The failure of the quantum regression hypothesis

Giacomo Guarnieri, Andrea Smirne, Bassano Vacchini

We explore the connection between two recently introduced notions of non-Markovian quantum dynamics and the validity of the so-called quantum regression theorem. While non-Markovianity of a quantum dynamics has been defined looking at the behavior in time of the statistical operator, which determines the evolution of mean values, the quantum regression theorem makes statements about the behavior of system correlation functions of order two and higher. The comparison relies on an estimate of the validity of the quantum regression hypothesis, which can be obtained exactly evaluating two-point correlation functions. To this aim we consider a qubit undergoing dephasing due to interaction with a bosonic bath, comparing the exact evaluation of the non-Markovianity measures with the violation of the quantum regression theorem for a class of spectral densities. We further study a photonic dephasing model, recently exploited for the experimental measurement of non-Markovianity. It appears that while a non-Markovian dynamics according to either definition brings with itself violation of the regression hypothesis, even Markovian dynamics can lead to a failure of the regression relation.

Simon Broadbent

G. W. Ford, Robert F. O’Connell

The Onsager regression hypothesis states that the regression of fluctuations is governed by macroscopic equations describing the approach to equilibrium. It is here asserted that this hypothesis fails in the quantum case. This is shown first by explicit calculation for the example of quantum Brownian motion of an oscillator and then in general from the fluctuation-dissipation theorem. It is asserted that the correct generalization of the Onsager hypothesis is the fluctuation-dissipation theorem.

Hung T. Nguyen, Le Si Dong

Leonardo Santos, Otfried Gühne, Stefan Nimmrichter

Temporal correlations are fundamental in quantum physics, yet their computation is often challenging. The regression theorem (or hypothesis) serves as a key tool in this context, offering a seemingly straightforward approach. However, it fails for systems strongly coupled to their surroundings, where memory effects become significant. Here, we extend the analysis of temporal correlations beyond the regression theorem, revealing what can be learned about open quantum systems when this hypothesis fails. We introduce robust, operationally meaningful methods to explore how the breakdown of the regression hypothesis can uncover fundamental quantum features of non-Markovian open systems, including entanglement, coherence, and quantum memory, namely, the fundamental impossibility of simulating memory in non-Markov processes with classical feedback mechanisms. Finally, we demonstrate how these quantum features are linked to microscopic properties such as the bath spectral density and heat flow.

Kerry Kuehn