In recent years, physicists have been trying to better understand how quantum information spreads in systems of interacting particles—a phenomenon often referred to as “scrambling.” Scrambling in closed systems, physical systems that can only exchange energy with degrees of freedom within the system, is a characteristic feature of chaotic many-body quantum dynamics.

In open systems, which can exchange both energy and matter with their surroundings, scrambling is influenced by various additional factors, including noise and errors. While the effects of these additional influences are well-documented, leading for example to decoherence, how they affect scrambling remains poorly understood.

Two researchers from the University of California Berkeley (UC Berkeley) and Harvard University recently introduced a new framework, published in Physical Review Letters, that provides a universal picture for how information scrambling occurs in open quantum systems. Their framework offers a particularly simple viewpoint on how to understand and model the propagation of errors in an open quantum system and might already help to explain some previously puzzling observations gathered in magnetic resonance experiments.

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.