In 1953, mathematician Alexander Grothendieck proved a theorem that became a cornerstone in the theory of tensor products. It states that for ordinary function spaces, two natural norms, the projective and the injective, are equivalent up to a universal constant. In other words, two ways of measuring the 'size' of combinations of vectors give similar results.
Decades passed, and mathematicians attempted to extend this idea to the more subtle world of operator spaces — structures where vectors carry something like a 'matrix grading.' Here, Grothendieck's theorem fails. In 2023, an explicit counterexample was found, but its connection to reality remained a mystery — until quantum games stepped in.
The authors of the new work translated the problem of the value of such a game into the language of operator spaces. The no-signaling value, what players can achieve using any correlations but without exchanging signals, turned out to be exactly equal to one of the tensor norms. And here something surprising emerged: that very counterexample to Grothendieck's theorem for operator spaces is simply a specific quantum game with a large gap between its no-signaling and quantum values. Pure mathematics and quantum information began speaking the same language.
This is not a neat coincidence. The authors obtained new upper bounds on the gap between the no-signaling and quantum values: that is, they showed how much stronger correlations without signals can be compared to those allowed by quantum mechanics. The bounds improve on everything known before and open the way to a deeper understanding of the limits of quantum correlations.
Grothendieck's theorem, proved in 1953, originally concerned Banach spaces and had nothing to do with quantum physics. The connection to quantum correlations was discovered only half a century later, and it still brings surprises.