Quantum Nonlocality under Latency Constraints

Abstract

Bell inequality violation is the phenomenon where multiple non-communicating parties can exhibit correlations using quantum resources that are impossible if they can only use classical resources. One way to enforce non-communication is to apply a latency constraint: the parties must all produce outputs after they receive their inputs within a time window shorter than the speed of light delay between any pair of parties. If this latency constraint is relaxed so that a subset of the parties can communicate, we can obtain a new set of inequalities on correlations that extends Bell inequalities in a very natural way. Moreover, with this relaxed latency constraint, we can also have quantum communication between a subset of parties and thereby achieve possible quantum violations of these new inequalities. We ultimately wish to answer the fundamental question: "What are the physically realizable correlations between multiple parties under varying latency constraints?" To answer this question, we introduce latency-constrained games, a mathematical framework that extends nonlocal games to the setting where a subset of parties can communicate. The notion of latency-constrained games can have real-world applications, including high frequency trading, distributed computing, computer architecture, and distributed control systems.

Figures (16)

• Figure 1: An illustration of the three parties $A,B,C$ in the LC game that are collinear on the $x$-axis and their respective light cones. We assume the three parties receive their inputs simultaneously at time $0$. They must produce their outputs within time $t$. Different latency constraints lead to inequivalent non-communication constraints.
• Figure 2: Parties $A$ and $B$ can have back-and-forth communication for a latency constraint that lies between those of the two partially communicating scenarios. Note that this is not an exhaustive list of all possible scenarios.
• Figure 3: A quantum strategy for a graph game, where the connectivity graph $G$ is the graph $v_1 \leftrightarrows v_2 \leftrightarrows v_3$.
• Figure 4: The distributed CHSH game.
• Figure 5: The magic square game. An optimal quantum strategy that has unit winning probability is to measure the observables on the right-side figure.

Theorems & Definitions (37)

• Definition 1
• Definition 2
• Proposition 3
• proof
• Proposition 4
• proof
• Definition 5
• Proposition 6
• proof
• Proposition 7