Hidden Quantum Memory: Is Memory There When Somebody Looks?

Philip Taranto1,2,3, Thomas J. Elliott4,5, and Simon Milz6,3,7

1Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo City, Tokyo 113-0033, Japan
2Atominstitut, Technische Universität Wien, 1020 Vienna, Austria
3Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
4Department of Physics & Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
5Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom
6School of Physics, Trinity College Dublin, Dublin 2, Ireland
7Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

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Abstract

In classical physics, memoryless dynamics and Markovian statistics are one and the same. This is not true for quantum dynamics, first and foremost because quantum measurements are invasive. Going beyond measurement invasiveness, here we derive a novel distinction between classical and quantum processes, namely the possibility of $\textit{hidden quantum memory}$. While Markovian statistics of classical processes can always be reproduced by a memoryless dynamical model, our main result shows that this is not true in quantum mechanics: We first provide an example of quantum non-Markovianity whose manifestation depends on whether or not a previous measurement is performed – an impossible phenomenon for memoryless dynamics; we then strengthen this result by demonstrating statistics that are Markovian independent of how they are probed, but are nonetheless $still$ incompatible with memoryless quantum dynamics. Thus, we establish the existence of Markovian statistics gathered by probing a quantum process that nevertheless $fundamentally$ require memory for their creation.

Quantum physics gives rise to many counter-intuitive phenomena that defy our classical understanding. At the heart of this is the fundamentally obtrusive nature of quantum measurement: the mere act of looking at a quantum system can change its state, blurring the line between system and observer. Here, we reveal a new consequence of this unavoidable invasiveness for quantum systems probed across multiple points in time — "hidden quantum memory" — where seemingly memoryless statistics nevertheless require dynamics with memory to realise.

The descriptors "Markovian" and "memoryless" are often used interchangeably to denote physical processes with particular memory properties. Markovian statistics emerge whenever the most recent observation alone captures all historical information relevant to the future, suggesting that any external memory is irrelevant to the dynamics. In the classical world, Markovianity is indeed equivalent to memorylessness of the process — i.e., a dynamical model comprising an independent sequence of state transformations between times that faithfully describes the measurement observations. Yet, as our results show, this equivalence is no longer true in the quantum realm, where different — albeit still Markovian — statistics can emerge depending upon historic interrogations. Such behaviour cannot be replicated by memoryless dynamics and therefore necessitates memory in the underlying process. As this phenomenon manifests first and foremost due to the inability to measure a quantum system without disturbing the state, such hidden quantum memory is a truly non-classical effect.

Our results go beyond the much lauded Leggett-Garg inequalities, in that even processes that violate LGIs do not generally display hidden quantum memory. Our work thus provides a deeper insight into complex temporal phenomena through the intricate interplay of measurement, invasiveness, and memory in quantum processes.

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