Kvantová tomografie a termodynamika

• Thesis title in Czech: Kvantová tomografie a termodynamika
• Thesis title in English: Quantum tomography and thermodynamics
• Key words: kvantová tomografie|termodynamika|kvantové technologie
• English key words: quantum tomography|thermodynamics|quantum technology
• Academic year of topic announcement: 2025/2026
• Thesis type: dissertation
• Department: Department of Condensed Matter Physics (32-KFKL)
• Supervisor: Ing. Dominik Šafránek, Ph.D.

Guidelines

Budou specifikovány v ISP.

References

Bude specifikován v průběhu řešení práce. Předběžně viz literatura uvedená v anglické upoutávce.

Preliminary scope of work in English

Identification of a quantum state, known as quantum tomography, is one of the most important tasks in emerging quantum technologies. This is because every quantum device is based on manipulating and utilizing a quantum state in order to exploit purely quantum phenomena such as superposition, entanglement, and tunneling to achieve tasks that are not possible classically. This includes quantum computers, sensors, cryptographic devices, and quantum batteries.

The current methods of quantum tomography suffer from several drawbacks. Linear inversion tomography returns non-physical states [1,2]. The second is maximum likelihood estimation (MLE) [3–6], which satisfies the physical constraints. However, it sometimes tends to overproduce eigenvalues that are exactly zero [1,2]. Bayesian state tomography (BME) [1,2,7–11] produces physical states and does not overproduce zeros; however, it is difficult to implement numerically and uses the classical Bayes theorem to identify quantum states.

The PhD project's primary focus is to develop a novel method for quantum tomography based on the Quantum Bayes theorem, which is a special case of the Petz recovery map [12–14] applied to the quantum-to-classical map. The goal is to develop a protocol that will reach the Gill–Massar bound [15,16] on the scaling of estimation, while performing better on “almost” pure states, which are characterized by a single dominant eigenvalue of their density matrix. This could be achieved by implementing an adaptive protocol, which could use, for example, the estimate for the next best possible measurements by finding those that overlap the least with already performed measurements [17].
A part of this project will be to determine the effectiveness of the algorithm by coding it in Python, Julia, or Mathematica and comparing it to existing methods. This could specifically include two adaptive BME [10,11] methods and one adaptive method based on self-guided tomography [2]. The PhD student will also assess the resources needed to perform this method of tomography, including the total number of measurements, the effective number of required measurement bases (depending on the purity of the state), energetic costs, other thermodynamic costs (such as entropy generation), and computational costs related to classical computations using experimental data to infer the quantum state.

[1] C. Granade, J. Combes, and D. Cory, New J. Phys. 18, 033024 (2016).
[2] C. Granade, C. Ferrie, and S. T. Flammia, New J. Phys. 19, 113017 (2017).
[3] Z. Hradil et al., Quantum State Estim. 59 (2004).
[4] Y. S. Teo et al., Phys. Rev. Lett. 107, 020404 (2011).
[5] J. A. Smolin, J. M. Gambetta, and G. Smith, Phys. Rev. Lett. 108, 070502 (2012).
[6] T. Baumgratz et al., New J. Phys. 15, 125004 (2013).
[7] R. Blume-Kohout, New J. Phys. 12, 043034 (2010).
[8] J. M. Lukens et al., New J. Phys. 22, 063038 (2020).
[9] F. Huszár and N. M. Houlsby, Phys. Rev. A 85, 052120 (2012).
[10] G. Struchalin et al., Phys. Rev. A 98, 032330 (2018).
[11] B. P. Williams and P. Lougovski, New J. Phys. 19, 043003 (2017).
[12] F. Buscemi, J. Schindler, and D. Šafránek, New J. Phys. 25, 053002 (2023).
[13] D. Petz, Commun. Math. Phys. 105, 123 (1986).
[14] D. Petz, Q. J. Math. 39, 97 (1988).
[15] R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000).
[16] Y. I. Bogdanov et al., Phys. Rev. A 84, 042108 (2011).
[17] D. Šafránek and D. Rosa, Phys. Rev. A 108, 022208 (2023).