Doctoral School of Mathematics and Computer Science 

Department of Analysis / Mathematical Institute

Supervisor: Dr. Dénes Petz

Noncommutative probability theory and quantum information theory 

Introducing the research area

My research field is non-commutative - or quantum - probability theory. Within this huge topic, my interest is focused on quantum entropies, non-commutative variances and some applications in quantum information theory.

 

Brief introduction of the research place

During the past few years, I worked with different supervisors and co-authors, hence I had the opportunity to work in different research groups.

 

I started my research work at the Department of Analysis, Mathematical Institute, in the group of Professor Dénes Petz, under the supervision of the young researcher László Ruppert. The results of this group in quantum entropy and quantum information theory is world-wide acknowledged.

A few month later, I started working under the supervision of Professor Katalin Hangos, the head of the Process Control Research Group, Institute for Computer Science and Control, HAS. Therefore, I was able to join in the top-level research work of the group.

I took part in the projects OTKA K83440 and OTKA K104206 which are supported by the Hungarian Scientific Research Fund.

 

History and context of the research

Nowadays, both quantum information theory and quantum probability theory are among the most important topics in mathematics, and the former one is really interdisciplinary because of its wide range of applications in physics and engineering.

I started studying the quantum Pauli channels – which form a wide class of quantum state transformations – at the Department of Analysis and at the Process Control Research Group under the supervision of László Ruppert and Prof. Katalin Hangos. Later – as a PhD student of Prof. Dénes Petz – I started investigating non-commutative (or matrix) variances and quantum entropies.

 

The research goal, open questions

The following three questions are in the focus of my research.

 

1. The entropy is a central notion in quantum information theory. One of the most important results in this topic is the strong subadditivity of the von Neumann entropy.

The Tsallis entropy is a one-parameter extension of the von Neumann entropy, which is not additive, but strongly subadditive in the case of classical probability distributions. However, for noncommutative probability distributions (density matrices) the strong subadditivity does not hold in general.

Therefore, two natural and important questions appear.

Can we describe the density matrices for which the Tsallis entropy is strongly subadditive?

Can we derive a sharp inequality for the Tsallis entropy, which is similar or strongly related to the strong subadditivity?

 

 

2. The tomography of the state transformations (or quantum process tomography) is a central topic in quantum information theory. The generalized Pauli channels form a wide class of the state transformations on finite dimensional systems. The goal is to describe the most efficient tomography scheme for the Pauli channels acting on n-qubit systems.

 

3. The decomposition of matrix variances is an interesting question not just because of its mathematical beauty, but from a physical point of view, as well. The aim is to give a necessary and sufficient condition for the existence of a decomposition of a matrix variance.

 

Methods

The strong subadditivity of the von Neumann entropy can be derived from the monotonicity of the relative entropy. Therefore, we define a function that can be considered as a relative Tsallis entropy. Some properties of this kind of relative entropy are similar to the monotonicity, hence we can derive relevant inequalities from these properties.

The question of the decomposition of matrix variances can be translated to a picturesque problem in finite-dimensional convex geometry. This translation seems to be a key step to the solution: after the translation, the efficient tools of convex geometry can be used to solve the problem.

Proving the optimality of a tomography method for a given quantum channel is difficult, in general, however, several efficient methods are available in this topic. In my works, both analytical and numerical methods (e. g. Monte-Carlo method) were applied, but the analytical ones are used more often.

Results

We provided optimal tomography schemes for a qubit Pauli channel with unknown channel directions. Different loss functions led to different tomography procedures. In order to prove the optimality, we used both analytical and numerical methods, but most of the theorems were proved analytically. This work is summarized in a paper that won the First Prize on the National Scientific Students’ Conference in 2013 in Applied Mathematics. Another publication that appeared in the Journal of Physics A in 2012 is based on this research, as well.

The problem of the decomposition of matrix variances was investigated in some journal articles in 2012 and 2013. These papers formulated sufficient conditions for the existence of a decomposition of a matrix variance. In my paper (which is a joint work with Dénes Petz) I provided a necessary and sufficient condition, that is, I characterized those sets of observables for which the induced matrix variance is decomposable. The former known results turn out to be special cases of this theorem. This paper is accepted for publication in the journal Acta Sci. Math. (Szeged).

 

One part of my Tsallis entropy related results is the proof of the strong subadditivity for a (rather tight) class of non-classical states. Another achievement is that I provided an equivalent formula for the strong subadditivity, which is an inequality of certain relative entropies. These results can be found here, the paper is accepted for publication in the journal Mathematical Inequalities and Applications.

 

I started investigating a more general question jointly with József Pitrik, namely the joint convexity of the Bregman divergence. In our publication we gave a necessary and sufficient condition for the joint convexity. This characterization theorem has an interesting consequence, which is a sharp inequality for Tsallis entropies and we recover the strong subadditivity of the von Neumann entropy in a special case.

 

Expected impact and further research 

The results of the research were published in relevant scientific journals (the impact factor of J. Phys. A is 1.766, while the impact factor of Math. Inequal. Appl. is 0.588).

A lot of open problems are known in these topics, let us mention two of them.

1. The optimal tomography scheme for Pauli channels acting on n-qubit systems is not known.

2. Finding the conditions of the joint convexity in the case of the operator valued  Bregman divergence seems challenging.

Publications, reference, link collection

Related own publications

 

Journal articles:

[1] L. Ruppert, D. Virosztek, K. M. Hangos, Optimal parameter estimation of Pauli

channels, J. Phys. A 45 (2012), no. 26, 265305, 14 pp.; MR2942597

 

[2]  D. Petz, D. Virosztek, A characterization theorem for matrix variances, to appear in Acta Sci. Math. (Szeged) in 2014. Available at arXiv:1311.3908

 

[3]  D. Petz, D. Virosztek, Some inequalities for quantum Tsallis entropy related to the strong subadditivity, to appear in Math. Inequal. Appl. in 2014. Available at arXiv:1403.7062

[4] J. Pitrik, D. Virosztek, On the joint convexity of the Bregman divergence of matrices, available at arXiv:1405.7885

 

Conference articles

[5] D. Virosztek, L. Ruppert and K. M. Hangos, Pauli channel tomography with unknown channel directions, 10th Central European Quantum Information Processing Workshop, 2013, Valtice, Czech Republic

 

[6] D. Virosztek, Decomposition of matrix variances and subadditivities of certain entropies,

16th Workshop on Non-commutative Harmonic Analysis: Random Matrices, representation theory and free probability with applications, 2014, Będlewo, Poland

 

 

Reference list

 

[1] E. Carlen, Trace inequalities and quantum entropy: an introductory course, Contemp.

Math. 529(2010), 73 - 140.

 

[2] R. Y. Chen, J. A. Tropp, Subadditivity of matrix ϕ-entropy and concentration of

random matrices. arXiv:1308.2952v1, 13 Aug., 2013.

 

 

[3]S. Furuichi, Information theoretical properties of Tsallis entropies, J. Math. Phys. 47,

023302 (2006)

 

[4]F. Hansen, Z. Zhang, Characterization of matrix entropies, arXiv:1402:2118v1, 10 Feb., 2014.

 

[5] D. Petz, H. Ohno: Generalizations of Pauli channels. Acta Math. Hungar., 124,

165-177, 2009.

 

[6] D. Petz, G. Toth, Matrix variances with projections, Acta Sci. Math. (Szeged),

78(2012), 683–688.