS scenario are considered. In the first variation, in addition to $(X^d,Y^d)$ the agents have access to a shared Bell state. The agents each measure their respective state, using a measurement of their choice, and use its classical output along with $(X^d,Y^d)$ to simulate the target distribution. This scenario is called the entanglement-assisted NISS (EA-NISS). In the second variation, the agents have access to a classical common random bit $Z$, in addition to $(X^d,Y^d)$. This scenario is called the classical common randomness NISS (CR-NISS). It is shown that for binary-output NISS scenarios, the set of feasible distributions for EA-NISS and CR-NISS are equal with each other. Hence, there is not quantum advantage in these EA-NISS scenarios. For non-binary output NISS scenarios, it is shown through an example that there are distributions that are feasible in EA-NISS but not in CR-NISS. This shows that there is a quantum advantage in non-binary output EA-NISS.

Figures (1)

• Figure 1: The EA-NISS and CR-NISS scenarios: (a) In the EA-NISS scenario, Alice and Bob share a Bell state and a sequence of IID random variables $X^d$ and $Y^d$, respectively, generated according to $P_{X,Y}$; (b) In the CR-NISS scenario, Alice and Bob receive one bit of common randomness and a sequence of IID random variables $X^d$ and $Y^d$, respectively, generated according to $P_{X,Y}$.

Theorems & Definitions (12)

• Definition 1: EA-NISS
• Definition 2: CR-NISS
• Proposition 1
• Proposition 2
• Corollary 1
• Proposition 3
• Corollary 2
• Theorem 1
• Example 1
• Theorem 2