Two coins are tossed simultaneously, and they always land the same way — heads with heads, tails with tails. Their behavior is strictly correlated. In the microworld, a similar connection between particles is called spin correlation. The authors of this work found a way to 'peek' at who exactly tossed these particles — more precisely, through which mediator they were born in an electron-positron collision.
The key turned out to be the spin density matrix — a mathematical object that describes all possible spin states of the pair. Its rank (the number of independent rows) directly reveals the properties of the mediator. If the particles arose through the exchange of a single scalar particle (e.g., the Higgs boson), the matrix has rank 1. This is like a single instruction that rigidly sets the spins. If the mediator was a photon, the rank is 2—a mixture of states appears, as if the recipe has two independent ingredients.
This simple rule, rank equals the number of degrees of freedom of the mediator, opens up powerful possibilities. When the rank is 1, physicists can measure the 'analyzing power' of the spin for each particle and their product, and from that infer CP violation (the difference between matter and antimatter). For the photon case (rank 2), it is possible to reconstruct the entire spin matrix from decay data, test quantum entanglement, and even circumvent a known no-go theorem. All this can already be applied to real experiments — for example, to the production of lambda hyperon pairs at BESIII or tau leptons at the Large Hadron Collider.
A spin density matrix of rank 1 describes a pure quantum state, the most ordered possible, without any admixture of randomness.