Quantum speed limits on operator flows and correlation functions

Nicoletta Carabba1, Niklas Hörnedal1,2, and Adolfo del Campo1,3

1Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg
2Fysikum, Stockholms Universitet, 106 91 Stockholm, Sweden
3Donostia International Physics Center, E-20018 San Sebastián, Spain

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Quantum speed limits (QSLs) identify fundamental time scales of physical processes by providing lower bounds on the rate of change of a quantum state or the expectation value of an observable. We introduce a generalization of QSL for unitary operator flows, which are ubiquitous in physics and relevant for applications in both the quantum and classical domains. We derive two types of QSLs and assess the existence of a crossover between them, that we illustrate with a qubit and a random matrix Hamiltonian, as canonical examples. We further apply our results to the time evolution of autocorrelation functions, obtaining computable constraints on the linear dynamical response of quantum systems out of equilibrium and the quantum Fisher information governing the precision in quantum parameter estimation.

The nature of time has always been one of the most debated subjects in human history, involving and relating different areas of human knowledge. In quantum physics, time, rather than being an observable as the position, is treated as a parameter. Accordingly, the Heisenberg uncertainty principle and time-energy uncertainty relation are of a deeply different nature. In 1945 the latter was refined by Mandelstam and Tamm as a quantum speed limit (QSL), that is, a lower bound on the time needed for the quantum state of a physical system to evolve into a distinguishable state. This new vision gave rise to a prolific series of works extending the notion of QSL to different kinds of quantum states and physical systems. Despite decades of research, QSL to date remains focused on quantum state distinguishability, natural for applications such as quantum computing and metrology. Yet, other applications involve operators flowing or evolving as a function of time. In this context, conventional QSL are inapplicable.

In this work we introduce a new class of QSL formulated for unitary operator flows. We generalize the celebrated Mandelstam-Tamm and Margolus-Levitin speed limits to operator flows, demonstrate their validity in simple and complex systems and illustrate their relevance to bound response functions in condensed matter physics. We expect our findings to find further applications including the dynamics of integrable systems, renormalization group and quantum complexity, among other examples.

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[12] Dimpi Thakuria, Abhay Srivastav, Brij Mohan, Asmita Kumari, and Arun Kumar Pati, "Generalised quantum speed limit for arbitrary time-continuous evolution", Journal of Physics A: Mathematical and Theoretical 57 2, 025302 (2024).

[13] Farha Yasmin and Jan Sperling, "Entanglement-assisted quantum speedup: Beating local quantum speed limits", arXiv:2211.14898, (2022).

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