Physicists were able to solve a long-standing mathematical problem using one of the key theorems of quantum information theory, Stein's generalized quantum lemma. This theorem underlies modern ideas about how to distinguish between quantum states and how to convert “quantum resources” into each other. The work was published in the journal Nature Physics.
Quantum information theory studies ways to store and process data in systems that obey the laws of quantum mechanics. In this direction, so-called resource theory is being developed – a formalism that describes what transformations are possible if only a limited set of operations is allowed. One of the cornerstones here is Stein's generalized quantum lemma, formulated in 2008, which describes how effectively a quantum state can be distinguished from a set of alternative states.
A few years ago, researchers discovered that there was a logical flaw in the original proof of the theorem. This raises questions about the correctness of some works using this result. Efforts to close the gap have been made before but have not been successful.
Now, the authors of the new paper have attempted to reconstruct the proof rigorously by providing additional mathematical arguments and removing controversial assumptions. According to them, this allows Stein's lemma to be reused as a reliable tool in quantum information theory.
One of the paper's authors explains: “The generalized quantum Stein lemma is a statement of the limits of the quantum hypothesis. The problem is that the previous proof did not take into account all the necessary conditions for the set of alternative states.”
During their research, scientists have shown that the transformation of quantum resources obeys a universal law similar to the second law of thermodynamics. This means that different types of quantum resources can be compared and converted into each other according to unified rules, based on the speed of that conversion.
“In effect, we have succeeded in reestablishing the ‘second law’ for quantum resource theories,” the researchers noted.
According to them, this result is important not only for fundamental physics but also for practical problems: it provides a more reliable mathematical basis for assessing the capabilities of quantum computers and other quantum devices.
In the future, the authors plan to develop this method to describe so-called dynamic quantum resources – processes and activities, not just states. This could help better understand the control limits of quantum systems and lead to the creation of more efficient quantum technologies.
