# Landmark Computer Science Proof Cascades Through Physics and Math

So if you see two players winning the game at unexpectedly high rates, you can conclude that they are using something other than classical physics to their advantage. Such Bell-type experiments are now called “nonlocal” games, in reference to the separation between the players. Physicists actually perform them in laboratories.

“People have run experiments over the years that really show this spooky thing is real,” said Yuen.

As when analyzing any game, you might want to know how often players can win a nonlocal game, provided they play the best they can. For example, with solitaire, you can calculate how often someone playing perfectly is likely to win.

But in 2016, William Slofstra proved that there’s no general algorithm for calculating the exact maximum winning probability for all nonlocal games. So researchers wondered: Could you at least approximate the maximum-winning percentage?

Computer scientists have homed in on an answer using the two models describing entanglement. An algorithm that uses the tensor product model establishes a floor, or minimum value, on the approximate maximum-winning probability for all nonlocal games. Another algorithm, which uses the commuting operator model, establishes a ceiling.

These algorithms produce more precise answers the longer they run. If Tsirelson’s prediction is true, and the two models really are equivalent, the floor and the ceiling should keep pinching closer together, narrowing in on a single value for the approximate maximum-winning percentage.

But if Tsirelson’s prediction is false, and the two models are not equivalent, “the ceiling and the floor will forever stay separated,” Yuen said. There will be no way to calculate even an approximate winning percentage for nonlocal games.

In their new work, the five researchers used this question — about whether the ceiling and floor converge and Tsirelson’s problem is true or false — to solve a separate question about when it’s possible to verify the answer to a computational problem.

## Entangled Assistance

In the early 2000s, computer scientists began to wonder: How does it change the range of problems you can verify if you interrogate two provers that share entangled particles?

Most assumed that entanglement worked against verification. After all, two suspects would have an easier time telling a consistent lie if they had some means of coordinating their answers.

But over the last few years, computer scientists have realized that the opposite is true: By interrogating provers that share entangled particles, you can verify a much larger class of problems than you can without entanglement.

“Entanglement is a way to generate correlations that you think might help them lie or cheat,” Vidick said. “But in fact you can use that to your advantage.”

To understand how, you first need to grasp the almost otherworldly scale of the problems whose solutions you could verify through this interactive procedure.

Imagine a graph — a collection of dots (vertices) connected by lines (edges). You might want to know whether it’s possible to color the vertices using three colors, so that no vertices connected by an edge have the same color. If you can, the graph is “three-colorable.”

If you hand a pair of entangled provers a very large graph, and they report back that it can be three-colored, you’ll wonder: Is there a way to verify their answer?

For very big graphs, it would be impossible to check the work directly. So instead, you could ask each prover to tell you the color of one of two connected vertices. If they each report a different color, and they keep doing so every time you ask, you’ll gain confidence that the three-coloring really works.

But even this interrogation strategy fails as graphs get really big — with more edges and vertices than there are atoms in the universe. Even the task of stating a specific question (“Tell me the color of XYZ vertex”) is more than you, the verifier, can manage: The amount of data required to name a specific vertex is more than you can hold in your working memory.

But entanglement makes it possible for the provers to come up with the questions themselves.

“The verifier doesn’t have to compute the questions. The verifier forces the provers to compute the questions for them,” Wright said.

The verifier wants the provers to report the colors of connected vertices. If the vertices aren’t connected, then the answers to the questions won’t say anything about whether the graph is three-colored. In other words, the verifier wants the provers to ask correlated questions: One prover asks about vertex ABC and the other asks about vertex XYZ. The hope is that the two vertices are connected to each other, even though neither prover knows which vertex the other is thinking about. (Just as Alice and Bob hope to fill in the same number in the same square even though neither knows which row or column the other has been asked about.)

If two provers were coming up with these questions completely on their own, there’d be no way to force them to select connected, or correlated, vertices in a way that would allow the verifier to validate their answers. But such correlation is exactly what entanglement enables.

“We’re going to use entanglement to offload almost everything onto the provers. We make them select questions by themselves,” Vidick said.

At the end of this procedure, the provers each report a color. The verifier checks whether they’re the same or not. If the graph really is three-colorable, the provers should never report the same color.

“If there is a three-coloring, the provers will be able to convince you there is one,” Yuen said.

As it turns out, this verification procedure is another example of a nonlocal game. The provers “win” if they convince you their solution is correct.

In 2012, Vidick and Tsuyoshi Ito proved that it’s possible to play a wide variety of nonlocal games with entangled provers to verify answers to at least the same number of problems you can verify by interrogating two classical computers. That is, using entangled provers doesn’t work against verification. And last year, Natarajan and Wright proved that interacting with entangled provers actually expands the class of problems that can be verified.

But computer scientists didn’t know the full range of problems that can be verified in this way. Until now.

## A Cascade of Consequences

In their new paper, the five computer scientists prove that interrogating entangled provers makes it possible to verify answers to unsolvable problems, including the halting problem.

“The verification capability of this type of model is really mind-boggling,” Yuen said.

But the halting problem can’t be solved. And that fact is the spark that sets the final proof in motion.

Imagine you hand a program to a pair of entangled provers. You ask them to tell you whether it will halt. You’re prepared to verify their answer through a kind of nonlocal game: The provers generate questions and “win” based on the coordination between their answers.

If the program does in fact halt, the provers should be able to win this game 100% of the time — similar to how if a graph is actually three-colorable, entangled provers should never report the same color for two connected vertices. If it doesn’t halt, the provers should only win by chance — 50% of the time.

That means if someone asks you to determine the approximate maximum-winning probability for a specific instance of this nonlocal game, you will first need to solve the halting problem. And solving the halting problem is impossible. Which means that calculating the approximate maximum-winning probability for nonlocal games is undecidable, just like the halting problem.

This in turn means that the answer to Tsirelson’s problem is no — the two models of entanglement are not equivalent. Because if they were, you could pinch the floor and the ceiling together to calculate an approximate maximum-winning probability.

“There cannot be such an algorithm, so the two [models] must be different,” said David Pérez-García of the Complutense University of Madrid.

The new paper proves that the class of problems that can be verified through interactions with entangled quantum provers, a class called MIP*, is exactly equal to the class of problems that are no harder than the halting problem, a class called RE. The title of the paper states it succinctly: “MIP* = RE.”

In the course of proving that the two complexity classes are equal, the computer scientists proved that Tsirelson’s problem is false, which, due to previous work, meant that the Connes embedding conjecture is also false.

For researchers in these fields, it was stunning that answers to such big problems would fall out from a seemingly unrelated proof in computer science.

“If I see a paper that says MIP* = RE, I don’t think it has anything to do with my work,” said Navascués, who co-authored previous work tying Tsirelson’s problem and the Connes embedding conjecture together. “For me it was a complete surprise.”

Quantum physicists and mathematicians are just beginning to digest the proof. Prior to the new work, mathematicians had wondered whether they could get away with approximating infinite-dimensional matrices by using large finite-dimensional ones instead. Now, because the Connes embedding conjecture is false, they know they can’t.

“Their result implies that’s impossible,” said Slofstra.

The computer scientists themselves did not aim to answer the Connes embedding conjecture, and as a result, they’re not in the best position to explain the implications of one of the problems they ended up solving.

“Personally, I’m not a mathematician. I don’t understand the original formulation of the Connes embedding conjecture well,” said Natarajan.

He and his co-authors anticipate that mathematicians will translate this new result into the language of their own field. In a blog post announcing the proof, Vidick wrote, “I don’t doubt that eventually complexity theory will not be needed to obtain the purely mathematical consequences.”

Yet as other researchers run with the proof, the line of inquiry that prompted it is coming to a halt. For more than three decades, computer scientists have been trying to figure out just how far interactive verification will take them. They are now confronted with the answer, in the form of a long paper with a simple title and echoes of Turing.

“There’s this long sequence of works just wondering how powerful” a verification procedure with two entangled quantum provers can be, Natarajan said. “Now we know how powerful it is. That story is at an end.”

This article was reprinted on Wired.com.