Abstract
When is the quantum speed limit (QSL) really quantum? While vanishing QSL times often indicate emergent classical behavior, it is still not entirely understood what precise aspects of classicality are at the origin of this dynamical feature. Here, we show that vanishing QSL times (or, equivalently, diverging quantum speeds) can be traced back to reduced uncertainty in quantum observables and can thus be understood as a consequence of emerging classicality for these particular observables. We illustrate this mechanism by developing a QSL formalism for continuous-variable quantum systems undergoing general Gaussian dynamics. For these systems, we show that three typical scenarios leading to vanishing QSL times, namely large squeezing, small effective Planck’s constant, and large particle number, can be fundamentally connected to each other. In contrast, by studying the dynamics of open quantum systems and mixed states, we show that the classicality that emerges due to incoherent mixing of states from the addition of classical noise typically increases the QSL time.
- Received 21 July 2021
- Accepted 10 November 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.040349
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
One of the most widely known building blocks of modern physics is Heisenberg’s uncertainty principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. This is particularly highlighted by the fact that there is no explicit Hermitian operator associated with a time measurement. It was several decades after Heisenberg’s suggestion before a mathematically rigorous and physically sound treatment was proposed by Mandelstam and Tamm, who kept time as a “classical” parameter and demonstrated that this indeterminacy principle was actually a fundamental statement about the intrinsic dynamical timescales of a system, ultimately providing the first quantum speed limit (QSL). The intimate relationship between a system’s energy spread and the ensuing speed of quantum evolution established this bound as an inherently quantum phenomena. However, it was subsequently demonstrated that this bound can be derived by purely geometric arguments, thus establishing that indistinguishability plays a crucial role.
Recently, this fact has been used to show that, by employing the geometric approach, similar speed limits for classical dynamics can be derived, thus leading to the question: “How quantum is a quantum speed limit?” In our work we tackle this question by developing a complete framework for QSLs for Gaussian systems that provide the ideal testbed for exploring both classical and quantum systems. With this formalism, we explore different limits in which the quantum speed diverges, and establish that a very precise notion of classicality, associated with the reduced uncertainty in particular observables, is linked to a vanishing QSL time. Our work therefore addresses a fundamental issue related to how one should interpret quantum speed limits while also providing a versatile framework to meaningfully define such bounds for continuous-variable systems.
