Characterising the Hierarchy of Multi-time Quantum Processes with Classical Memory

Philip Taranto1, Marco Túlio Quintino2, Mio Murao1, and Simon Milz3,4

1Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo City, Tokyo 113-0033, Japan
2Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
3School of Physics, Trinity College Dublin, Dublin 2, Ireland
4Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, D02YN67, Ireland

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Abstract

Memory is the fundamental form of temporal complexity: when present but uncontrollable, it manifests as non-Markovian noise; conversely, if controllable, memory can be a powerful resource for information processing. Memory effects arise from/are transmitted via interactions between a system and its environment; as such, they can be either classical or quantum. From a practical standpoint, quantum processes with classical memory promise near-term applicability: they are more powerful than their memoryless counterpart, yet at the same time can be controlled over significant timeframes without being spoiled by decoherence. However, despite practical and foundational value, apart from simple two-time scenarios, the distinction between quantum and classical memory remains unexplored. Here, we analyse multi-time quantum processes with memory mechanisms that transmit only classical information forward in time. Complementing this analysis, we also study two related – but simpler to characterise – sets of processes that could also be considered to have classical memory from a structural perspective, and demonstrate that these lead to remarkably distinct phenomena in the multi-time setting. Subsequently, we systematically stratify the full hierarchy of memory effects in quantum mechanics, many levels of which collapse in the two-time setting, making our results genuinely multi-time phenomena.

Talk presented at International Conference on Quantum Energy (Melbourne, 2023):

Memory plays a vital role in various natural and engineered processes, from predicting weather patterns to financial markets and computational tasks. When memory is present but uncontrollable, it leads to complex non-Markovian noise, which is challenging to model accurately. On the other hand, a controlled memory becomes a powerful tool for information processing, as seen in systems like quantum dots, where tunable memory can enhance properties like charge transport and emission spectra, potentially benefiting technologies like photovoltaic cells and communication protocols.

In the same vein as processes themselves, memory effects can be quantum or classical. They arise from interactions between a system of interest and its environment, with the latter acting as the memory carrier. Rapid dissipation of information leads to simple, memoryless processes; on the other hand, strong interactions with low dissipation often result in non-classical multi-time correlations. In between the two extremes of memorylessness and coherent quantum memory lies a class of quantum processes that offer significant application: quantum processes with classical memory. These are more powerful than memoryless ones and can be controlled over extended timeframes without being spoiled by decoherence or errors.

Despite their potential, understanding the distinction between quantum and classical memory has thus far remained largely unexplored. Here, we systematically characterise the hierarchy of multi-time memory effects in quantum mechanics, in particular demonstrating the distinct behaviour between various types of possible memory effects. Many levels of the engendered hierarchy only emerge as discernible beyond the two-time setting, making our results genuinely multi-time phenomena. On the practical side, since noise in quantum devices—and thus the observed memory effects—will predominately be classical in the near future, our work provides a methodological framework upon which efficient and reliable quantum devices can be built.

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