## Quantum Thermodynamics

Quantum Thermodynamics (QT) is concerned with the investigation of phenomena such as thermalization and equilibration, heat engines, and foundations of thermodynamics in the quantum regime.The study of thermodynamics in the quantum domain provides an amazing example for the cross-fertilization between different areas of physics, involving classical thermodynamics and solid state physics, as well as quantum information and quantum field theory. For recent reviews see [Goold, Huber, Riera, del Rio, Skrzypczyk, J. Phys. A: Math. Theor.

My current reseach interest within QT lies in fundamental and practical restrictions for thermodynamic processes in quantum optics. In particular, I am interested in learning about the usefulness (and restrictiveness) of Gaussian operations for tasks such as energy extraction, storage, and distribution.

**49**, 143001 (2016)], [Vinjanampathy, Anders, Contemp. Phys.**57**, 1 (2016)] and [Millen, Xuereb, New J. Phys.**18**, 011002 (2016)].My current reseach interest within QT lies in fundamental and practical restrictions for thermodynamic processes in quantum optics. In particular, I am interested in learning about the usefulness (and restrictiveness) of Gaussian operations for tasks such as energy extraction, storage, and distribution.

**Energy Cost of Measurements**

The third law of thermodynamics in the quantum regime tells us that no quantum system can be cooled to the ground state (which, in non-degenerate
cases, is a pure state) in finite time and with finite resources. This is in apparent contradiction to the projection postulate of quantum mechanics, suggesting that an ideal projective measurement leaves the system in a pure state. How is it that an ideal, error-free, measurement leaves the system in a state forbidden by the laws of thermodynamics? This question has been addressed in a recent article [Guryanova, Friis, Huber, Quantum 4, 222 (2020)], where it is shown that thermodynamics indeed does not allow for ideal projective measurements to be performed with finite resources (time, energy, number of controlled systems).We identify three properties of ideal projective measurements, which are faithful, unbiased, and noninvasive, and we show that exactly satisfying all of three criteria simultaneously is impossible using finite resources. At the same time, we show that one may approximate such ideal measurements using finite resources for non-ideal measurements. Our model is based on the interaction of a system (to be measured) with a pointer, and the quantification of the correlations in their joint post-interaction state.In [Debarba, Manzano, Guryanova, Huber, Friis, New J. Phys.
21, 113002 (2019), arXiv:1902.08568] we investigate the consequences of non-ideal measurements for the estimation of work in non-equilibrium processes. We show that obtaining work estimates and their statistical moments at finite work cost implies an imperfection of the estimates themselves: more accurate estimates incur higher costs. Our results provide a qualitative relation between the cost of obtaining information about work and the trustworthiness of this information. Moreover, we show that Jarzynski's equality can be maintained exactly at the expense of a correction that depends only on the system's energy scale, while the more general fluctuation relation due to Crooks no longer holds when the cost of the work estimation procedure is finite. We show that precise links between dissipation and irreversibility can be extended to the non-ideal situation. |

**Gaussian Quantum Thermodynamics**

In QT, energy is one of the primary resources, but not all energy may be freely extracted or converted to be used at leisure for tasks at hand. In particular, so-called passive states cannot yield any work in cyclic Hamiltonian processes. Viewing QT as a resource theory of work extraction it is hence tempting to view passive states as being freely available (and non-passive states as valuable). On the other hand, the unitaries needed to extract work from an arbitrary non-passive state may in practice be complicated and not feasibly realizable.

This motivates the notion of Gaussian passivity [Brown, Friis, Huber, New J. Phys.

**18**, 113028 (2016)], i.e., a quantum state’s property of containing work extractable by Gaussian operations, which are typically practically implementable in many quantum optical setups. Once work has been extracted, one would also like to put it to use, e.g., to prepare specific states for quantum computation or quantum metrology. It is hence expected that practical limitations applying to work extraction will be of relevance for quantum technologies. Following the full characterization of Gaussian passivity [Brown, Friis, Huber, New J. Phys.

**18**, 113028 (2016)], I hence wish to investigate practical limitations, in particular, the restriction to Gaussian operations, for other tasks in QT.

One such task is storing (previously extracted) work in a suitable battery. Here, quantum systems in thermal equilibrium with the environment come to mind, since they can be considered empty in the sense that they do not contain any extractable work. In [Friis, Huber, Quantum
2, 61 (2018)]. we investigate the usefulness and limitations when using Gaussian unitaries to charge such batteries. In particular, we derive the fundamental bounds on the charging precision and work fluctuations during the charging process, and compare these with the optimal and worst Gaussian operations. |

Energy Cost of CorrelationsAnother reseach interest within QT is the relationship of entanglement and correlations with thermodynamic quantities, that is, how energy can be converted into (quantum) correlations in the presence of background temperature (entropy). Interestingly, every amount of correlations implies extractable work [Perarnau-Llobet, Hovhannisyan, Huber, Skrzypczyk, Brunner, Acín, Phys. Rev. X 5, 041011 (2015)], and conversely, correlations come at the expense of energy [Huber, Perarnau-Llobet, Hovhannisyan, Skrzypczyk, Klöckl, Brunner, Acín, New J. Phys. 17, 065008 (2015)]. |

We have studied the optimal conversion between resources of thermodynamics and quantum information tasks for non-interacting [Bruschi, Perarnau-Llobet, Friis, Hovhannisyan, Huber, Phys. Rev. E

**91**, 032118 (2015)], and interacting systems [Friis, Huber, Perarnau-Llobet, Phys. Rev. E**93**, 042135 (2016)]. A review of this topic and information on interesting open problems in this area can be found in [Vitagliano, Klöckl, Huber, Friis, arXiv:1803.06884] and a presentation about this research can be found here. In our most recent work [F. Bakhshinezhad, F. Clivaz, G. Vitagliano, P. Erker, A. T. Rezakhani, M. Huber, and N. Friis, J. Phys. A Math. Theor.**52**, 465303 (2019), arXiv:1904.07942] on this topic, we present mathematical tools for determining the existence of symmetrically thermalizing unitaries necessary saturate entropic bounds on the energy cost of correlations.