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Proof at the nexus of pure mathematics and algorithms puts ‘quantum weirdness’ on a whole new level.

Albert Einstein famously said that quantum mechanics should allow two objects to affect each other’s behaviour instantly across vast distances, something he
dubbed “spooky action at a distance”
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. Decades after his death, experiments confirmed this, but to this day, it remains unclear exactly how much coordination nature allows between distant objects. Now, five researchers say they have solved a theoretical problem that shows that the answer is, in principle, unknowable.

The team’s 165-page paper appeared on the arXiv preprint repository
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, and has yet to be peer reviewed. If it checks out, it solves a number of related problems in pure mathematics, quantum mechanics and a branch of computer science known as complexity theory in one fell swoop. In particular, it answers a mathematical question that had been unsolved for more than 40 years.

If their proof holds up, “it’s a super-beautiful result” says Stephanie Wehner, a theoretical quantum physicist at Delft University of Technology in the Netherlands.

At the heart of the paper is a proof of a theorem in complexity theory, which is concerned with efficiency of algorithms. Earlier studies had shown this problem to be mathematically equivalent to the question of spooky action at a distance — also known as quantum entanglement.

The theorem concerns a game-theory problem, with a team of two players who are able to coordinate their actions through quantum entanglement, even though they are not allowed to talk to each other. This enables both players to ‘win’ much more often than they would without quantum entanglement. But it is intrinsically impossible for the two players to calculate an optimal strategy, the authors show. This implies that it is impossible to calculate how much coordination they could theoretically reach. “There is no algorithm that is going to tell you what is the maximal violation you can get in quantum mechanics,” says co-author Thomas Vidick of the California Institute of Technology in Pasadena.

“What’s amazing is that quantum complexity theory has been the key to the proof,” says Toby Cubitt, a quantum-information theorist at University College London.

News of the paper spread quickly through social media after the work was posted on 14 January, sparking excitement. “I thought it would turn out to be one of those complexity-theory questions that might take 100 years to answer,”
tweeted Joseph Fitzsimons
, chief executive of Horizon Quantum Computing, a start-up company in Singapore.

“I’m shitting bricks here,”
commented another physicist
, Mateus Araújo of the Austrian Academy of Sciences in Vienna. “I never thought I’d see this problem being solved in my lifetime.”

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Observable properties

On the pure-mathematics side, the problem was known as the Connes embedding problem, after the French mathematician and Fields medalist Alain Connes. It is a question in the theory of operators, a branch of maths that itself arose from efforts to provide the foundations of quantum mechanics in the 1930s. Operators are matrices of numbers that can have either a finite or an infinite number of rows and columns. They have a crucial role in quantum theory, whereby each operator encodes an observable property of a physical object.

In a 1976 paper
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, using the language of operators, Connes asked whether quantum systems with infinitely many measurable variables could be approximated by simpler systems that have a finite number.

But the paper by Vidick and collaborators shows that the answer is no: there are, in principle, quantum systems that cannot be approximated by ‘finite’ ones. According to work by physicist Boris Tsirelson
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, who reformulated the problem, this also means that it is impossible to calculate the amount of correlation that two such systems can display across space when entangled.

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Disparate fields

The proof has come as a surprise to much of the community. “I was sure that Tsirelson’s problem had a positive answer,” wrote Araújo in his comments, adding that the result shook his basic conviction that “Nature is in some vague sense fundamentally finite.”

But researchers have barely begun to grasp the implications of the results. Quantum entanglement is at the heart of the nascent fields of quantum computing and quantum communications, and could be used to make super-secure networks. In particular, measuring the amount of correlation between entangled objects across a communication system can provide proof that it is safe from eavesdropping. But the results probably do not have technological implications, Wehner says, because all applications use quantum systems which are ‘finite’. In fact, it could be difficult to even conceive an experiment that could test quantum weirdness on an intrinsically ‘infinite’ system, she says.

The confluence of complexity theory, quantum information and mathematics means that there are very few researchers who say that they are able to grasp all the facets of this paper. Connes himself told
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Nature
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that he was not qualified to comment. But he added that he was surprised by how many ramifications it turned out to have. “It is amazing that the problem went so deep and I never foresaw that!”