My research
Broad scientific interests
I am interested in all areas of quantum information theory, with a particular emphasis on quantum thermodynamics, decoherence theory, and quantum computing.
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 singlethemed series of 15 publications entitled Optimizing quantum information processing under constraints.
Current focus
Currently I am working on quantum algorithms for faulttolerant quantum computing.
Areas of research
(To see a full list of publications and presentations in a chronological order check my Scientific CV)
Quantum thermodynamics
Coherence and symmetry
Other quantum information
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 resourcetheoretic 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 resourcetheoretic) transformations of finitesize systems, and in particular the effect this nonasymptotic regime has on reversible transformations.
− Publications

Quantum dichotomies and coherent thermodynamics beyond firstorder asymptotics
Patryk LipkaBartosik, Christopher T. Chubb, Joseph M. Renes, Marco Tomamichel, Kamil Korzekwa
arXiv:2303.05524 (2023)PDF
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+ Popular Summary
Abstract: We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel \(\cal{E}\) mapping \(\rho_1^{\otimes n}\) into \(\rho_2^{\otimes R_nn}\) with an error \(\epsilon_n\) (measured by trace distance) and \(\sigma_1^{\otimes n}\) into \(\sigma_2^{\otimes R_n n}\) exactly, for a large number \(n\). We derive secondorder asymptotic expressions for the optimal transformation rate \(R_n\) in the small, moderate, and large deviation error regimes, as well as the zeroerror regime, for an arbitrary pair \((\rho_1,\sigma_1)\) of initial states and a commuting pair \((\rho_2,\sigma_2)\) of final states. We also prove that for \(\sigma_1\) and \(\sigma_2\) given by thermal Gibbs states, the derived optimal transformation rates in the first three regimes can be attained by thermal operations. This allows us, for the first time, to study the secondorder asymptotics of thermodynamic state interconversion with fully general initial states that may have coherence between different energy eigenspaces. Thus, we discuss the optimal performance of thermodynamic protocols with coherent inputs and describe three novel resonance phenomena allowing one to significantly reduce transformation errors induced by finitesize effects. What is more, our result on quantum dichotomies can also be used to obtain, up to secondorder asymptotic terms, optimal conversion rates between pure bipartite entangled states under local operations and classical communication.
Popular summary: Statistical inference forms the backbone of scientific progress. Knowing which statistical model is more informative, giving us better insight into reality, is thus of paramount importance. Although classical statistics is a wellestablished field, with the recent progress of quantum technologies the demand for understanding quantum statistical models is growing. In our work, we address this pressing issue by developing a rigorous mathematical framework that allows one to compare the informativeness of quantum dichotomies, which can be seen as basic building blocks of more complex statistical models. As a direct application, we show how to apply our results in the field of quantum thermodynamics to study the fundamental limitations on the thermodynamic processing of quantum systems prepared in coherent superpositions of different energy states. To assess whether one quantum dichotomy is more informative than the other, we verify whether the latter can be deduced from the former via the means of quantum processing. More precisely, we find the optimal rate at which many instances of the latter model can be reconstructed from instances of the former model with a given reconstruction error. We then translate these statistical results into the language of thermodynamics, which allows us to study fully quantum versions of thermodynamic protocols. In particular, we discover a novel coherent resonance phenomenon that exploits quantum coherence to increase the reversibility of a thermodynamic transformation. This contribution paves the way for further harnessing the power of quantum statistical inference tools for the development of quantum technologies, especially for the optimal design of quantum thermodynamic protocols.

Thermal recall: Memoryassisted Markovian thermal processes
Jakub Czartowski, A. de Oliveira Junior, Kamil Korzekwa
PRX Quantum 4, 040304 (2023)PDF
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+ Popular Summary
Abstract: We develop a resourcetheoretic framework that allows one to bridge the gap between two approaches to quantum thermodynamics based on Markovian thermal processes (which model memoryless dynamics) and thermal operations (which model arbitrarily nonMarkovian dynamics). Our approach is built on the notion of memoryassisted Markovian thermal processes, where memoryless thermodynamic processes are promoted to nonMarkovianity by explicitly modelling ancillary memory systems initialised in thermal equilibrium states. Within this setting, we propose a family of protocols composed of sequences of elementary twolevel thermalisations that approximate all transitions between energyincoherent states accessible via thermal operations. We prove that, as the size of the memory increases, these approximations become arbitrarily good for all transitions in the infinite temperature limit, and for a subset of transitions in the finite temperature regime. Furthermore, we present solid numerical evidence for the convergence of our protocol to any transition at finite temperatures. We also explain how our framework can be used to quantify the role played by memory effects in thermodynamic protocols such as work extraction. Finally, our results show that elementary control over two energy levels at a given time is sufficient to generate all energyincoherent transitions accessible via thermal operations if one allows for ancillary thermal systems.
Popular summary: The traditional approach to thermodynamics assumes a “drop of water vs. ocean” situation  a small system embedded within a large heat bath, which is virtually unaffected by the evolution of the former. Since the state of the bath is stationary, it cannot depend on the system’s past, thus rendering it memoryless. However, fundamentally, there is a possibility of building up arbitrarily strong correlations between the system and the bath, leading to memory effects. This view has been recently explored in the quest for finding ultimate limits of thermodynamic processing of quantum systems. Therefore, there are currently two disjoint approaches to quantum thermodynamics (with no memory and with a total recall), and our contribution provides a framework that bridges the gap between the two. We approached the problem with a simple assumption in mind  it is enough to have additional control of some part of the heat bath, treating it as a finitesize memory, and perform memoryless operations on such a tandem. Based on this and using both analytical and numerical methods we demonstrated how the range of possible thermodynamic transformations grows as the size of the memory increases. As a result, our framework allows one to investigate the role played by memory effects in thermodynamic tasks, and to quantify the advantage brought by it. We believe that our work can serve as a cornerstone for future investigations into the tradeoff between the cost of memory control and performance improvements in thermodynamic processes, and thus can help in designing optimal quantum thermal machines.

Geometric structure of thermal cones
A. de Oliveira Junior, Jakub Czartowski, Karol Życzkowski, Kamil Korzekwa
Phys. Rev. E 106, 064109 (2022)PDF
+ Abstract
Abstract: The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The socalled 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 majorisationbased 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)PDF
+ Abstract
Editors' Suggestion
Featured in Physics
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 outofequilibrium 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)PDF
+ Abstract
Editors' Suggestion
Featured in Physics
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 nonequilibrium 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.

Fluctuationdissipation relations for thermodynamic distillation processes
Tanmoy Biswas, A. de Oliveira Junior, Michał Horodecki, Kamil Korzekwa
Phys. Rev. E 105, 054127 (2022)PDF
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+ Popular Summary
Abstract: The fluctuationdissipation 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 resourcetheoretic 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 energyincoherent systems, and allow not only for a state transformation, but also for the change of Hamiltonian. The fluctuationdissipation relations we derive enable us to find the optimal performance of thermodynamic protocols such as work extraction, information erasure and thermodynamicallyfree communication, up to secondorder 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 fluctuationdissipation 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 nonequilibrium 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 informationtheoretic tools can be potentially used to explore fluctuationdissipation relations for entanglement and other quantum resources.

Work fluctuations due to partial thermalizations in twolevel systems
Maria Quadeer, Kamil Korzekwa, Marco Tomamichel
Phys. Rev. E 103, 042141 (2021)PDF
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Abstract: We study work extraction processes mediated by finitetime interactions with an ambient bath — partial thermalizations — as continuous time Markov processes for twolevel 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 twolevel 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 fluctuationfree 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 fluctuationdissipation 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 finitetime 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)PDF
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+ Popular Summary
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 finitesize 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 tradeoff 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 intermediatescale 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 finitesize 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 finitesize 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 intermediatescale quantum devices and thermal machines developed in the near term.

Moderate deviation analysis of majorisationbased resource interconversion
Christopher T. Chubb, Marco Tomamichel, Kamil Korzekwa
Phys. Rev. A 99, 032332 (2019)PDF
+ Abstract
Abstract: We consider the problem of interconverting a finite amount of resources within all theories whose singleshot 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 energyincoherent transformations). When only finite resources are available we expect to see a nontrivial tradeoff 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 tradeoff in the socalled 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: finitesize corrections to state interconversion rates
Christopher T. Chubb, Marco Tomamichel, Kamil Korzekwa
Quantum 2, 108 (2018)
PDF
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+ Popular Summary
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 finitesize systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energyincoherent 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 socalled secondorder asymptotics provide a bridge between the extreme cases of singleshot 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 finitesize 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 finitesize 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 socalled 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 informationtheoretic framework describing thermodynamic transformations of finitesize 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 finitesize 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)
PDF
+ Abstract
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 infinitetemperature 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 twodimensional 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)
PDF
+ Abstract
Highlights of 2016
Abstract: The interplay between quantummechanical 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 quantummechanical 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, TimeTranslation Symmetry, and Thermodynamics
Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph
Phys. Rev. X 5, 021001 (2015)
PDF
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+ Popular Summary
Featured in Physics
Abstract: The first law of thermodynamics imposes not just a constraint on the energycontent of systems in extreme quantum regimes, but also symmetryconstraints 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 thermomajorization relations on blockdiagonal 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 eversmaller 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 timetranslation symmetry. This enables quantification of the way coherence can play an active role, facilitating the otherwiseimpossible 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.
+ Oral Presentations

Quantum dichotomies and coherent thermodynamics beyond first order asymptotics
Slides 1
Slides 2
+ Presented at
 [Invited] 48th Congress of Polish Physicists, Gdańsk, Poland (2023)
 Asian Quantum Information Science Conference, Seoul, South Korea (2023)

Coherent thermodynamics beyond first order asymptotics
Slides
+ Presented at
 TEAMNET Workshop, Kroczyce, Poland (2022)

Finitesize effects in quantum thermodynamics
Slides 1
Slides 2
+ Presented at
 [Invited] Quantum Resources Workshop, Singapore (2023)
 CoherenceCorrelationsComplexity seminar, Wrocław University of Technology, Poland (2023)
 Center for Theoretical Physics seminar, Polish Academy of Sciences, Poland (2023)
 Quantum Information Seminar, University of Warsaw, Warsaw, Poland (2022)
 [Invited] DPG Meeting of the Condensed Matter Section, Regensburg, Germany (2022)

Optimizing thermalizations
Slides
Talk
+ Presented at
 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
Slides 1
Talk 1
Slides 2
Talk 2
+ Presented at
 [Invited] Quantum Optics X, Toruń, Poland (2021)
 [Invited] UTS Centre for Quantum Software and Information Seminar, Sydney, Australia (2021)

Resourcetheoretic approach to the thermodynamic arrow of time
(Blackboard talk)
+ Presented at
 [Invited] Quantum Information Theory and Mathematical Physics Workshop, Budapest, Hungary (2019)

Avoiding irreversibility: lossless interconversion of quantum resources
Slides
+ Presented at
 [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
Slides 1
Slides 2
Slides 3
+ Presented at
 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
Slides
+ Presented at
 Scientific meeting of PhD students, Wrocław University of Technology, Poland (2016)

Quantum information and thermodynamics: a resourcetheoretic approach
Slides
+ Presented at
 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)
 CoherenceCorrelationsComplexity Seminar, Wrocław University of Technology, Poland (2015)

Quantum Coherence, TimeTranslation Symmetry, and Thermodynamics
Slides
+ Presented at
 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)
+ Poster Presentations

Quantum dichotomies and coherent thermodynamics beyond first order asymptotics
Poster
+ Presented at
 18th Conference on the Theory of Quantum Computation, Communication and Cryptography, Aveiro, Portugal (2023)
 26th Conference on Quantum Information Processing, Ghent, Belgium (2023)

Avoiding irreversibility: resonant conversion of quantum resources
Poster
+ Presented at
 EQUS Annual Workshop, Perth, Australia (2018)

Work extraction from quantum coherence
Poster
+ Presented at
 3rd Quantum Thermodynamics Conference, Porquerolles, France (2015)
 Postgraduate Research Symposium at Imperial College London, London, United Kingdom (2015)

Quantum Coherence, TimeTranslation Symmetry, and Thermodynamics
Poster
+ Presented at
 18th Conference on Quantum Information Processing, Sydney, Australia (2015)
 2nd Quantum Thermodynamics Conference, Mallorca, Spain (2015)
Although quantum mechanics owes its name to the quantisation of energy, postulated at the beginning of the 20th century by the old quantum theory, its counterintuitive “weirdness” comes from the superposition principle that was discovered a quarter of a century later. Concepts such as Schrödinger’s cat, quantum teleportation and quantum computing all stem from this fundamental rule of the quantum realm: given two valid states of a system, e.g., spinup and spindown, their coherent superposition is also a valid state.
Beyond the role that quantum coherence plays in thermodynamics, I am also interested in characterizing it as a resource in general quantum information scenarios. This involves studies on optimal processing of coherence (i.e., finding the ultimate limits of avoiding decoherence processes that deteriorate quantum information), understanding how it affects the nature of reversible processes and investigating the role quantum coherence plays in distinguishing physical states and processes.
− Publications

Perfect quantum protractors
Michał Piotrak, Marek Kopciuch, Arash Dezhang Fard, Magdalena Smolis, Szymon Pustelny, Kamil Korzekwa
arXiv:2310.13045 (2023)PDF
+ Abstract
Abstract: In this paper we introduce and investigate the concept of a perfect quantum protractor, a pure quantum state \(\psi\rangle\in\mathcal{H}\) that generates three different orthogonal bases of \(\mathcal{H}\) under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variancebased measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a welldefined total angular momentum \(j\), and we prove that they do not exist for \(j\in\{1/2,2,5/2\}\), but they do exist for \(j\in\{1,3/2,3\}\) (with numerical evidence for their existence when \(j=7/2\). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not \(\textit{a priori}\) known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium87, where we prepare a perfect quantum protractor for a spin1 system, let it precess around \(x\), \(y\) or \(z\) axis, and then employ it to optimally estimate the rotation angle.

Quantumembeddable stochastic matrices
Fereshte Shahbeigi, Christopher T. Chubb, Ryszard Kukulski, Łukasz Pawela, Kamil Korzekwa
arXiv:2305.17163 (2023)PDF
+ Abstract
Abstract: The classical embeddability problem asks whether a given stochastic matrix \(T\), describing transition probabilities of a \(d\)level system, can arise from the underlying homogeneous continuoustime Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given \(T\). More precisely, we aim at characterising the set of quantumembeddable stochastic matrices that arise from memoryless continuoustime quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantumembeddable but not classicallyembeddable, as well as families of stochastic matrices that are not quantumembeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a nonzero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantumembeddable.

Resource engines
Hanna WojewódkaŚciążko, Zbigniew Puchała, Kamil Korzekwa
Quantum 8, 1222 (2024)PDF
+ Abstract
Abstract: In this paper we aim to push the analogy between thermodynamics and quantum resource theories one step further. Previous inspirations were based on thermodynamic considerations concerning scenarios with a single heat bath, neglecting an important part of thermodynamics that studies heat engines operating between two baths at different temperatures. Here, we investigate the performance of resource engines, which replace the access to two heat baths at different temperatures with two arbitrary constraints on state transformations. The idea is to imitate the action of a twostroke heat engine, where the system is sent to two agents (Alice and Bob) in turns, and they can transform it using their constrained sets of free operations. We raise and address several questions, including whether or not a resource engine can generate a full set of quantum operations or all possible state transformations, and how many strokes are needed for that. We also explain how the resource engine picture provides a natural way to fuse two or more resource theories, and we discuss in detail the fusion of two resource theories of thermodynamics with two different temperatures, and two resource theories of coherence with respect to two different bases.

Encoding classical information into quantum resources
Kamil Korzekwa, Zbigniew Puchała, Marco Tomamichel, Karol Życzkowski
IEEE Trans. Inf. Theory 68, 4518 (2022)
PDF
+ Abstract
Abstract: We introduce and analyse the problem of encoding classical information into different resources of a quantum state. More precisely, we consider a general class of communication scenarios characterised by encoding operations that commute with a unique resource destroying map and leave free states invariant. Our motivating example is given by encoding information into coherences of a quantum system with respect to a fixed basis (with unitaries diagonal in that basis as encodings and the decoherence channel as a resource destroying map), but the generality of the framework allows us to explore applications ranging from superdense coding to thermodynamics. For any state, we find that the number of messages that can be encoded into it using such operations in a oneshot scenario is upperbounded in terms of the information spectrum relative entropy between the given state and its version with erased resources. Furthermore, if the resource destroying map is a twirling channel over some unitary group, we find matching oneshot lowerbounds as well. In the asymptotic setting where we encode into many copies of the resource state, our bounds yield an operational interpretation of resource monotones such as the relative entropy of coherence and its corresponding relative entropy variance.

Dephasing superchannels
Zbigniew Puchała, Kamil Korzekwa, Roberto Salazar, Paweł Horodecki, Karol Życzkowski
Phys. Rev. A 104, 052611 (2021)
PDF
+ Abstract
Editors' Suggestion
Abstract: We characterise a class of environmental noises that decrease coherent properties of quantum channels by introducing and analysing the properties of dephasing superchannels. These are defined as superchannels that affect only nonclassical properties of a quantum channel \(\mathcal{E}\), i.e., they leave invariant the transition probabilities induced by \(\mathcal{E}\) in the distinguished basis. We prove that such superchannels \(\Xi_C\) form a particular subclass of Schurproduct supermaps that act on the Jamiołkowski state \(J(\mathcal{E})\) of a channel \(\mathcal{E}\) via a Schur product, \(J'=J\circ C\). We also find physical realizations of general \(\Xi_C\) through a pre and postprocessing employing dephasing channels with memory, and show that memory plays a nontrivial role for quantum systems of dimension \(d>2\). Moreover, we prove that coherence generating power of a general quantum channel is a monotone under dephasing superchannels. Finally, we analyse the effect dephasing noise can have on a quantum channel \(\mathcal{E}\) by investigating the number of distinguishable channels that \(\mathcal{E}\) can be mapped to by a family of dephasing superchannels. More precisely, we upper bound this number in terms of hypothesis testing channel divergence between \(\mathcal{E}\) and its fully dephased version, and also relate it to the robustness of coherence of \(\mathcal{E}\).

Quantum advantage in simulating stochastic processes
Kamil Korzekwa, Matteo Lostaglio
Phys. Rev. X 11, 021019 (2021)
PDF
+ Abstract
+ Popular Summary
Popular summary: For the past three decades, scientists have been harnessing the quantum features of nature to challenge the bestknown classical methods of computing, metrology, and cryptography. In all these cases, quantum superposition allows one to perform certain tasks faster, more precisely, or more securely. Here, we report and investigate a novel quantum advantage that reduces the amount of memory and time needed to implement certain dynamical processes, such as thermodynamic cooling or information processing. In our work, we introduce a unifying framework to theoretically analyze memory and time costs from a mathematical, computational, and thermodynamical point of view, and we apply it to study three distinct scenarios where quantum advantages arise. First, we show that there exist stochastic processes that cannot be simulated classically without memory but can be implemented “quantumly” in a memoryless fashion. Second, we prove that for processes that require memory even in the quantum regime, there is an improvement over classical computers in the number of times the memory must be accessed or in its size. Third, we demonstrate that memoryless quantum processes allow for much better control of the system’s state than their classical counterparts. Our work provides new tools to quantify the extent to which quantum superposition can replace the role that classically is played by memory. This shows that quantum mechanics provides powerful models to simulate stochastic processes and paves the way to the investigation of practical advantages for quantum information processing.
Abstract: We investigate the problem of simulating classical stochastic processes through quantum dynamics, and present three scenarios where memory or time quantum advantages arise. First, by introducing and analysing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory. Second, by extending the notion of spacetime cost of a stochastic process \(P\) to the quantum domain, we prove an advantage of the quantum cost of simulating \(P\) over the classical cost. Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to a potential advantage in cooling protocols.

Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory
Cristina Cirstoiu, Kamil Korzekwa, David Jennings
Phys. Rev. X 10, 041035 (2020)
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+ Abstract
+ Popular Summary
Abstract: To what extent does Noether's principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law, and show that this relates to physically impossible transformations in quantum theory, such as timereversal and spininversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group \(G\) becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel. In particular, we investigate in detail the U(1) and SU(2) symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin\(j_A\) system can be used to polarise a larger spin\(j_B\) system, and on how much one can invert spin polarisation using a rotationallysymmetric operation. Finally, we also establish novel links between unitarity, complementary channels and purity that are of independent interest.
Popular summary: Noether’s theorem is a celebrated result showing that continuous symmetries are intimately related to conservation laws. For example, the fact that no point in time is special leads to energy conservation, while rotational symmetry results in conservation of angular momentum. These important relations placed symmetries at the forefront of modern physics. However, their applicability is mostly constrained to classical and quantum systems that are isolated. In this work we address the fundamental question: to what degree does Noether’s principle hold for open (nonisolated) quantum systems that possess a continuous symmetry? Consequently, we quantify the admissible disconnects that occur in quantum theory between conservation laws and symmetries of open systems, and uncover their highly nontrivial structure that is related to physically impossible quantum processes, such as spininversion and timereversal. Our analysis strongly relates to core results from entanglement theory and recent theories of coherent resources, and uses techniques from quantum information theory that have been remarkably successful in quantifying general limitations on quantum processing tasks. We obtain novel structural results that describe the set of all quantum operations respecting a symmetry principle. Employing these, we derive general tradeoff relations between deviations from conservation laws and openness of quantum dynamics that obeys a given symmetry. Besides providing fundamental insights into the structure of quantum theory, our results are of relevance for a range of research areas, such as open system dynamics, quantum information science and the development of quantum technologies. Moreover, our technical results on reversibility of quantum dynamics can find applications in randomized benchmarking of quantum information processing devices and decoherence theory.

Distinguishing classically indistinguishable states and channels
Kamil Korzekwa, Stanisław Czachórski, Zbigniew Puchała, Karol Życzkowski
J. Phys. A: Math. Theor. 52, 475303 (2019)
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+ Abstract
Abstract: We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between \(M\) quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of \(M\) quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to timeenergy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of \(M\) perfectly distinguishable states (channels) that are classically indistinguishable.

Coherifying quantum channels
Kamil Korzekwa, Stanisław Czachórski, Zbigniew Puchała, Karol Życzkowski
New J. Phys. 20, 043028 (2018)
PDF
+ Abstract
Abstract: Is it always possible to explain random stochastic transitions between states of a finitedimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel \(\Phi\), i.e., we look for channels \(\Phi^{\mathcal{C}}\) that induce the same classical transitions \(T\), but are "more coherent". To quantify the coherence of a channel \(\Phi\) we measure the coherence of the corresponding Jamiołkowski state \(J_{\Phi}\). We show that the classical transition matrix \(T\) can be coherified to reversible unitary dynamics if and only if \(T\) is unistochastic. Otherwise the Jamiołkowski state \(J_\Phi^{\mathcal{C}}\) of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To asses the extent to which an optimal process \(\Phi^{\mathcal{C}}\) is indeterministic we find explicit bounds on the entropy and purity of \(J_\Phi^{\mathcal{C}}\), and relate the latter to the unitarity of \(\Phi^{\mathcal{C}}\). We also find optimal coherifications for several classes of channels, including all onequbit channels. Finally, we provide a nonoptimal coherification procedure that works for an arbitrary channel \(\Phi\) and reduces its rank (the minimal number of required Kraus operators) from \(d^2\) to \(d\).

Markovian evolution of quantum coherence under symmetric dynamics
Matteo Lostaglio, Kamil Korzekwa, Antony Milne
Phys. Rev. A 96, 032109 (2017)
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+ Abstract
+ Popular Summary
Abstract: Both conservation laws and practical restrictions impose symmetry constraints on the dynamics of open quantum systems. In the case of timetranslation symmetry, which arises naturally in many physically relevant scenarios, the quantum coherence between energy eigenstates becomes a valuable resource for quantum information processing. In this work we identify the minimum amount of decoherence compatible with this symmetry for a given population dynamics. This yields a generalisation to higherdimensional systems of the relation \(T_2\leq 2T_1\) for qubit decoherence and relaxation times. It also enables us to witness and assess the role of nonMarkovianity as a resource for coherence preservation and transfer. Moreover, we discuss the relationship between ergodicity and the ability of Markovian dynamics to indefinitely sustain a superposition of different energy states. Finally, we establish a formal connection between the resourcetheoretic and the master equation approaches to thermodynamics, with the former being a nonMarkovian generalisation of the latter. Our work thus brings the abstract study of quantum coherence as a resource towards the realm of actual physical applications.
Popular summary: Quantum bits differ substantially from classical ones since, like Schrödinger’s cat, they can exist in a superposition of two different states. This wavelike nature is a key ingredient of quantum computing, but it is very fragile and can easily be destroyed. Whereas one could keep track of the intermediate steps of computation performed on a classical computer without interfering with the calculation itself, in a quantum computer superpositions must be hidden until the final result is obtained. A major goal is then to prevent the leakage of quantum information from the computer, a process known as "decoherence". Here we provide a general framework to quantify the minimal decoherence experienced by a quantum mechanical system. Our main result is a minimal decoherence theorem that gives a fundamental upper bound on the amount of superposition that can be preserved when energy leaks from a system into the surrounding environment. The theorem has a range of consequences, since it generalises the notion of relaxation and decoherence times used in NMR spectroscopy, allows us to understand the role of memory effects in decoherence processes, and identifies what kind of energy flows are compatible with the preservation of superpositions. In this work we show how novel quantum information tools characterising superposition as a resource can be employed to study decoherence processes, thus bridging the gap between abstract theory and the realm of physical applications.
+ Oral Presentations

Quantum advantage in simulating stochastic processes
Slides
+ Presented at
 Distinguished talk at the Annual Ingarden session on Quantum Information, Sopot, Poland (2020)
 Quantum Information & Chaos Seminar, Jagiellonian University, Kraków, Poland (2020)
 [Invited] The Quantum and Complexity Science Initiative Seminar, Nanyang Technological University, Singapore (2020)

Encoding classical information in quantum resources
Slides
Talk
+ Presented at
 CTP Quantum Information Days, Warsaw, Poland (2021)
 Beyond IID in Information Theory 8, Palo Alto, USA (2020)
 15th Conference on the Theory of Quantum Computation, Communication and Cryptography, Riga, Latvia (2020)
 Quantum Information & Chaos Seminar, Jagiellonian University, Kraków, Poland (2020)

Coherifying quantum states and channels
Slides
+ Presented at
 Centre for Quantum Software and Information Seminar, University of Technology Sydney, Sydney, Australia (2018)
 Monash Quantum Information Science Seminar, Monash University, Melbourne Australia (2018)
 QSciTech Research Group Seminar, Macquarie University, Sydney, Australia (2018)

On time evolution of coherences and populations
Slides
+ Presented at
 Quantum Information & Chaos Seminar, Jagiellonian University, Kraków, Poland (2016)
+ Poster Presentations

Coherifying quantum channels
Poster
+ Presented at
 13th Conference on the Theory of Quantum Computation, Communication and Cryptography, Sydney, Australia (2018)

Markovian evolution of quantum coherence under symmetric dynamics
Poster
+ Presented at
 21st Conference on Quantum Information Processing, Delft, Netherlands (2018)
 EQuS Annual Workshop, Hunter Valley, Australia (2017)
During my career I worked on a number of quantum information projects that are not directly related to quantum thermodynamics or coherence/symmetry considerations. At the earliest stage I was developing models for spin dynamics in pdoped semiconductor nanostructures subject to a magnetic field. Apart from the general interest in understanding the often nontrivial kinetics of spin precession and decoherence, the optical studies of such structures are motivated by possible applications in spintronics and spinbased quantum information processing. I also got interested in the problem of quantum state transfer in spin chains, which were proposed as potential candidates for shortrange communication channels providing interqubit communication during more complex quantum computation processes.
Later, I investigated novel forms of quantum uncertainty relations, focusing mainly on the interplay between quantum and classical uncertainty. The motivation behind this line of research is that most quantum states arising in theoretical considerations (e.g., in entanglement theory) and the ones used to describe quantum systems prepared in laboratories are mixed, and so a proper understanding of the interplay between classical lack of knowledge and purely quantum uncertainty coming from coherence is crucial for quantum information based technologies.
More recently, I spent some time studying the structure of unistochastic matrices, i.e., trying to understand what kind of transition matrices can be induced by reversible unitary dynamics. This abstract mathematical problem not only allows one to characterise random processes with a potentially deterministic cause, but its solution could also be employed to the studies of quantum walks on graphs. I also started working on designing classical algorithms to simulate quantum circuits. On the application side, this can provide efficient tools to verify the performance of nearterm intermediate scale (NISQ) quantum devices; and on fundamental side, research into the classical simulation of quantum circuits is a means of studying and quantifying the distinction between the computational power of quantum and classical computers.
− Publications

Improved simulation of quantum circuits dominated by free fermionic operations
Oliver ReardonSmith, Michał Oszmaniec, Kamil Korzekwa
arXiv:2307.12702 (2023)PDF
+ Abstract
Featured in Quanta
Abstract: We present a classical algorithm for simulating universal quantum circuits composed of "free" nearestneighbour matchgates or equivalently fermioniclinearoptical (FLO) gates, and "resourceful" nonGaussian gates. We achieve the promotion of the efficiently simulable FLO subtheory to universal quantum computation by gadgetizing controlled phase gates with arbitrary phases employing nonGaussian resource states. Our key contribution is the development of a novel phasesensitive algorithm for simulating FLO circuits. This allows us to decompose the resource states arising from gadgetization into free states at the level of statevectors rather than density matrices. The runtime cost of our algorithm for estimating the Bornrule probability of a given quantum circuit scales polynomially in all circuit parameters, except for a linear dependence on the newly introduced FLO extent, which scales exponentially with the number of controlledphase gates. More precisely, as a result of finding optimal decompositions of relevant resource states, the runtime doubles for every maximally resourceful (e.g., swap or CZ) gate added. Crucially, this cost compares very favourably with the best known prior algorithm, where each swap gate increases the simulation cost by a factor of approximately 9. For a quantum circuit containing arbitrary FLO unitaries and \(k\) controlledZ gates, we obtain an exponential improvement \(O(4.5k)\) over the prior stateoftheart.

Fast estimation of outcome probabilities for quantum circuits
Hakop Pashayan, Oliver ReardonSmith, Kamil Korzekwa, Stephen D. Bartlett
PRX Quantum 3, 020361 (2022)PDF
+ Abstract
+ Popular Summary
Abstract: We present two classical algorithms for the simulation of universal quantum circuits on \(n\) qubits constructed from \(c\) instances of Clifford gates and \(t\) arbitraryangle \(Z\)rotation gates such as \(T\) gates. Our algorithms complement each other by performing best in different parameter regimes. The \(\tt{Estimate}\) algorithm produces an additive precision estimate of the Born rule probability of a chosen measurement outcome with the only source of runtime inefficiency being a linear dependence on the stabilizer extent (which scales like \(\approx 1.17t\) for \(T\) gates). Our algorithm is stateoftheart for this task: as an example, in approximately 25 hours (on a standard desktop computer), we estimated the Born rule probability to within an additive error of 0.03, for a 50 qubit, 60 nonClifford gate quantum circuit with more than 2000 Clifford gates. The \(\tt{Compute}\) algorithm calculates the probability of a chosen measurement outcome to machine precision with runtime \(O(2t−r(t−r)t)\) where \(r\) is an efficiently computable, circuitspecific quantity. With high probability, \(r\) is very close to \(\min\{t,n−w\}\) for random circuits with many Clifford gates, where \(w\) is the number of measured qubits. \(\tt{Compute}\) can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+\(T\) circuits with \(n=55\), \(w=5\), \(c=105\) and \(t=80\) \(T\) gates, and then compute the Born rule probability with a runtime consistently less than 104 seconds using a single core of a standard desktop computer. We provide a C+Python implementation of our algorithms.
Popular summary: All known methods of simulating quantum mechanics using classical computers require exponential resources. It is widely believed that this difference is a fundamental one, and that quantum computers can efficiently solve problems for which no efficient classical algorithm exists. However, classical computers are cheaper, faster, more accessible and more reliable than modern quantum computers, and so classical simulation algorithms continue to play a significant role in assessing and benchmarking the performance of quantum devices. In this paper we provide stateoftheart classical algorithms for estimating the outcome probabilities that characterize the output of a quantum computer. Recent works on classical simulations of quantum computers have determined what appears to be a fundamental limit on the cost of running such algorithms, scaling exponentially not in the number of qubits, but in a quantity called "magic", which describes how far a particular operation is from a classical one. We present a classical simulation algorithm that in certain previously inaccessible parameter regimes, permits practical simulation runtimes for ‘typical quantum circuits’. For the cases where this result doesn’t apply e.g., for an adversarial choice of quantum circuit, we develop novel tools that allow us to achieve orders of magnitude improvements in simulation runtime for practically relevant parameter regimes. It is increasingly important to have methods for verifying and validating the outputs of quantum devices and assessing proposals for applications of nearterm quantum devices using trusted classical methods. We expect our algorithms to be useful in this setting.

Algebraic and geometric structures inside the Birkhoff polytope
Grzegorz RajchelMieldzioć, Kamil Korzekwa, Zbigniew Puchała, Karol Życzkowski
J. Math. Phys. 63, 012202 (2022)PDF
+ Abstract
Abstract: The Birkhoff polytope \(\mathcal{B}_d\) consisting of all bistochastic matrices of order \(d\) assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset \(\mathcal{U}_d\) of unistochastic matrices, determined by squared moduli of unitary matrices, is of a particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantised. In order to investigate the problem of unistochasticity we introduce the set \(\mathcal{L}_d\) of bracelet matrices that forms a subset of \(\mathcal{B}_d\), but a superset of \(\mathcal{U}_d\). We prove that for every dimension \(d\) this set contains the set of factorisable bistochastic matrices \(\mathcal{F}_d\) and is closed under matrix multiplication by elements of \(\mathcal{F}_d\). Moreover, we prove that both \(\mathcal{L}_d\) and \(\mathcal{F}_d\) are starshaped with respect to the flat matrix. We also analyse the set of \(d\times d\) unistochastic matrices arising from circulant unitary matrices, and show that their spectra lie inside \(d\)hypocycloids on the complex plane. Finally, applying our results to small dimensions, we fully characterise the set of circulant unistochastic matrices of order \(d\leq 4\), and prove that such matrices form a monoid for \(d=3\).

Classical noise and the structure of minimal uncertainty states
Kamil Korzekwa, Matteo Lostaglio
Phys. Rev. A 93, 062347 (2016)
PDF
+ Abstract
Abstract: Which quantum states minimise the unavoidable uncertainty arising from the noncommutativity of two observables? The immediate answer to such a question is: it depends. Due to the plethora of uncertainty measures there are many answers. Here, instead of restricting our study to a particular measure, we present plausible axioms for the set \(\mathcal{F}\) of bonafide informationtheoretic uncertainty functions. Then, we discuss the existence of states minimising uncertainty with respect to all members of \(\mathcal{F}\), i.e., universal minimum uncertainty states (MUS). We prove that such states do not exist within the full state space and study the effect of classical noise on the structure of minimum uncertainty states. We present an explicit example of a qubit universal MUS that arises when purity is constrained by introducing a threshold amount of noise. For higher dimensional systems we derive several nogo results limiting the existence of noisy universal MUS. However, we conjecture that universality may emerge in an approximate sense. We conclude by discussing connections with thermodynamics, and highlight the privileged role that nonequilibrium free energy \(F_2\) plays close to equilibrium.

Quantum and classical entropic uncertainty relations
Kamil Korzekwa, Matteo Lostaglio, David Jennings, Terry Rudolph
Phys. Rev. A 89, 042122 (2014)
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+ Abstract
Abstract: How much of the uncertainty in predicting measurement outcomes for noncommuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two noncommuting observables into a classical component, and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any stateindependent quantities, but instead admits "puritybased" lower bounds that generalize entropic formulations such as the MaassenUffink relation. These relations reveal a nontrivial interplay between quantum and classical randomness in any finitedimensional state.

Operational constraints on statedependent formulations of quantum errordisturbance tradeoff relations
Kamil Korzekwa, David Jennings, Terry Rudolph
Phys. Rev. A 89, 052108 (2014)
PDF
+ Abstract
+ Popular Summary
Abstract: We argue for an operational requirement that all statedependent measures of disturbance should satisfy. Motivated by this natural criterion, we prove that in any \(d\)dimensional Hilbert space and for any pair of noncommuting operators, \(A\) and \(B\), there exists a set of at least \(2^{d−1}\) zeronoise, zerodisturbance (ZNZD) states, for which the first observable can be measured without noise and the second will not be disturbed. Moreover, we show that it is possible to construct such ZNZD states for which the expectation value of the commutator \([A,B]\) does not vanish. Therefore any statedependent errordisturbance relation, based on the expectation value of the commutator as a lower bound, must violate the operational requirement. We also discuss Ozawa's statedependent errordisturbance relation in light of our results and show that the disturbance measure used in this relation exhibits unphysical properties. We conclude that the tradeoff is inevitable only between stateindependent measures of error and disturbance.
Popular summary: The uncertainty principle lies at the very heart of quantum physics. Its best known formulation, the HeisenbergRobertson inequality, states that two physical properties of any quantum system that are represented by noncommuting observables, cannot simultaneously have well defined values. It has been speculated for a number of years that the socalled errordisturbance relation is a direct consequence of this wellestablished inequality. In its statedependent form it states that given a physical system in a known quantum state, one cannot sequentially perform measurements of two noncommuting observables perfectly. In other words: either the first measurement has to be imperfect (noisy) or the second one has to be disturbed. In this paper, assuming only that a noisy or disturbed measurement should be experimentally distinguishable from a perfect one, we prove that this is actually not the case. We show that for every quantum system there exist quantum states for which it is possible to perform perfect, i.e., noiseless and undisturbed, measurements of two noncommuting observables. This demonstrates that it is not only the measurement process that is responsible for the disturbance to subsequent measurements, but also the lack of knowledge about the initial quantum state. We conclude that in order to properly understand errordisturbance relation, one should seek for a stateindependent formulation that takes into account this lack of knowledge.

Quantumstate transfer in spin chains via isolated resonance of terminal spins
Kamil Korzekwa, Paweł Machnikowski, Paweł Horodecki
Phys. Rev. A 89, 062301 (2014)
PDF
+ Abstract
Abstract: We propose a quantumstate transfer protocol in a spin chain that requires only the control of the spins at the ends of the quantum wire. The protocol is to a large extent insensitive to inhomogeneity caused by local magnetic fields and perturbation of exchange couplings. Moreover, apart from the free evolution regime, it allows one to induce an adiabatic spin transfer, which provides the possibility of performing the transfer on demand. We also show that the amount of information leaking into the central part of the chain is small throughout the whole transfer process (which protects the information sent from being eavesdropped) and can be controlled by the magnitude of the external magnetic field.

Spin dynamics in \(p\)doped semiconductor nanostructures subject to a magnetic field tilted from the Voigt geometry
K. Korzekwa, C. Gradl, M. Kugler, S. Furthmeier, M. Griesbeck, M. Hirmer, D. Schuh, W. Wegscheider, T. Kuhn, C. Schüller, T. Korn, P. Machnikowski
Phys. Rev. B 88, 155303 (2013)
PDF
+ Abstract
Editors' Suggestion
Abstract: We develop a theoretical description of the spin dynamics of resident holes in a pdoped semiconductor quantum well (QW) subject to a magnetic field tilted from the Voigt geometry. We find the expressions for the signals measured in timeresolved Faraday rotation (TRFR) and resonant spin amplification (RSA) experiments and study their behavior for a range of system parameters. We find that an inversion of the RSA peaks can occur for long hole spin dephasing times and tilted magnetic fields. We verify the validity of our theoretical findings by performing a series of TRFR and RSA experiments on a \(p\)modulation doped GaAs/Al\(_{0.3}\)Ga\(_{0.7}\)As single QW and showing that our model can reproduce experimentally observed signals.

Spin dynamics in twodimensional electron and hole systems revealed by resonant spin amplification
T. Korn, M. Griesbeck, M. Kugler, S. Furthmeier, C. Gradl, M. Hirmer, D. Schuh, W. Wegscheider, K. Korzekwa, P. Machnikowski, T. Kuhn, M.M. Glazov, E.Ya. Sherman,C. Schüller
Proc. SPIE 8461, Spintronics V, 84610O (2012)
PDF
+ Abstract
Abstract: Understanding and controlling the spin dynamics in semiconductor heterostructures is a key requirement for the design of future spintronics devices. In GaAsbased heterostructures, electrons and holes have very different spin dynamics. Some control over the spinorbit fields, which drive the electron spin dynamics, is possible by choosing the crystallographic growth axis. Here, (110)grown structures are interesting, as the Dresselhaus spinorbit fields are oriented along the growth axis and therefore, the typically dominant DyakonovPerel mechanism is suppressed for spins oriented along this axis, leading to long spin depasing times. By contrast, hole spin dephasing is typically very rapid due to the strong spinorbit interaction of the plike valence band states. For localized holes, however, most spin dephasing mechanisms are suppressed, and long spin dephasing times may be observed. Here, we present a study of electron and hole spin dynamics in GaAsAlGaAsbased quantum wells. We apply the resonant spin amplification (RSA) technique, which allows us to extract all relevant spin dynamics parameters, such as \(g\) factors and dephasing times with high accuracy. A comparison of the measured RSA traces with the developed theory reveals the anisotropy of the spin dephasing in the (110)grown twodimensional electron systems, as well as the complex interplay between electron and hole spin and carrier dynamics in the twodimensional hole systems.

Tunneling transfer protocol in a quantum dot chain immune to inhomogeneity
Kamil Korzekwa, Paweł Machnikowski
Acta Phys. Pol. A, 120, 859861 (2011)
PDF
+ Abstract
Abstract: We propose a quantum dot (QD) implementation of a quantum state transfer channel. The proposed channel consists of \(N\) vertically stacked QDs with the nearest neighbor tunnel coupling, placed in an axial electric field. We show that the system supports highfidelity transfer of the state of a terminal dot both by free evolution and by adiabatic transfer. The protocol is to a large extent insensitive to inhomogeneity of the energy parameters of the dots and requires only a global electric field.

Decoherenceassisted initialization of a resident hole spin polarization in a \(p\)doped semiconductor quantum well
M. Kugler, K. Korzekwa, P. Machnikowski, C. Gradl, S. Furthmeier, M. Griesbeck, M. Hirmer, D. Schuh, W. Wegscheider, T. Kuhn, C. Schüller, T. Korn
Phys. Rev. B 84, 085327 (2011)
PDF
+ Abstract
Abstract: We investigate spin dynamics of resident holes in a \(p\)modulationdoped GaAs/Al\(_{0.3}\)Ga\(_{0.7}\)As single quantum well. Timeresolved Faraday and Kerr rotation, as well as resonant spin amplification, are utilized in our study. We observe that nonresonant or highpower optical pumping leads to a resident hole spin polarization with opposite sign with respect to the optically oriented carriers, while lowpower resonant optical pumping only leads to a resident hole spin polarization if a sufficient inplane magnetic field is applied. The competition between two different processes of spin orientation strongly modifies the shape of resonant spin amplification traces. Calculations of the spin dynamics in the electronhole system are in good agreement with the experimental Kerr rotation and resonant spin amplification traces and allow us to determine the hole spin polarization within the sample after optical orientation, as well as to extract quantitative information about spin dephasing processes at various stages of the evolution.
+ Oral Presentations

Fast estimation of outcome probabilities for quantum circuits
Slides
+ Presented at
 Terhal Group Seminar, QuTech, Delft, Netherlands (2021)

Classical simulations of quantum circuits
Slides
+ Presented at
 International Centre for Theory of Quantum Technologies Seminar, University of Gdańsk, Gdańsk, Poland (2020)
 Quantum Information & Chaos Seminar, Jagiellonian University, Kraków, Poland (2020)
 Krakow Quantum Informatics Seminar, Krakow, Poland (2020)

Quantum state transfer via spin chains
Slides
+ Presented at
 CoherenceCorrelationsComplexity Seminar, Wrocław University of Technology, Poland (2013)

Decoherencedriven mechanism for initialization of hole spins in a pdoped semiconductor quantum well
Slides
Slides (shortened)
+ Presented at
 41st “Jaszowiec” International School & Conference on the Physics of Semiconductors, KrynicaZdrój, Poland (2012)
 Optical Spectroscopy of Semiconductor Quantum Structures Seminar, Universität Regensburg, Germany (2011)
 CoherenceCorrelationsComplexity Seminar, Wrocław University of Technology, Poland (2011)
+ Poster Presentations

Towards quantum technologies in Quantum Resources Group
Poster
+ Presented at
 4th Interdisciplinary FNP Conference, Warsaw, Poland (2022)

Noise and disturbance of quantum measurement: measures and uncertainty relations
Poster
+ Presented at
 Summer School on Quantum Information, Computing and Control, London, United Kingdom (2013)

Spin dynamics in pdoped semiconductor nanostructures subject to magnetic fields tilted from the Voigt geometry
Poster
+ Presented at
 16th International Conference on Modulated Semiconductor Structures, Wrocław, Poland (2013)

Ondemand quantum state transfer in spin chains with limited control and high resilience against imperfections
Poster
+ Presented at
 Summer School on Quantum Information, Computing and Control, Aberystwyth, United Kingdom (2012)

Theoretical modelling of magnetooptical experiments in pdoped nanostructures
Poster
+ Presented at
 PolishGerman Workshop on the Optical Properties of Nanostructures, Münster (2012)

Theory of Kerr rotation and resonant spin amplification in pdoped nanostructures
Poster
+ Presented at
 The international conference on Optics of Excitons in Confined Systems, Paris (2011)
 PolishGerman Workshop on the Optical Properties of Nanostructures, Wrocław (2011)

Tunneling Transfer Protocol in a Quantum Dots Chain Immune to Inhomogeneity
Poster
+ Presented at
 40th “Jaszowiec” Conference on the Physics of Semiconductors, Krynica Zdrój (2011)