# My research

I am interested in all areas of quantum information theory, with a particular emphasis on quantum thermodynamics and decoherence theory.

#### Current focus

Currently I am working mainly on three fronts that roughly correspond to the three research objectives of my research group (see the Research Agenda). First, I am looking for potential quantum advantages within thermodynamics and I am studying the feasibility of achieving them. This means that, on the one hand, I am looking for the improvements in the performance of thermodynamic protocols that arise from purely quantum effects such as the superposition principle (in particular focusing on the ways to reduce free energy dissipation). On the other hand, I am working on bridging the gap between the somewhat abstract resource-theoretic formulation of quantum thermodynamics and experimentally feasible thermodynamic setups. Second, I am trying to sharpen the foundational and operational understanding of the differences between classical stochastic dynamics and open quantum dynamics. Here, I am particularly interested in the interplay between memory effects and quantum coherence, as well as in the extent to which an observed random transformation can be explained via the underlying deterministic and coherent process. Third, I am developing classical simulation algorithms for universal quantum circuits in order to employ them to verify and validate near-term quantum devices, but also to improve our understating of what kind of quantum computations can be efficiently simulated classically.

A detailed description of most of my work within the last five years can be found in my Summary of Professional Accomplishments that I prepared as a part of the habilitation procedure in Poland and that summarises a single-themed series of 15 publications entitled Optimizing quantum information processing under constraints.

#### Areas of research

(To see a full list of publications and presentations in a chronological order check my Scientific CV)

When speaking of thermodynamics one inevitably thinks of concepts such as heat flows, thermal machines and pressure, which seem to be far removed from the ideas of quantum information theory. However, on a more abstract level, thermodynamics can be seen as a field studying the accessibility/inaccessibility of one physical state from another. The first and second laws of thermodynamics are fundamental constraints on state transformations, forcing thermodynamic processes to conserve the overall energy and forbidding free conversion of heat into work. Hence, the resource-theoretic machinery originally developed to study entanglement is also perfectly suited to shed light on thermodynamics.

In my research I mostly focus on the role that superposition principle plays in thermodynamic considerations. More precisely, I am interested in thermodynamic limitations on processing quantum coherence, the way it affects the thermodynamic arrow of time and the possibility of exploiting coherence to enhance the performance of heat engines. These foundational questions may be of interest for future advancements in nanotechnology, as interference effects are particularly relevant at scales we are increasingly able to control. Recently I am also interested in the problem of thermodynamic (and general resource-theoretic) transformations of finite-size systems, and in particular the effect this non-asymptotic regime has on reversible transformations.

• Geometric structure of thermal cones
A. de Oliveira Junior, Jakub Czartowski, Karol Życzkowski, Kamil Korzekwa
arXiv:2207.02237 (2022)

Abstract: The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future and incomparable regions. In this work, we analyse the structure of the resulting thermal cones, i.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a $$d$$-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the incomparable region. Moreover, we provide a detailed analysis of their behaviour based on newly introduced thermodynamic monotones given by the volumes of thermal cones. Finally, we point out that our results also apply to other majorisation-based resource theories (such as that of entanglement and coherence), since the partial ordering describing allowed state transformations is then the opposite of the thermodynamic order in the infinite temperature limit.

• Optimizing thermalizations
Kamil Korzekwa, Matteo Lostaglio
Phys. Rev. Lett. 129, 040602 (2022)

Abstract: We present a rigorous approach, based on the concept of continuous thermomajorisation, to algorithmically characterise the full set of energy occupations of a quantum system accessible from a given initial state through weak interactions with a heat bath. The algorithm can be deployed to solve complex optimization problems in out-of-equilibrium setups and it returns explicit elementary control sequences realizing optimal transformations. We illustrate this by finding optimal protocols in the context of cooling, work extraction and catalysis. The same tools also allow one to quantitatively assess the role played by memory effects in the performance of thermodynamic protocols. We obtained exhaustive solutions on a laptop machine for systems with dimension $$d\leq 7$$, but with heuristic methods one could access much higher $$d$$.

• Continuous thermomajorization and a complete set of laws for Markovian thermal processes
Matteo Lostaglio, Kamil Korzekwa
Phys. Rev. A 106, 012426 (2022)

Abstract: The standard dynamical approach to quantum thermodynamics is based on Markovian master equations describing the thermalization of a system weakly coupled to a large environment, and on tools such as entropy production relations. Here we introduce a new framework overcoming the limitations that the current dynamical and information theory approaches encounter when applied to this setting. More precisely, based on a newly introduced notion of continuous thermomajorization, we obtain necessary and sufficient conditions for the existence of a Markovian thermal process transforming between given initial and final energy distributions of the system. These lead to a complete set of generalized entropy production inequalities including the standard one as a special case. Importantly, these conditions can be reduced to a finitely verifiable set of constraints governing non-equilibrium transformations under master equations. What is more, the framework is also constructive, i.e., it returns explicit protocols realizing any allowed transformation. These protocols use as building blocks elementary thermalizations, which we prove to be universal controls. Finally, we also present an algorithm constructing the full set of energy distributions achievable from a given initial state via Markovian thermal processes and provide a $$\texttt{Mathematica}$$ implementation solving $$d=6$$ on a laptop computer in minutes.

• Fluctuation-dissipation relations for thermodynamic distillation processes
Tanmoy Biswas, A. de Oliveira Junior, Michał Horodecki, Kamil Korzekwa
Phys. Rev. E 105, 054127 (2022)

Abstract: The fluctuation-dissipation theorem is a fundamental result in statistical physics that establishes a connection between the response of a system subject to a perturbation and the fluctuations associated with observables in equilibrium. Here we derive its version within a resource-theoretic framework, where one investigates optimal quantum state transitions under thermodynamic constraints. More precisely, we first characterise optimal thermodynamic distillation processes, and then prove a relation between the amount of free energy dissipated in such processes and the free energy fluctuations of the initial state of the system. Our results apply to initial states given by either asymptotically many identical pure systems or arbitrary number of independent energy-incoherent systems, and allow not only for a state transformation, but also for the change of Hamiltonian. The fluctuation-dissipation relations we derive enable us to find the optimal performance of thermodynamic protocols such as work extraction, information erasure and thermodynamically-free communication, up to second-order asymptotics in the number $$N$$ of processed systems. We thus provide a first rigorous analysis of these thermodynamic protocols for quantum states with coherence between different energy eigenstates in the intermediate regime of large but finite $$N$$.

Popular summary: Almost two centuries ago, Robert Brown observed that pollen seeds immersed in water move randomly in erratic motion. It was not until the 1905 papers by Einstein and Smoluchowski that people understood that this “Brownian” motion is induced by the bombardment of pollen particles by water molecules. Crucially, by noting that these collisions would also create friction for the particle being pulled through the fluid, Einstein realised that the two processes, fluctuations of particle’s position and dissipation of its energy, have the same origin and thus must be related. Over the years, physicists generalized and formalized this observation into fundamental fluctuation-dissipation relations describing the behavior of systems driven out of equilibrium. Here, we provide a novel formulation of these relations in the quantum information realm, where fluctuations may not only be thermal in nature but can also arise from quantum superpositions. In our work, we investigate the behavior of quantum systems driven, in the presence of a thermal bath, from one non-equilibrium state to another. We prove that the minimal amount of free energy dissipated in this process is directly related to the fluctuations of the free energy content of the initial state of the system. Our analysis relies heavily on quantum information tools, while our results allow us to study the performance of thermal machines whose operation depends on quantum interference effects. This contribution paves the way for further studies on how quantum effects can be harnessed to minimise dissipation in thermodynamic processes. Moreover, our information-theoretic tools can be potentially used to explore fluctuation-dissipation relations for entanglement and other quantum resources.

• Work fluctuations due to partial thermalizations in two-level systems
Maria Quadeer, Kamil Korzekwa, Marco Tomamichel
Phys. Rev. E 103, 042141 (2021)

Abstract: We study work extraction processes mediated by finite-time interactions with an ambient bath — partial thermalizations — as continuous time Markov processes for two-level systems. Such a stochastic process results in fluctuations in the amount of work that can be extracted and is characterized by the rate at which the system parameters are driven in addition to the rate of thermalization with the bath. We analyze the distribution of work for the case where the energy gap of a two-level system is driven at a constant rate. We derive analytic expressions for average work and lower bound for the variance of work showing that such processes cannot be fluctuation-free in general. We also observe that an upper bound for the Monte Carlo estimate of the variance of work can be obtained using Jarzynski's fluctuation-dissipation relation for systems initially in equilibrium. Finally, we analyse work extraction cycles by modifying the Carnot cycle, incorporating processes involving partial thermalizations and obtain efficiency at maximum power for such finite-time work extraction cycles under different sets of constraints.

• Avoiding irreversibility: engineering resonant conversions of quantum resources
Kamil Korzekwa, Christopher T. Chubb, Marco Tomamichel
Phys. Rev. Lett. 122, 110403 (2019)

Abstract: We identify and explore the intriguing property of resource resonance arising within resource theories of entanglement, coherence and thermodynamics. While the theories considered are reversible asymptotically, the same is generally not true in realistic scenarios where the available resources are bounded. The finite-size effects responsible for this irreversibility could potentially prohibit small quantum information processors or thermal machines from achieving their full potential. Nevertheless, we show here that by carefully engineering the resource interconversion process any such losses can be greatly suppressed. Our results are predicted by higher order expansions of the trade-off between the rate of resource interconversion and the achieved fidelity, and are verified by exact numerical optimizations of appropriate approximate majorization conditions.

Popular summary: Due to the rapid progress in experimental control of intermediate-scale quantum systems, we may soon witness the emergence of new technologies that will utilize quantum resources to overcome current technological constraints. From a theoretical perspective, it is then crucial to understand the potential and limitations of manipulating and interconverting these resources in realistic scenarios, when only finite amounts of resources are available. In our work we address this pressing issue by developing a rigorous mathematical framework that allows one to investigate resource interconversion of finite-size systems within resource theories of entanglement, coherence and thermodynamics. This allows us to quantitatively analyse the irreversibility (and thus the unavoidable loss) of the conversion process arising from finite-size effects. Although this could potentially prohibit small quantum information processors or thermal machines from achieving their full potential, we show that by carefully engineering the resource interconversion process any such losses can be greatly suppressed. More precisely, we identify and explore the intriguing property of resource resonance that ensures that certain pairs of resources can be interconverted at greatly reduced loss. By analysing its applications within quantum thermodynamics and entanglement theory, we further explain how the resonance phenomenon can be employed to enhance the performance of intermediate-scale quantum devices and thermal machines developed in the near term.

• Moderate deviation analysis of majorisation-based resource interconversion
Christopher T. Chubb, Marco Tomamichel, Kamil Korzekwa
Phys. Rev. A 99, 032332 (2019)

Abstract: We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorisation relation, e.g. the resource theories of entanglement and coherence (for pure state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a non-trivial trade-off between the rate $$r_n$$ at which $$n$$ copies of a resource state $$\rho$$ can be transformed into $$nr_n$$ copies of another resource state $$\sigma$$, and the error level $$\epsilon_n$$ of the interconversion process, as a function of $$n$$. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion $$r_n$$ approaches its optimum in the asymptotic limit of unbounded resources ($$n\to\infty$$), while the error $$\epsilon_n$$ vanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behaviour which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite $$n$$ regime.

• Beyond the thermodynamic limit: finite-size corrections to state interconversion rates
Christopher T. Chubb, Marco Tomamichel, Kamil Korzekwa
Quantum 2, 108 (2018)

Abstract: Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.

Popular summary: Thermodynamics is one of the most versatile physical theories, finding applications in almost all fields of science, from cosmology and astrophysics to chemistry and the theory of computation. Its strength comes from the fact that it provides a universal framework that uses statistical tools to study physical phenomena in the so-called thermodynamic limit, i.e., when the number of involved systems is very large. However, our increasing ability to manipulate and control systems at smaller and smaller scales allows us to build novel nanodevices operating well beyond the thermodynamic limit. Therefore, in order to understand the thermodynamic properties of such devices, we need to formulate a theory that is not constrained to the study of macroscopic systems. In this paper we achieve this by developing an information-theoretic framework describing thermodynamic transformations of finite-size systems. One immediate application of our theoretical results is to the study of irreversible processes in the nanoscale regime. In particular, we show how the amount of ordered energy needed to drive a small system out of equilibrium is larger than the amount one could obtain in a reverse process. This affects reversibility of thermodynamic cycles and, in turn, deteriorates performance of nanoengines. Despite these negative finite-size effects, we find that in specially engineered conditions nanoscale engines can still achieve the ultimate limit of efficiency. Our results expand the realm of applicability of thermodynamics beyond the constraint of macroscopic systems, and thus provide new tools to study the universe at the smallest scale.

• Structure of the thermodynamic arrow of time in classical and quantum theories
Kamil Korzekwa
Phys. Rev. A 95, 052318 (2017)

Abstract: In this work we analyse the structure of the thermodynamic arrow of time, defined by transformations that leave the thermal equilibrium state unchanged, in classical (incoherent) and quantum (coherent) regimes. We note that in the infinite-temperature limit the thermodynamic ordering of states in both regimes exhibits a lattice structure. This means that when energy does not matter and the only thermodynamic resource is given by information, the thermodynamic arrow of time has a very specific structure. Namely, for any two states at present there exists a unique state in the past consistent with them and with all possible joint pasts. Similarly, there also exists a unique state in the future consistent with those states and with all possible joint futures. We also show that the lattice structure in the classical regime is broken at finite temperatures, i.e., when energy is a relevant thermodynamic resource. Surprisingly, however, we prove that in the simplest quantum scenario of a two-dimensional system, this structure is preserved at finite temperatures. We provide the physical interpretation of these results by introducing and analysing the history erasure process, and point out that quantum coherence may be a necessary resource for the existence of an optimal erasure process.

• The extraction of work from quantum coherence
Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, David Jennings
New J. Phys. 18, 023045 (2016)

Abstract: The interplay between quantum-mechanical properties, such as coherence, and classical notions, such as energy, is a subtle topic at the forefront of quantum thermodynamics. The traditional Carnot argument limits the conversion of heat to work; here we critically assess the problem of converting coherence to work. Through a careful account of all resources involved in the thermodynamic transformations within a fully quantum-mechanical treatment, we show that there exist thermal machines extracting work from coherence arbitrarily well. Such machines only need to act on individual copies of a state and can be reused. On the other hand, we show that for any thermal machine with finite resources not all the coherence of a state can be extracted as work. However, even bounded thermal machines can be reused infinitely many times in the process of work extraction from coherence.

• Quantum Coherence, Time-Translation Symmetry, and Thermodynamics
Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph
Phys. Rev. X 5, 021001 (2015)

Abstract: The first law of thermodynamics imposes not just a constraint on the energy-content of systems in extreme quantum regimes, but also symmetry-constraints related to the thermodynamic processing of quantum coherence. We show that this thermodynamic symmetry decomposes any quantum state into mode operators that quantify the coherence present in the state. We then establish general upper and lower bounds for the evolution of quantum coherence under arbitrary thermal operations, valid for any temperature. We identify primitive coherence manipulations and show that the transfer of coherence between energy levels manifests irreversibility not captured by free energy. Moreover, the recently developed thermo-majorization relations on block-diagonal quantum states are observed to be special cases of this symmetry analysis.

Popular summary: The remarkable discovery that energy at microscopic scales often comes in discrete chunks originated in Planck's attempt to understand the way that hot bodies glow. Thus began the long and intimate relationship between the field of thermodynamics, which explores our ability to manipulate heat and other energy transfers between macroscopic systems, and quantum mechanics, which explains the dynamics of individual microscopic systems. Even as both our technology and our theoretical investigations have extended to ever-smaller devices, our understanding of quantum effects on thermodynamics has remained almost exclusively limited to the quantized nature of energy. There is much more, however, to quantum theory than energy quantization; here our focus has been the property of quantum coherence the ability of quantum systems to emulate Schroedinger's cat and somehow be neither "dead and alive" nor "dead or alive" but something completely different altogether. We have discovered how to simplify our understanding of the thermal processing of coherence by using the fact that thermodynamical processes obey time-translation symmetry. This enables quantification of the way coherence can play an active role, facilitating the otherwise-impossible unlocking of energy from certain systems. We have conversely found fundamental limitations on how coherence can be irreversibly manipulated, limitations related to those on energy transfer as dictated by the Second Law of thermodynamics. It has long been appreciated that understanding of thermodynamics must be accompanied by an understanding of information theory. Our work provides evidence that to apply the laws of thermodynamics to the smallest systems around us necessitates an understanding of quantum information theory.

• Finite-size effects in quantum thermodynamics
• [Invited] DPG Meeting of the Condensed Matter Section, Regensburg, Germany (2022)
• Optimizing thermalizations
• Quantum Thermodynamics Conference 2022, Belfast, Ireland (2022)
• 25th Annual Conference on Quantum Information Processing, Pasadena, USA (2022)
• Fundamental constraints of quantum thermodynamics in the Markovian regime
• [Invited] Quantum Optics X, Toruń, Poland (2021)
• [Invited] UTS Centre for Quantum Software and Information Seminar, Sydney, Australia (2021)
• Resource-theoretic approach to the thermodynamic arrow of time

(Blackboard talk)

• [Invited] Quantum Information Theory and Mathematical Physics Workshop, Budapest, Hungary (2019)
• Avoiding irreversibility: lossless interconversion of quantum resources
• [Invited] X Jubilee Symposium KCIK, Sopot, Poland (2019)
• Quantum Information & Chaos Seminar, Jagiellonian University, Kraków, Poland (2018)
• AIP Congress, Perth, Australia (2018)
• Island Physics Conference, Magnetic Island, Australia (2018)
• Beyond the thermodynamic limit
• Asian Quantum Information Science Conference, Nagoya University, Japan (2018)
• Center for Theoretical Physics Seminar, Polish Academy of Sciences, Poland (2017)
• Quantum Information & Chaos Seminar, Jagiellonian University, Poland (2017)
• [Invited] Quantum Foundations and Beyond Symposium, National Quantum Information Centre, Sopot, Poland (2017)
• Quantum Science Group Seminar, University of Sydney, Australia (2017)
• The extraction of work from quantum coherence
• Scientific meeting of PhD students, Wrocław University of Technology, Poland (2016)
• Quantum information and thermodynamics: a resource-theoretic approach
• Quantum Optics and Laser Science Group Seminar, Imperial College London, United Kingdom (2016)
• Takahiro Sagawa's Group Seminar, University of Tokyo, Japan (2016)
• Quantum Science Group Seminar, University of Sydney, Australia (2016)
• Coherence-Correlations-Complexity Seminar, Wrocław University of Technology, Poland (2015)
• Quantum Coherence, Time-Translation Symmetry, and Thermodynamics
• APS March Meeting, Baltimore, USA (2016)
• 4th International Workshop on the Optical Properties of Nanostructures, Wrocław, Poland (2016)
• Quantum Information Theory Seminar, ICFO Barcelona, Spain (2016)
• Symposium on Quantum Coherence, University of Ulm, Germany (2015)
• Quantum Information Theory Seminar, ETH Zurich, Switzerland (2015)
• 7th Colleges of London Quantum Information Meeting, Imperial College London, United Kingdom (2014)
• Avoiding irreversibility: resonant conversion of quantum resources
• EQUS Annual Workshop, Perth, Australia (2018)
• Work extraction from quantum coherence
• 3rd Quantum Thermodynamics Conference, Porquerolles, France (2015)
• Postgraduate Research Symposium at Imperial College London, London, United Kingdom (2015)
• Quantum Coherence, Time-Translation Symmetry, and Thermodynamics
• 18th Conference on Quantum Information Processing, Sydney, Australia (2015)
• 2nd Quantum Thermodynamics Conference, Mallorca, Spain (2015)