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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 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., Comment: 7 + 9 pages, 5 figures

Thermodynamics connects our knowledge of the world to our capability to manipulate and thus to control it. This crucial role of control is exemplified by the third law of thermodynamics, Nernst's unattainability principle, which states that infinite resources are required to cool a system to absolute zero temperature. But what are these resources and how should they be utilized And how does this relate to Landauer's principle that famously connects information and thermodynamics We answer these questions by providing a framework for identifying the resources that enable the creation of pure quantum states. We show that perfect cooling is possible with Landauer energy cost given infinite time or control complexity. However, such optimal protocols require complex unitaries generated by an external work source. Restricting to unitaries that can be run solely via a heat engine, we derive a novel Carnot-Landauer limit, along with protocols for its saturation. This generalizes Landauer's principle to a fully thermodynamic setting, leading to a unification with the third law and emphasizes the importance of control in quantum thermodynamics., PRX Quantum, 4 (1), ISSN:2691-3399

Practical implementations of quantum technologies require preparation of states with a high degree of purity---or, in thermodynamic terms, very low temperatures. Given finite resources, the Third Law of thermodynamics prohibits perfect cooling; nonetheless, attainable upper bounds for the asymptotic ground state population of a system repeatedly interacting with quantum thermal machines have been derived. These bounds apply within a memoryless (Markovian) setting, in which each refrigeration step proceeds independently of those previous. Here, we expand this framework to study the effects of memory on quantum cooling. By introducing a memory mechanism through a generalized collision model that permits a Markovian embedding, we derive achievable bounds that provide an exponential advantage over the memoryless case. For qubits, our bound coincides with that of heat-bath algorithmic cooling, which our framework generalizes to arbitrary dimensions. We lastly describe the adaptive step-wise optimal protocol that outperforms all standard procedures., Comment: 4.5+13 pages, 9 figures. Close to published version

The duration, strength and structure of memory effects are crucial properties of physical evolution. Due to the invasive nature of quantum measurement, such properties must be defined with respect to the probing instruments employed. Here, using a photonic platform, we experimentally demonstrate this necessity via two paradigmatic processes: future-history correlations in the first process can be erased by an intermediate quantum measurement; for the second process, a noisy classical measurement blocks the effect of history. We then apply memory truncation techniques to recover an efficient description that approximates expectation values for multi-time observables. Our proof-of-principle analysis paves the way for experiments concerning more general non-Markovian quantum processes and highlights where standard open systems techniques break down., Comment: 4.5+7 pages; 7 figures; 62 references

Understanding temporal processes and their correlations in time is of paramount importance for the development of near-term technologies that operate under realistic conditions. Capturing the complete multi-time statistics defining a stochastic process lies at the heart of any proper treatment of memory effects. In this thesis, using a novel framework for the characterisation of quantum stochastic processes, we first solve the long standing question of unambiguously describing the memory length of a quantum processes. This is achieved by constructing a quantum Markov order condition that naturally generalises its classical counterpart for the quantification of finite-length memory effects. As measurements are inherently invasive in quantum mechanics, one has no choice but to define Markov order with respect to the interrogating instruments that are used to probe the process at hand: different memory effects are exhibited depending on how one addresses the system, in contrast to the standard classical setting. We then fully characterise the structural constraints imposed on quantum processes with finite Markov order, shedding light on a variety of memory effects that can arise through various examples. Lastly, we introduce an instrument-specific notion of memory strength that allows for a meaningful quantification of the temporal correlations between the history and the future of a process for a given choice of experimental intervention. These findings are directly relevant to both characterising and exploiting memory effects that persist for a finite duration. In particular, immediate applications range from developing efficient compression and recovery schemes for the description of quantum processes with memory to designing coherent control protocols that efficiently perform information-theoretic tasks, amongst a plethora of others., Masters Thesis. Includes work related to 3 original papers: arXiv:1805.11341, arXiv:1810.10809 and arXiv:1907.12583. 142 pages + 27 pages appendices; 41 figures; 190 references

We formally extend the notion of Markov order to open quantum processes by accounting for the instruments used to probe the system of interest at different times. Our description recovers the classical Markov order property in the appropriate limit: when the stochastic process is classical and the instruments are non-invasive, \emph{i.e.}, restricted to orthogonal, projective measurements. We then prove that there do not exist non-Markovian quantum processes that have finite Markov order with respect to all possible instruments; the same process exhibits distinct memory effects with respect to different probing instruments. This naturally leads to a relaxed definition of quantum Markov order with respect to specified sequences of instruments. The memory effects captured by different choices of instruments vary dramatically, providing a rich landscape for future exploration., Comment: 4.5+2 pages, 3 figures

Generic non-Markovian quantum processes have infinitely long memory, implying an exact description that grows exponentially in complexity with observation time. Here, we present a finite memory ansatz that approximates (or recovers) the true process with errors bounded by the strength of the non-Markovian memory. The introduced memory strength is an operational quantity and depends on the way the process is probed. Remarkably, the recovery error is bounded by the smallest memory strength over all possible probing methods. This allows for an unambiguous and efficient description of non-Markovian phenomena, enabling compression and recovery techniques pivotal to near-term technologies. We highlight the implications of our results by analyzing an exactly solvable model to show that memory truncation is possible even in a highly non-Markovian regime., Comment: 8 pages, 7 pages of appendices, 5 figures. Close to the published version

In a generalized framework for the Landauer erasure protocol, we study bounds on the heat dissipated in typical nonequilibrium quantum processes. In contrast to thermodynamic processes, quantum fluctuations are not suppressed in the nonequilibrium regime and cannot be ignored, making such processes difficult to understand and treat. Here we derive an emergent fluctuation relation that virtually guarantees the average heat produced to be dissipated into the reservoir either when the system or reservoir is large (or both) or when the temperature is high. The implication of our result is that for nonequilibrium processes, heat fluctuations away from its average value are suppressed independently of the underlying dynamics exponentially quickly in the dimension of the larger subsystem and linearly in the inverse temperature. We achieve these results by generalizing a concentration of measure relation for subsystem states to the case where the global state is mixed., Comment: Rewritten to focus on fluctuation theorem result, new application of Levy's lemma to unitary orbits of mixed states; 4+2.5 pages; 2 figures

Non-Markovian quantum processes exhibit different memory effects when measured in different ways; an unambiguous characterization of memory length requires accounting for the sequence of instruments applied to probe the system dynamics. This instrument-specific notion of quantum Markov order displays stark differences to its classical counterpart. Here, we explore the structure of quantum stochastic processes with finite length memory in detail. We begin by examining a generalized collision model with memory, before framing this instance within the general theory. We detail the constraints that are placed on the underlying system-environment dynamics for a process to exhibit finite Markov order with respect to natural classes of probing instruments, including deterministic (unitary) operations and sequences of generalized quantum measurements with informationally-complete preparations. Lastly, we show how processes with vanishing quantum conditional mutual information form a special case of the theory. Throughout, we provide a number of representative, pedagogical examples to display the salient features of memory effects in quantum processes., Comment: 15.5+8 pages; 11 figures

Understanding the demarcation line between classical and quantum is an important issue in modern physics. The development of such an understanding requires a clear picture of the various concurrent notions of `classicality' in quantum theory presently in use. Here, we focus on the relationship between Kolmogorov consistency of measurement statistics -- the foundational footing of classical stochastic processes in standard probability theory -- and the commutativity (or absence thereof) of measurement operators -- a concept at the core of quantum theory. Kolmogorov consistency implies that the statistics of sequential measurements on a (possibly quantum) system could be explained entirely by means of a classical stochastic process, thereby providing an operational notion of classicality. On the other hand, commutativity of measurement operators is a structural property that holds in classical physics and its breakdown is the origin of the uncertainty principle, a fundamentally quantum phenomenon. Here, we formalise the connection between these two a priori independent notions of classicality, demonstrate that they are distinct in general and detail their implications for memoryless multi-time quantum processes., 14.5 pages, 2 figures. Close to published version