Solving Tsirelson’s problem

Alice and Bob are on a game show. They each sit in isolation booths at either ends of the set and can’t communicate in any way. The game show host asks a number of questions. Neither knows what questions are being asked, or the answers the other gives. The judges are shocked that they provide the exact same answer more often than they should. From the judges’ perspectives, Alice and Bob appear to read each others’ minds.

If you've read about quantum mechanics, you may suggest that quantum entanglement can explain the lack of mistakes. Entanglement describes how our knowledge of distant elements of a quantum state can change immediately once we learn the local outcome. It’s not what we experience in our every day, but it is something that mathematicians, computer scientists and physicists study.

Entanglement does not allow Alice and Bob to read each others’ minds, but it does potentially allow Alice and Bob to gain an advantage in scenarios where they cannot see each other or communicate. Researchers call games built around these types of scenarios, like our hypothetical game show, “non-local games.” 

Faculty from the Institute for Quantum Computing (IQC) and the David R. Cheriton School of Computer Science at the University of Waterloo, Richard Cleve and John Watrous, are two of the founding fathers of the idea that entanglement is a type of computational resource that can be used to succeed in non-local games than what would normally be possible.

Although we know a lot about the mathematics of entanglement, there are still many questions about entanglement and non-local games that we don’t know how to answer. William Slofstra, a research assistant professor at IQC, recently announced a solution for one of these problems, called the strong Tsirelson problem, which gets at the nature of how we model entanglement. The pre-print, Tsirelson’s Problem and an Embedding Theorem for Groups Arising From Non-local Games, focuses on whether two ways of explaining, or modeling, Alice and Bob’s actions in the game show are the same. More specifically, Tsirelson’s problem asks whether commuting operator models for two-party quantum correlations are equivalent to tensor product models.

To resolve this problem, Slofstra had to come up with a complicated game for Alice and Bob to play. If Alice and Bob are allowed to use the type of entanglement allowed in one set of models, the commuting operator models, then they can play this complicated game successfully. But they cannot win if they are restricted to the type of entanglement used in tensor-product models. This shows that the two types of models are not the same.

Sloftstra was first introduced to Tsirelson’s problem while visiting IQC shortly after finishing his undergraduate studies in the departments of pure mathematics and combinatorics and optimization at Waterloo. But Slofstra's approach to the problem started with a much more recent visit to IQC in 2014, while a postdoctoral fellow at the University of California, Davis. Meeting with Cleve and Watrous, Slofstra learned about some open problems for a new class of games proposed by Cleve. Eventually this work led back to Tsirelson’s problem.

Currently, it’s not clear what Slofstra’s proof could lead to. According to Slofstra, “We currently struggle to create systems with just a few qubits of entanglement, and we don't know if we'll ever be able to do an experiment where we observe the difference between these two models.”

According to Watrous, “it’s a step in answering one of the extremely difficult open questions in mathematics.” Watrous explains by sharing Andrew Wiles’ analogy of mathematics and walking around a dark mansion. Suppose you enter a dark room, and, after finding a light, you find a significant piece of furniture. This can help make assumptions about the mansion without getting the complete picture of the mansion.

Slofstra’s findings can help mathematicians understand the possible ways in which two spatially separated systems behave. For now, as the paper moves through the standard peer review process, Slofstra is working with other mathematicians to explain and test the logic. “We now know that the commuting operator models and tensor product models are not the same,” said Slofstra. “The main question becomes, is this something we could hypothetically test?”