![]() They have a crucial role in quantum theory, whereby each operator encodes an observable property of a physical object. Operators are matrices of numbers that can have either a finite or an infinite number of rows and columns. 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. On the pure-maths side, the problem was known as the Connes embedding problem, after the French mathematician and Fields medalist Alain Connes. “I never thought I’d see this problem being solved in my lifetime.” Observable properties “I’m shitting bricks here,” commented another physicist, Mateus Araújo at the Austrian Academy of Sciences in Vienna. “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. News of the paper spread quickly through social media after the work was posted, sparking excitement. “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. “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 at the California Institute of Technology in Pasadena. This implies that it is impossible to calculate how much coordination they could theoretically achieve. But it is intrinsically impossible for the two players to calculate an optimal strategy, the authors show. This enables both players to ‘win’ much more often than they would without 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. ![]() Earlier studies had shown this problem to be mathematically equivalent to the question of spooky action at a distance - also known as quantum entanglement 3. If their proof checks out, “it’s a super-beautiful result” says Stephanie Wehner, a theoretical quantum physicist at Delft University of Technology in the Netherlands.Īt the heart of the paper is a proof of a theorem in complexity theory, which is concerned with efficiency of algorithms. Machine learning leads mathematicians to unsolvable problem In particular, it will answer a mathematical question that has been unsolved for more than 40 years. If it holds up, it will solve in one fell swoop a number of related problems in pure mathematics, quantum mechanics and a branch of computer science known as complexity theory. The team’s proof, presented in a 165-page paper, was posted on on the arXiv preprint repository on 14 January 2, and has yet to be peer reviewed. Now, five researchers say they have solved a theoretical problem that shows that the answer is, in principle, unknowable. But, to this day, it remains unclear exactly how much coordination nature allows between distant objects. Decades after his death, experiments confirmed this. Credit: Victor De Schwanberg/Science Photo LibraryĪlbert 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” 1. ![]() Quantum entanglement is at the centre of a mathematical proof. ![]()
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