Quantum Entanglement with Just a Little Algebra

(Attention Conservation Notice: this post contains algebra! You’ve been warned.)

Imagine two people, Alice and Bob, are implicated in a crime and questioned in separate rooms with no way to communicate.

Trying to seem lenient, the investigators say they will be set free if they can corroborate each other’s story on more than 75% of the questions they are asked.

They have two alibis, Charlie and Diane. Alice and Bob know they will be asked one of two questions: “Were you with Charlie?” or “Were you with Diane?”

They also know from an informant, Eve, that the investigators are trying to trap them. Eve has told them that to corroborate each other’s story, they must answer precisely the same if either of them is asked if they were with Charlie, but differently if they are both asked if they were with Diane.

So, Alice and Bob do the obvious thing and devise a strategy so that their answers will be correlated correctly. Alice points out four possible question scenarios: Charlie-Charlie, Charlie-Diane, Diane-Charlie, or Diane-Diane. She writes them on paper as CC, CD, DC, and DD. She goes on to say that the pair could also give four possible answers to any question: yes-yes, yes-no, no-yes, or no-no. In total, there are sixteen possible ways the investigation could…