A quantum state is like mountain weather: it changes with every glance. To estimate its uncertainty, physicists make copies and take measurements. Each copy is a thermometer that breaks after use. The question is: how many thermometers do you need to know the temperature with the required precision?

The new work closes almost the entire gap. For Rényi and Tsallis entropies, measures of chaos that generalize ordinary entropy, the authors constructed algorithms requiring exactly as many copies as the lower bound prescribes. Up to polylogarithmic factors, which in practice hardly matter.

How did they manage it? They combined two tools. The first is quantum tomography: from several copies, they reconstruct a rough portrait of the state. The second is random purification: any mixed state can be seen as part of a larger pure state, and then measurements become more precise. By combining these approaches and carefully tuning the weights, the authors achieved near-optimal efficiency.

The result is not just a theoretical triumph. Now engineers have a precise benchmark: how many copies of a state are needed to verify a quantum device or assess randomness in quantum cryptography. And if someone proposes an algorithm requiring fewer copies, it would violate the laws of quantum mechanics.

So what has changed? Before this work, there were upper and lower bounds, and no one knew where the truth lay. Now we know: the truth lies at the lower bound. All that remains is to remove the polylogarithmic factors, but that is a matter of engineering polish, not fundamental physics.