Synopsis

Uncertainty beyond the Uncertainty Principle

Physics 16, s150
According to a new extension to an old theory, a particle’s position cannot be measured precisely even if its momentum is not measured simultaneously.
Y. Kuramochi and H. Tajima [1]

Heisenberg’s uncertainty principle limits the precision with which two observables that do not commute with each other can be simultaneously measured. The Wigner-Araki-Yanase (WAY) theorem goes further. If observables A and B do not commute, and if observable A is conserved, observable B cannot be measured with arbitrary precision even if A is not measured at all. In its original 1960 formulation, the WAY theorem applied only to observables, such as spin, whose possible values are discrete and bounded. Now Yui Kuramochi of Kyushu University and Hiroyasu Tajima of the University of Electro-Communications—both in Japan—have proven that the WAY theorem also encompasses observables, such as position, that are continuous and unbounded [1]. Besides resolving the decades-long problem of how to deal with such observables, the extension will likely find practical applications in quantum optics.

The difficulty of extending the WAY theorem arose from how an unbounded observable L is represented: as an infinite-dimensional matrix with unbounded eigenvalues. To tame the problem, Kuramochi and Tajima avoided considering L directly. Instead, they looked at an exponential function of L, which forms a one-parameter unitary group. Although the exponential function is also unbounded, its spectrum of eigenvalues is contained within the complex plane’s unit circle. Thanks to that boundedness, Kuramochi and Tajima could go on to use off-the-shelf techniques from quantum information to complete their proof.

Because momentum is conserved, the extended WAY theorem implies that a particle’s position cannot be measured with arbitrary precision even if its momentum is not measured simultaneously. Similar pairs of observables crop up in quantum optics. Kuramochi and Tajima anticipate that their theorem could be useful in setting limits on the extent to which quantum versions of transmission protocols can outperform the classical ones.

–Charles Day

Charles Day is a Senior Editor for Physics Magazine.

References

  1. Y. Kuramochi and H. Tajima, “Wigner-Araki-Yanase theorem for continuous and unbounded conserved observables,” Phys. Rev. Lett. 131, 210201 (2023).

Subject Areas

Quantum Physics

Related Articles

Quantum “Torch” Begins Its Relay
Quantum Physics

Quantum “Torch” Begins Its Relay

A quantum light source is touring European labs in preparation for the 2025 International Year of Quantum Science and Technology. Read More »

Quantum Machine Learning Goes Photonic
Quantum Physics

Quantum Machine Learning Goes Photonic

Measuring a photon’s angular momentum after it passes through optical devices teaches an algorithm to reconstruct the properties of the photon’s initial quantum state. Read More »

Shielding Quantum Light in Space and Time
Quantum Physics

Shielding Quantum Light in Space and Time

A way to create single photons whose spatiotemporal shapes do not expand during propagation could limit information loss in future photonic quantum technologies. Read More »

More Articles