Doctoral School of Mathematics and Computer Science  

Department of Analysis / Mathematical Institute

Supervisor: Dr. Petz Dénes

Noncommutative Probability Theory, Quantum Information Theory and Preserver Problems

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. In addition, I am interested in preserver problems, as well. The aim of these investigations is to determine the maps on a base set (which is mainly the cone of positive matrices) which preserve certain algebraic operations or generalized distance measures.

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.

The MTA-DE "Lendület" ("Momentum") Functional Analysis Research Group of the Hungarian Academy of Sciences plays a leading role in the Hungarian functional analysis scene. The main topics investigated by the group are quantum structures, preserver problems and transformations of operator algebras. I start working in this research group as an assistant research fellow on the first of July, 2015.

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. Nowadays, my interest is focused on the properties of generalized relative entropies (so-called Bregman divergences) and on various preserver problems. This part of my research is a joint work with József Pitrik and Lajos Molnár.

The goal of the research and open questions

The following 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 arise.

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.

 

4. Recently, we managed to describe the Jordan triple endomorphisms of the cone of 2x2 positive definite matrices. Among others, by this result we were in the position to determine all the algebraic endomorphisms of the three dimensional real unit ball equipped with the relativistic sum (this structure is called the Einstein gyrogroup). It seems to be challenging to describe the endomorphisms of two other gyrogroups, namely the Mobius gyrogroup and the “Proper Velocity” gyrogroup. On the other hand, the case of the Einstein gyrogroup of dimension greater than three also seems to be interesting.

 

5. We determined the invariance-transformation group of the positive matrices for a wide class of Bregman and Jensen divergences. It is an open question whether the square root of quantum Jensen-Shannon divergence is a true metric or not for nxn positive matrices with n>=3. For 2x2 matrices, the answer is positive, therefore the following question arise. Are there other Jensen-type divergences which determine true metric?

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 the key of the solution: after the translation the efficient tools of convex geometry can be used to solve problem.

To prove the optimality of a tomography method for a given quantum channel is not easy in general. However, a lot of efficient methods are used 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.

In order to determine the Jordan triple endomorphisms of the positive 2x2 matrices we used tricky linear algebraic observations and some analytic calculations rather than deep and abstract mathematical tools. We utilized that the structure of 2x2 matrices is essentially different than the structure of nxn matrices for n>=3. Similar methods and tricks were applied by describing the endomorphisms of the Einstein-gyrogroup.

The key step of the description of maps which preserve Bregman and Jensen divergences is the observation that these transformations are order automorphisms by necessity.

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 has been published in the journal Acta Sci. Math. (Szeged).

 

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

 

We started to investigate a more general question 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. Our paper has been published in the journal Letters in Mathematical Physics.

 

We determined the Jordan triple endomorphisms of the 2x2 positive definite matrices. Two immediate consequences of this result appeared. Namely, by this result the description of maps on nxn positive matrices with n>=3 which preserve certain generalized distance measures turned out to be valid for 2x2 matrices, as well. (The original work of Lajos Molnár and Patrícia Szokol is ref. [16].) The other corollary is the description of the sequential endomorphisms of the effect algebra over the two-dimensional Hilbert space. This problem was presented as an open question in ref. [17].

 

We described the algebraic endomorphisms of the three dimensional Einstein gyrogroup.

 

In our latest manuscript we managed to describe the transformations of the positive cone which preserve Bregman and Jensen divergences for a large class of generating functions [7].

 

Expected impact and further research 

The results about Pauli channels and Tsallis entropies 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).

Our paper which characterizes the joint convexity of the Bregman divergences was published in the journal Letters in Mathematical Physics, which is a leading journal of the topic (impact factor: 1.939).

The preprints [5] and [6] were submitted to the Journal of Mathematical Analysis and Applications (impact factor: 1.119) and to the Journal of Mathematical Physics (impact factor: 1.234), respectively. Both journals are among the most important journals of the corresponding research field.

 

The four published research article have 13 independent citations by now.

 

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

3. Using the description of the transformations of the state space which preserve quantum f-divergences, it is reasonable to try to describe  the transformations of the positive cone which preserve quantum f-divergences (or more generally, the transformations which preserve quasi-entropies).

 

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, Acta Sci. Math. (Szeged) 80 (2014), 681–687.

 

[3]  D. Petz, D. Virosztek, Some inequalities for quantum Tsallis entropy related to the strong subadditivity, Math. Inequal. Appl. 18(2) (2015), 555–568.

 

 [4] J. Pitrik, D. Virosztek, On the joint convexity of the Bregman divergence of matrices, Lett. Math. Phys. 105(5) (2015), 675–692.

 

Preprints, manuscripts

 

[5]  L. Molnár, D. Virosztek, Continuous Jordan triple endomorphisms of P_2. Submitted to J. Math. Anal. Appl. (2015). Available at arXiv:1506.06223.

 

[6] L. Molnár, D. Virosztek, On algebraic endomorphisms of the Einstein gyrogroup. Submitted to J. Math. Phys. (2015). Available at arXiv:1506.06225.

 

[7]  L. Molnár, J. Pitrik, D. Virosztek, Maps on positive definite matrices preserving Bregman and Jensen divergences. Manuscript, (2015).

 

Conference articles

 

[8] 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

 

[9] 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.

 

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

Math. 529(2010), 73 – 140.

 

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

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

 

 

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

023302 (2006)

 

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

 

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

165–177, 2009.

 

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

78(2012), 683–688.

 

[16] L. Molnár and P. Szokol, Transformations on positive definite matrices preserving generalized distance measures, Linear Algebra Appl. 466 (2015), 141–159.

 

[17] G. Dolinar and L. Molnár, Sequential endomorphisms of finite dimensional Hilbert space effect algebras, J. Phys. A: Math. Theor. 45 (2012), 065207.