Resource theories of multi-time processes: A window into quantum non-Markovianity

Graeme D. Berk1, Andrew J. P. Garner2,3, Benjamin Yadin4, Kavan Modi1, and Felix A. Pollock1

1School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
2Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
3School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore
4School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.

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We investigate the conditions under which an uncontrollable background processes may be harnessed by an agent to perform a task that would otherwise be impossible within their operational framework. This situation can be understood from the perspective of resource theory: rather than harnessing 'useful' quantum states to perform tasks, we propose a resource theory of quantum processes across multiple points in time. Uncontrollable background processes fulfil the role of resources, and a new set of objects called $\textit{superprocesses}$, corresponding to operationally implementable control of the system undergoing the process, constitute the transformations between them. After formally introducing a framework for deriving resource theories of multi-time processes, we present a hierarchy of examples induced by restricting quantum or classical communication within the superprocess — corresponding to a client-server scenario. The resulting nine resource theories have different notions of quantum or classical memory as the determinant of their utility. Furthermore, one of these theories has a strict correspondence between non-useful processes and those that are Markovian and, therefore, could be said to be a true 'quantum resource theory of non-Markovianity'.

In the olden times, sailors navigated the seas at the mercy of an uncontrolled background process, namely, the weather. Winds were their primary resource, and they developed sophisticated ways to harness winds to their advantage. In the modern era, another class of energy resources has risen to prominence, which unlike the wind, are directly controllable; these include fossil fuels, nuclear energy, and electric batteries.

Nascent quantum technologies are more akin to the sailboats of a bygone era as they are inevitably influenced by the environment they exist within. Our paper asks to what degree one can harness useful properties of this environment to their advantage. What is the quantum analogue of sailing? In particular, can we make use of patterns in the environment, i.e. the temporal correlations therein known as non-Markovianity?

Using a mathematical toolbox known as resource theories, we present a general framework for answering these types of questions, uncovering families of monotones — functions that measure the usefulness of resources. Additionally, we analysed how these rules play out for a hierarchy of nine example scenarios, varying the level of capability to harness an uncontrolled process. In one of these example scenarios we show that the uncontrolled processes that are useful correspond precisely to the aforementioned non-Markovianity property.

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