Holographic connections between quantum fields, information, and gravity

This project is funded by a Sonata Bis grant of the Polish National Science Centre (NCN) under the grant number 2021/42/E/ST2/00234.


Principal Investigator: Dr. Mario Flory

2 PhDs will be hired soon! Dr. Aranya Bhattacharya is expected to join the group as postdoc later this year.

First PhD position:

We are currently looking for a PhD student to engage in a 4-year thesis project on "Quantum information aspects of holography”. A second PhD student for a different topic will be hired at a later date.

Project description:
In mathematical physics, the AdS/CFT correspondence is a conjecture according to which certain quantum theories (conformal quantum field theories or short CFTs, called the ”boundary”) can be described in a mathematically equivalent way in terms of theories of gravitation (called the ”bulk”) involving a negative curvature of spacetime (Anti-De Sitter spaces, AdS). This makes it an extremely powerful tool that allows translating questions about one side of the duality into questions about the theory on the other side of the duality, where they may be easier to solve with the available mathematical methods. For example, calculating the amount of quantum entanglement present in a physical system can be quite mathematically demanding, however, AdS/CFT maps this task to a geometrical problem conceptually similar to calculating the shape of a soap bubble in the bulk.
In this PhD project, we will research the importance of concepts from quantum information theory in AdS/CFT. Specifically, we will explore (multipartite) quantum entanglement and what role it plays in the AdS/CFT correspondence and in identifying its range of applicability. Another main focus of this project will be the task of analysing measures for the "complexity" of a given state or operator, where complexity is defined as some notion of distance between a given target state and a simple reference state in the space of states, or between a given operator and the identity operator in a group manifold respectively. This allows the application of methods of differential geometry to problems in quantum information and computation theory. Questions that can be tackled in this project concern the definition and calculation of complexity in strongly coupled quantum field theories, testing which calculations in the bulk correspond to a calculation of complexity on the boundary in AdS/CFT, and how such calculations can shed light on the fundamental mechanisms behind the AdS/CFT correspondence.

Example literature:

Candidate’s profile:
- M.Sc. in Theoretical or Mathematical Physics
- experience in Quantum Mechanics and Quantum Field Theory
- strong interest (and ideally already experience) in General Relativity, the AdS/CFT correspondence, Quantum Information Theory, and/or numerical computations
- fluent English in writing and speaking (minimum B2 level)

Schedule of the competition:
- Announcement on the School's website: 20.01.2023
- Opening of the competition (means: opening admission in the Online Application System): 03.02.2023
- Application submission deadline: (means: closing the Online Application System): 06.03.2023
- Entrance exams: 13.03-15.03.2023
- Announcement of results: 17.03.2023
- Enrollment: 20.03.2023 - 20.06.2023

Application procedure:
The application takes place via the online application system of Jagiellonian University. In the sign up form, the candidate uploads the the documents required in the admission process merged into a single PDF. Documents issued in a language other than Polish or English have to be enclosed with a certified translation into Polish or English. Detailed information on the application process can be found in the official announcement and the documents linked there.

To summarize, at the application stage the following will be needed:
- A motivation letter (1-2 pages ideally);
- Curriculum vitae (CV) with emphasis on academic achievements;
- Supporting documents confirming academic achievements detailed in the curriculum vitae where applicable;
- A certificate of average grade in first cycle studies (i.e. bachelor) and a certificate of average grade in the first year of supplementary master studies (i.e. second cycle) or for candidates who have completed five-year master studies a certificate of average grade in the last year of studies excluding the last year of studies;
- Diploma supplement or extract from the course of study (transcript of records);
- For candidates that have not completed their Master degree (the case referred to in Article 186, section 2 of the Act on Higher Education and Science): Two recommendations confirming the high quality of the candidates research work and the high level of advancement of such work, issued by: a research fellow being an employee of a foreign higher education institution or a research institution or holding at least the academic degree of doctor habilitated, who has significant achievements in academic issues related to the curriculum must be provided;
- An opinion of a research fellow holding a degree or the academic degree of doctor habilitated on the possibility of supervising the doctoral student in the event of admission to the Doctoral School: This document has to be provided by us, so all candidates should contact mflory@th.if.uj.edu.pl by February 28th at the latest to express their interest in applying and to receive this document in order to include it in their application.

After the application stage, the selected candidate will have to provide further information or documentation (e.g. Master certificate) in order to enter the Doctoral School of Exact and Natural Sciences.

Jagiellonian University is the oldest and also one of the most prestigious universities in Poland. The physics department is located in the very spacious and modern campus south of the city centre, easily reachable by public transport from anywhere in the city. Kraków (also known as Cracow) is a beautiful city that attracts millions of tourists from all over the world every year. It offers many attractions, a safe environment and a high quality of life.


Quantum field theory is one of the central concepts in theoretical physics, however, in the strong coupling regime where perturbation theory is not reliable, few analytical approaches exist that help understand the physics of the system. One such approach is the so called Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT) or gauge/gravity duality. As a concrete realisation of the more general concept of holography, this duality posits that certain strongly coupled quantum field theories (the "boundary") can be described in a mathematically equivalent way in terms of weakly coupled gravitational theories in higher dimensions (the "bulk"). It is hence a powerful concept that allows translating questions concerning one of these fields into the language of the other, where a new perspective on the problem can be gained, or where different mathematical methods may be at one's disposal.

The importance of this "holographic duality" is that it allows to derive predictions which are expected to be universal for strongly coupled quantum matter. This shows that holography as a general concept can help understanding real-world systems, or conversely, that real world systems can have properties that are holographic in nature. The overarching goal of this project will be to understand how holographic dualities can arise as a form of emergent gravity in strongly coupled states of matter, and how this in turn can help our understanding of such forms of matter. Specifically, we will deal with the following topics:

Applications of AdS/CFT to strongly coupled systems

AdS/CFT models lend themselves to applications to strongly coupled systems because they implement a duality between a weakly coupled bulk theory and a strongly coupled boundary theory. An important aspect of these applications, even in nonequilibrium situations, is universality, i.e. the fact that many phenomena observed in strongly coupled systems are qualitatively independent of their precise realisation at the level of underlying degrees of freedom.


For example, in our recent paper Critical and near-critical relaxation of holographic superfluids we numerically studied exactly critical and near-critical quenches in holographic models of superconductors. We found that these systems relax towards their new equilibrium state in a way that exhibits a discrete type of scale invariance.

We further proposed a phenomenological Landau-Ginzburg like equation which, after numerically fixing its free parameters, was able to provide analytic predictions for the behaviour of the system not just at late, but even at early and intermediate times.

Fundamental aspects of the holographic principle


The AdS/CFT correspondence can be interpreted as giving rise to a form of emergent gravity, and holographic conceptions have often played a role in proposing more general scenarios of emergent gravity. In the AdS/CFT correspondence specifically, it has been demonstrated that Einstein’s equations in the bulk can be mathematically derived from the so-called first law of entanglement in the boundary theory.

Quantum information aspects of holography


The crucial role of boundary entanglement in the derivation of Einstein’s equations mentioned above is not coincidental, in fact aspects of quantum information theory lie at the very heart of the AdS/CFT correspondence. This result is based on the famed Ryu-Takayanagi (RT) formula which states that entanglement entropy of the dual theory (a measure of entanglement) is exactly encoded in the areas of certain extremal hypersurfaces embedded in the bulk spacetime, generalising the Bekenstein-Hawking formula for black hole entropy. This means that at least aspects of the bulk geometry can be reconstructed from boundary entanglement data. Certain inequalities of entanglement entropy that are automatically satisfied by systems for which the RT formula holds, forming the ”holographic entropy cone”, were later derived. This cone is only a subset of the more general ”quantum entropy cone”, a set of inequalities of entanglement entropy that any quantum system can be proven to obey. Hence these inequalities allow checking whether a given quantum state can in principle have a holographic description in terms of the RT formula or if it lies outside of the holographic entropy cone and can thus not have such a holographic description.

Another recent development concerns the idea that a measure of complexity of a field theory state could be calculated by holographic methods. In quantum information theory, especially assuming finite dimensional Hilbert spaces, measures of complexity have been introduced as notions of geodesic distances on group manifolds. On the quantum field theory side of the holographic duality, complexity is conjectured to be a notion of distance between the CFT state and a simple reference state, however, concrete tractable calculations along these lines are mostly restricted to free field theories. In the bulk, the ability to compute such measures of complexity is important as it may give insight into the storage and scrambling of information by black holes.

List of Publications

- M. Flory, S. Grieninger, S. Morales-Tejera; Critical and near-critical relaxation of holographic superfluids