Extending the uncertainty principle using unbounded operators

the study published in diary physical review letter This study by Japanese researchers solved a long-standing problem in quantum physics by redefining the uncertainty principle.

Werner Heisenberg’s Uncertainty Principle is an important and surprising feature of quantum mechanics, for which he can thank Hay Fever. In the summer of 1925, miserable in Berlin, the young German physicist vacationed on the remote and rocky island of Helgoland in the North Sea off the coast of northern Germany. His allergies improved and he was able to continue his research in trying to understand the intricacies of Bohr’s atomic model, creating tables of internal atomic properties such as energy, position, and momentum.

When he returned to Göttingen, his advisor Max Born realized that each of these tables could be formed into a matrix (essentially a two-dimensional table of values). Together with his 22-year-old Pascal and his Jordan, they refined their research into matrix mechanics, the first successful theory of quantum mechanics, the physical laws that describe small objects such as atoms and electrons.

Matrix mechanics would be replaced within a few years by Schrödinger’s wave function and its equations, which gave Heisenberg the insight to formulate the uncertainty principle. There are limits to our ability to accurately determine the position and momentum of quantum systems, typically particles. It was measured.

The limit for the product of measurement uncertainties of two quantities is h/4π. Here h is Planck’s constant, which is very small but still non-zero. That is, it is not possible to measure both the position and momentum of a quantum object with arbitrary precision. Measuring one with greater precision means that the other can only be measured with less precision.

From a physical point of view, suppose we want to measure the position and momentum of an electron. To measure the properties of a system, we must shine some light on the system. Light is quantized as photons with non-zero energy. When an electron is irradiated with a photon, the electron is inevitably disturbed from its original state. In quantum mechanics, the mere act of measurement imposes limits on measurement accuracy.

Similar uncertainties apply to measurements of time and energy, angular position and angular momentum. Also, in general, operator In strict quantum mechanics.

Decades later, the uncertainty principle was refined by physicists Eugene Wigner, then Fujihiro Araki, and Mutsuo Yanase to become the Wigner-Araki-Yanase (WAY) theorem. This theorem states that for two observable quantities q and p, p is conserved (such as the momentum of a system), then q can be measured with arbitrary precision even if p is not measured at all. You can not.

“As a result of the WAY theorem, we see that (in some sense) it is impossible to measure the position q of a particle; all that can be measured is the position of the particle relative to the device, q—Q,” the mathematician said. says John Baez. The University of California, Riverside writtenwhere Q is the position of the measuring device.

However, the WAY theorem only applies to quantities such as particle rotation, and can only take on discrete and bounded quantities.

Now, Yui Kuramochi of Kyushu University and Hiroyasu Tajima of the University of Electro-Communications in Japan have shown that the WAY theorem also applies to continuous (rather than discrete) or unlimited observable quantities such as position, problem solved.

“According to the uncertainty principle, it is not possible to accurately measure position and momentum at the same time,” Kuramochi says. “Our results give us a further limitation: as long as we are using natural measurements that satisfy conservation of momentum, we cannot precisely measure only the position itself.” Let’s look at the “unbounded operator”, which is a physical quantity that can take.

Strictly speaking, the result requires a certain condition, called the Yanase condition, which is the basis of the WAY theorem. Although highly technical, it essentially specifies the compatibility of unlimited variables of equipment with stored quantities. Although the Yanase condition is mathematical, it seems desirable for applications to real-world physical systems.

“The WAY theorem predicts that under conservation laws, physical quantities that cannot be exchanged with conserved charges cannot be measured without error,” Kuramochi continues. “This corresponds to an answer to a problem that has remained unsolved for 60 years. This new result is particularly useful in fields such as quantum optics, where extensions of the new theorem are likely to be applicable. It solves a decades-old problem of how to approach observations.” ”

Although the original WAY theorem forbids the measurement error to be zero, it is a qualitative theorem and does not specify a measurement limit or even whether there is a lower bound greater than zero. The same applies to this extension of his WAY theorem by Kuramochi and Tajima.

The authors write in their paper that it is still an open question whether the original WAY theorem for repeated measures can be generalized to unrestricted conserved observables.

Proposing a new direction of research on extensions of the WAY theorem, the research team would like to generalize the results to energy-constrained states, as current results are limited to state-independent and approximate cases. One potential application is to set limits on how quantum network transmission protocols can perform better than classical limits.