##### Leader of the research group: Lukasz Rudnicki

Post-doc: Fattah Sakuldee, Stefano Cusumano

PhD student: Tomasz Linowski, Otavio Augusto Dantas Molitor

The aim of the group is to explore intersections of quantum optics, quantum thermodynamics and open system dynamics. The group’s research interests lay within standard quantum optics, currently, looked at from the perspective of open quantum systems and thermodynamics. We also extensively collaborate with experimental groups concerned with quantum technologies, with emphasis recently put on metrology.

## Activity

Open Quantum Systems and Quantum Optics

Among various topics of our interest there are: bosonic systems at a mesoscopic scale, open quantum evolution of Gaussian systems, limitations concerning quantum batteries, non-Markovian dynamics, interplays between work and coherence, as well as potential thermodynamic advantages of indefinite causal order.

Quantum Metrology

We coordinate the QuantERA project “Application-ready superresolution in space and frequency” (ApresSF), and take part in Horizon 2020, FET Open project “Spectral-Temporal Metrology with Tailored Quantum Measurements” (Stormytune). Both projects are devoted to superresolution in quantum metrology.

## Publications

### 2021

- Stefano Cusumano and Łukasz Rudnicki. Comment on “Fluctuations in Extractable Work Bound the Charging Power of Quantum Batteries”.
*Physical Review Letters*, 127(2):28901, jul 2021. arXiv: 2102.05627 doi:10.1103/PhysRevLett.127.028901

[BibTeX] [Abstract] [Download PDF]

In the abstract of{\textasciitilde}[Phys. Rev. Lett. \{{\textbackslash}bf 125\}, 040601 (2020)] one can read that: […]\{{\textbackslash}it to have a nonzero rate of change of the extractable work, the state \${\textbackslash}rho_{\textbackslash}mathcal\{W\}\$ of the battery cannot be an eigenstate of a “free energy operator”, defined by \${\textbackslash}mathcal\{F\}=H_{\textbackslash}mathcal\{W\}+{\textbackslash}beta{\textasciicircum}\{-1\}{\textbackslash}log {\textbackslash}rho_{\textbackslash}mathcal\{W\}\$, where \$H_{\textbackslash}mathcal\{W\}\$ is the Hamiltonian of the battery and \${\textbackslash}beta\$ is the inverse temperature\} […]. Contrarily to what is presented below Eq.{\textasciitilde}(17) of the paper, we observe that the above conclusion does not hold when the battery is subject to nonunitary dynamics.

`@Article{cusumano_comment_2021, author = {Cusumano, Stefano and Rudnicki, Łukasz}, journal = {Physical {R}eview {L}etters}, title = {Comment on "{Fluctuations} in {Extractable} {Work} {Bound} the {Charging} {Power} of {Quantum} {Batteries}"}, year = {2021}, issn = {0031-9007, 1079-7114}, month = jul, note = {arXiv: 2102.05627}, number = {2}, pages = {028901}, volume = {127}, abstract = {In the abstract of{\textasciitilde}[Phys. Rev. Lett. \{{\textbackslash}bf 125\}, 040601 (2020)] one can read that: [...]\{{\textbackslash}it to have a nonzero rate of change of the extractable work, the state \${\textbackslash}rho\_{\textbackslash}mathcal\{W\}\$ of the battery cannot be an eigenstate of a "free energy operator", defined by \${\textbackslash}mathcal\{F\}=H\_{\textbackslash}mathcal\{W\}+{\textbackslash}beta{\textasciicircum}\{-1\}{\textbackslash}log {\textbackslash}rho\_{\textbackslash}mathcal\{W\}\$, where \$H\_{\textbackslash}mathcal\{W\}\$ is the Hamiltonian of the battery and \${\textbackslash}beta\$ is the inverse temperature\} [...]. Contrarily to what is presented below Eq.{\textasciitilde}(17) of the paper, we observe that the above conclusion does not hold when the battery is subject to nonunitary dynamics.}, doi = {10.1103/PhysRevLett.127.028901}, keywords = {Quantum Physics}, url = {http://arxiv.org/abs/2102.05627}, urldate = {2021-07-28}, }`

- Stefano Cusumano and Łukasz Rudnicki. Thermodynamics of Reduced State of the Field.
*Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies*, 23(9):Paper No. 1198, 2021. doi:10.3390/e23091198

[BibTeX] [Download PDF]`@Article{Cusumano2021, author = {Cusumano, Stefano and Rudnicki, Łukasz}, journal = {Entropy. {A}n {I}nternational and {I}nterdisciplinary {J}ournal of {E}ntropy and {I}nformation {S}tudies}, title = {Thermodynamics of {R}educed {S}tate of the {F}ield}, year = {2021}, number = {9}, pages = {Paper No. 1198}, volume = {23}, doi = {10.3390/e23091198}, keywords = {81}, mrnumber = {4320432}, url = {https://www.mdpi.com/1099-4300/23/9/1198/pdf}, }`

- Łukasz Rudnicki. Quantum speed limit and geometric measure of entanglement.
*Physical Review A*, 104(3):32417, sep 2021. doi:10.1103/PhysRevA.104.032417

[BibTeX] [Abstract] [Download PDF]

Using the approach offered by quantum speed limit, we show that geometric measure of multipartite entanglement for pure states [T.-C. Wei and P. M. Goldbart, Phys. Rev. A 68, 042307 (2003), 10.1103/PhysRevA.68.042307] can be interpreted as the minimal time necessary to unitarily evolve a given quantum state to a separable one.

`@Article{Rudnicki2021, author = {Rudnicki, Łukasz}, journal = {Physical {R}eview {A}}, title = {Quantum speed limit and geometric measure of entanglement}, year = {2021}, month = sep, number = {3}, pages = {032417}, volume = {104}, abstract = {Using the approach offered by quantum speed limit, we show that geometric measure of multipartite entanglement for pure states [T.-C. Wei and P. M. Goldbart, Phys. Rev. A 68, 042307 (2003), 10.1103/PhysRevA.68.042307] can be interpreted as the minimal time necessary to unitarily evolve a given quantum state to a separable one.}, archiveprefix = {arXiv}, doi = {10.1103/PhysRevA.104.032417}, eid = {032417}, eprint = {2107.11877}, groups = {Rudnicki}, keywords = {Quantum Physics}, primaryclass = {quant-ph}, url = {https://ui.adsabs.harvard.edu/abs/2021PhRvA.104c2417R}, }`

- Klaus Liegener and Łukasz Rudnicki. Algorithmic approach to cosmological coherent state expectation values in loop quantum gravity.
*Classical and Quantum Gravity*, 38(20):Paper No. 205001, 39, 2021. doi:10.1088/1361-6382/ac226f

[BibTeX] [Download PDF]`@Article{Liegener2021, author = {Liegener, Klaus and Rudnicki, Łukasz}, journal = {Classical and {Q}uantum {G}ravity}, title = {Algorithmic approach to cosmological coherent state expectation values in loop quantum gravity}, year = {2021}, issn = {0264-9381}, number = {20}, pages = {Paper No. 205001, 39}, volume = {38}, doi = {10.1088/1361-6382/ac226f}, keywords = {83C45 (83C27)}, mrnumber = {4318548}, url = {https://iopscience.iop.org/article/10.1088/1361-6382/ac226f/pdf}, }`

- Łukasz Rudnicki and Stephen P. Walborn. Entropic uncertainty relations for mutually unbiased periodic coarse-grained observables resembling their discrete counterparts.
*Physical Review A*, 104(4):Paper No. 042210, 2021. doi:10.1103/physreva.104.042210

[BibTeX] [Download PDF]`@Article{Rudnicki2021a, author = {Rudnicki, Łukasz and Walborn, Stephen P.}, journal = {Physical {R}eview {A}}, title = {Entropic uncertainty relations for mutually unbiased periodic coarse-grained observables resembling their discrete counterparts}, year = {2021}, issn = {2469-9926}, number = {4}, pages = {Paper No. 042210}, volume = {104}, doi = {10.1103/physreva.104.042210}, keywords = {81S07}, mrnumber = {4339485}, url = {https://journals.aps.org/pra/pdf/10.1103/PhysRevA.104.042210}, }`

### 2020

- Tomasz Linowski, Grzegorz Rajchel-Mieldzioć, and Karol Życzkowski. Entangling power of multipartite unitary gates.
*Journal of Physics A: Mathematical and Theoretical*, 53(12):125303, 2020. doi:10.1088/1751-8121/ab749a

[BibTeX] [Download PDF]`@Article{linowski_entangling_2020, author = {Linowski, Tomasz and Rajchel-Mieldzioć, Grzegorz and Życzkowski, Karol}, journal = {Journal of {P}hysics {A}: {M}athematical and {T}heoretical}, title = {Entangling power of multipartite unitary gates}, year = {2020}, issn = {1751-8113, 1751-8121}, month = mar, number = {12}, pages = {125303}, volume = {53}, doi = {10.1088/1751-8121/ab749a}, url = {https://iopscience.iop.org/article/10.1088/1751-8121/ab749a}, urldate = {2020-04-22}, }`

- Simon Milz, Fattah Sakuldee, Felix A. Pollock, and Kavan Modi. Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories.
*Quantum*, 4:255, apr 2020. doi:10.22331/q-2020-04-20-255

[BibTeX] [Abstract] [Download PDF]

In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes – classical, quantum or beyond – that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.

`@Article{milz_kolmogorov_2020, author = {Milz, Simon and Sakuldee, Fattah and Pollock, Felix A. and Modi, Kavan}, journal = {Quantum}, title = {Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories}, year = {2020}, issn = {2521-327X}, month = apr, pages = {255}, volume = {4}, abstract = {In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.}, doi = {10.22331/q-2020-04-20-255}, language = {en}, url = {https://quantum-journal.org/papers/q-2020-04-20-255/}, urldate = {2020-04-22}, }`

- Ł. Rudnicki, L. L. Sánchez-Soto, G. Leuchs, and R. W. Boyd. Fundamental quantum limits in ellipsometry.
*Optics Letters*, 45(16):4607, 2020. doi:10.1364/OL.392955

[BibTeX] [Download PDF]`@Article{rudnicki_fundamental_2020, author = {Rudnicki, Ł. and Sánchez-Soto, L. L. and Leuchs, G. and Boyd, R. W.}, journal = {Optics {L}etters}, title = {Fundamental quantum limits in ellipsometry}, year = {2020}, issn = {0146-9592, 1539-4794}, month = aug, number = {16}, pages = {4607}, volume = {45}, doi = {10.1364/OL.392955}, language = {en}, url = {https://www.osapublishing.org/abstract.cfm?URI=ol-45-16-4607}, urldate = {2021-05-10}, }`

- Tomasz Linowski, Clemens Gneiting, and Łukasz Rudnicki. Stabilizing entanglement in two-mode Gaussian states.
*Physical Review A*, 102(4):42405, oct 2020. doi:10.1103/PhysRevA.102.042405

[BibTeX] [Download PDF]`@Article{linowski_stabilizing_2020, author = {Linowski, Tomasz and Gneiting, Clemens and Rudnicki, Łukasz}, journal = {Physical {R}eview {A}}, title = {Stabilizing entanglement in two-mode {Gaussian} states}, year = {2020}, issn = {2469-9926, 2469-9934}, month = oct, number = {4}, pages = {042405}, volume = {102}, doi = {10.1103/PhysRevA.102.042405}, language = {en}, url = {https://link.aps.org/doi/10.1103/PhysRevA.102.042405}, urldate = {2021-05-10}, }`

### 2019

- Alejandro Pozas-Kerstjens, Rafael Rabelo, Łukasz Rudnicki, Rafael Chaves, Daniel Cavalcanti, Miguel Navascués, and Antonio Acín. Bounding the sets of classical and quantum correlations in networks.
*Physical Review Letters*, 123(14):140503, 2019. doi:10.1103/PhysRevLett.123.140503

[BibTeX] [Download PDF]`@Article{pozas-kerstjens_bounding_2019, author = {Pozas-Kerstjens, Alejandro and Rabelo, Rafael and Rudnicki, Łukasz and Chaves, Rafael and Cavalcanti, Daniel and Navascués, Miguel and Acín, Antonio}, journal = {Physical {R}eview {L}etters}, title = {Bounding the sets of classical and quantum correlations in networks}, year = {2019}, issn = {0031-9007, 1079-7114}, month = oct, number = {14}, pages = {140503}, volume = {123}, doi = {10.1103/PhysRevLett.123.140503}, groups = {Rudnicki}, language = {en}, url = {https://link.aps.org/doi/10.1103/PhysRevLett.123.140503}, urldate = {2020-04-22}, }`

## arXiv preprints

### 2021

- Lucas Chibebe Céleri and Łukasz Rudnicki. Gauge invariant quantum thermodynamics: consequences for the first law.
*Arxiv e-prints*, pages arXiv:2104.10153, apr 2021.

[BibTeX] [Abstract] [Download PDF]

Universality of classical thermodynamics rests on the central limit theorem, due to which, measurements of thermal fluctuations are unable to reveal detailed information regarding the microscopic structure of a macroscopic body. When small systems are considered and fluctuations become important, thermodynamic quantities can be understood in the context of classical stochastic mechanics. A fundamental assumption behind thermodynamics is therefore that of coarse-graning, which stems from a substantial lack of control over all degrees of freedom. However, when quantum systems are concerned, one claims a high level of control. As a consequence, information theory plays a major role in the identification of thermodynamic functions. Here, drawing from the concept of gauge symmetry, essential in all modern physical theories, we put forward a new possible, intermediate route. Working within the realm of quantum thermodynamics we explicitly construct physically motivated gauge transformations which encode a gentle variant of coarse- graining behind thermodynamics. As a consequence, we reinterpret quantum work and heat, as well as the role of quantum coherence.

`@Article{ChibebeCeleri2021, author = {Chibebe Céleri, Lucas and Rudnicki, Łukasz}, journal = {arXiv e-prints}, title = {Gauge invariant quantum thermodynamics: consequences for the first law}, year = {2021}, month = apr, pages = {arXiv:2104.10153}, abstract = {Universality of classical thermodynamics rests on the central limit theorem, due to which, measurements of thermal fluctuations are unable to reveal detailed information regarding the microscopic structure of a macroscopic body. When small systems are considered and fluctuations become important, thermodynamic quantities can be understood in the context of classical stochastic mechanics. A fundamental assumption behind thermodynamics is therefore that of coarse-graning, which stems from a substantial lack of control over all degrees of freedom. However, when quantum systems are concerned, one claims a high level of control. As a consequence, information theory plays a major role in the identification of thermodynamic functions. Here, drawing from the concept of gauge symmetry, essential in all modern physical theories, we put forward a new possible, intermediate route. Working within the realm of quantum thermodynamics we explicitly construct physically motivated gauge transformations which encode a gentle variant of coarse- graining behind thermodynamics. As a consequence, we reinterpret quantum work and heat, as well as the role of quantum coherence.}, archiveprefix = {arXiv}, eid = {arXiv:2104.10153}, eprint = {2104.10153}, keywords = {Quantum Physics}, primaryclass = {quant-ph}, url = {https://ui.adsabs.harvard.edu/abs/2021arXiv210410153C}, }`

- Tomasz Linowski and Łukasz Rudnicki. Dissipative evolution of covariance matrix beyond quadratic order.
*Arxiv e-prints*, pages arXiv:2105.12644, may 2021.

[BibTeX] [Abstract] [Download PDF]

The covariance matrix contains the complete information about the second-order expectation values of the mode quadratures (position and momentum operators) of the system. Due to its prominence in studies of continuous variable systems, most significantly Gaussian states, special emphasis is put on time evolution models that result in self-contained equations for the covariance matrix. So far, despite not being explicitly implied by this requirement, virtually all such models assume a so- called quadratic, or second-order case, in which the generator of the evolution is at most second-order in the mode quadratures. Here, we provide an explicit model of covariance matrix evolution of infinite order. Furthermore, we derive the solution, including stationary states, for a large subclass of proposed evolutions. Our findings challenge the contemporary understanding of covariance matrix dynamics and may give rise to new methods and improvements in quantum technologies employing continuous variable systems.

`@Article{Linowski2021a, author = {Linowski, Tomasz and Rudnicki, Łukasz}, journal = {arXiv e-prints}, title = {Dissipative evolution of covariance matrix beyond quadratic order}, year = {2021}, month = may, pages = {arXiv:2105.12644}, abstract = {The covariance matrix contains the complete information about the second-order expectation values of the mode quadratures (position and momentum operators) of the system. Due to its prominence in studies of continuous variable systems, most significantly Gaussian states, special emphasis is put on time evolution models that result in self-contained equations for the covariance matrix. So far, despite not being explicitly implied by this requirement, virtually all such models assume a so- called quadratic, or second-order case, in which the generator of the evolution is at most second-order in the mode quadratures. Here, we provide an explicit model of covariance matrix evolution of infinite order. Furthermore, we derive the solution, including stationary states, for a large subclass of proposed evolutions. Our findings challenge the contemporary understanding of covariance matrix dynamics and may give rise to new methods and improvements in quantum technologies employing continuous variable systems.}, archiveprefix = {arXiv}, eid = {arXiv:2105.12644}, eprint = {2105.12644}, keywords = {Quantum {P}hysics}, primaryclass = {quant-ph}, url = {https://ui.adsabs.harvard.edu/abs/2021arXiv210512644L}, }`

- Tomasz Linowski and Łukasz Rudnicki. Classical description of bosonic quantum fields in terms of the reduced-state-of-the-field framework.
*Arxiv e-prints*, pages arXiv:2107.03196, jul 2021.

[BibTeX] [Abstract] [Download PDF]

We discuss compatibility between various quantum aspects of bosonic fields, relevant for quantum optics and quantum thermodynamics, and the mesoscopic formalism of reduced state of the field (RSF). In particular, we derive exact conditions under which Gaussian and Bogoliubov-type evolutions can be cast into the RSF framework. In that regard, special emphasis is put on Gaussian thermal operations. To strengthen the link between the RSF formalism and the notion of classicality for bosonic quantum fields, we prove that RSF contains no information about entanglement in two-mode Gaussian states. For the same purpose, we show that the entropic characterisation of RSF by means of the von Neumann entropy is qualitatively the same as its description based on the Wehrl entropy. Our findings help bridge the conceptual gap between quantum and classical mechanics.

`@Article{Linowski2021, author = {Linowski, Tomasz and Rudnicki, Łukasz}, journal = {arXiv e-prints}, title = {Classical description of bosonic quantum fields in terms of the reduced-state-of-the-field framework}, year = {2021}, month = jul, pages = {arXiv:2107.03196}, abstract = {We discuss compatibility between various quantum aspects of bosonic fields, relevant for quantum optics and quantum thermodynamics, and the mesoscopic formalism of reduced state of the field (RSF). In particular, we derive exact conditions under which Gaussian and Bogoliubov-type evolutions can be cast into the RSF framework. In that regard, special emphasis is put on Gaussian thermal operations. To strengthen the link between the RSF formalism and the notion of classicality for bosonic quantum fields, we prove that RSF contains no information about entanglement in two-mode Gaussian states. For the same purpose, we show that the entropic characterisation of RSF by means of the von Neumann entropy is qualitatively the same as its description based on the Wehrl entropy. Our findings help bridge the conceptual gap between quantum and classical mechanics.}, archiveprefix = {arXiv}, eid = {arXiv:2107.03196}, eprint = {2107.03196}, keywords = {Quantum Physics}, primaryclass = {quant-ph}, url = {https://ui.adsabs.harvard.edu/abs/2021arXiv210703196L}, }`

### 2020

- Thais L. Silva, Łukasz Rudnicki, Daniel S. Tasca, and Stephen P. Walborn. Periodic discretized continuous observables are neither continuous nor discrete.
*Arxiv:2009.05062 [quant-ph]*, sep 2020.

[BibTeX] [Abstract] [Download PDF]

Most of the fundamental characteristics of quantum mechanics, such as non-locality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous-variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number “$d$” of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on $d$, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with $d = 3$ outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.

`@Article{Silva2020, author = {Thais L. Silva and Łukasz Rudnicki and Daniel S. Tasca and Stephen P. Walborn}, journal = {arXiv:2009.05062 [quant-ph]}, title = {Periodic discretized continuous observables are neither continuous nor discrete}, year = {2020}, month = sep, abstract = {Most of the fundamental characteristics of quantum mechanics, such as non-locality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous-variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number "$d$" of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on $d$, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with $d = 3$ outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.}, archiveprefix = {arXiv}, eprint = {2009.05062}, file = {:Silva2020 - Periodic Discretized Continuous Observables Are Neither Continuous nor Discrete.pdf:PDF}, keywords = {quant-ph}, primaryclass = {quant-ph}, url = {https://arxiv.org/pdf/2009.05062.pdf}, }`

## Group members

###### Get to know the people behind ICTQT.

## Former members

__Keywords:__ open system dynamics, quantum optics, indefinite causal order, quantum metrology.