Introduction to my research thema (short version)

If you are interested in the above research thema and want to study the thema with me, please contact me.

E-mail: m18003b (at) math.nagoya-u.ac.jp

Foundation of Quantum Theory for Practical Quantum Physics

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If you are interested in the above research thema and want to study the thema with me, please contact me.

E-mail: m18003b (at) math.nagoya-u.ac.jp

Foundation of Quantum Theory for Practical Quantum Physics

My research topics address the foundation of quantum theory. However, my motivation is beyond the foundation. My ultimate motivation is to predict the efficient way to design qubit systems on physical systems by using the information theoretical method. For this ultimate aim, I mainly study the possibility/impossibility or accessibility/difficulty of mathematical models of physical systems. In order to discuss such possibilities, I am mainly working on General Probabilistic Theories.

General Probabilistic Theories
General Probabilistic Theories (GPTs) are models that describe physical systems with states and measurements.
The definition of GPTs includes only the assumption that the relation between states and measurements is probabilistic and the minimum mathematical structure to discuss the assumption.
Moreover, in GPTs, we define information theory that can be designed in the corresponding physical systems, and we mainly discuss the relation between information properties and physical properties.
The typical examples of GPTs are classical probability theory and quantum theory, whose corresponding information theories are known as classical information theory and quantum information theory.

While we know only classical theory and quantum theory in physics, there are many models in GPTs.
Besides, some models in GPTs have unlikely physical properties.
For example, a model called PR-box breaks Tsirelson's bound [1].
Therefore, we study some principles that derive physical theory from GPTs.
Recently, preceding studies have remarked performance of fundamental information tasks [2] and entropic quantity [3].

If you want an introduction to GPTs, see the review paper here [4].

Models in pioneering studies of GPTs (for example of PR-box) are quite different from our physical systems. Therefore, we can easily denied possibilities of such models (for example, PR-box is denied by Information Causality [5]). On the other hand, there are quantum-like models that cannot be denied by experimental verification of standard quantum theory. Our recent study [6] has clarified the existence of infinitely many such models called Pseudo Standard Entanglement Structure except for the standard quantum theory. Moreover, we have clarified that such models have superiority for perfect state discrimination to standard quantum theory (detailed information is given the paper).

Quantum Information Theory
The above approach originates from quantum information theory.
Quantum information theory has clarified that performance of quantum information tasks is strongly concerned with physical properties such as non-locality, entanglement, entropy, and so on.
Besides, quantum information theory has many mathematical similarities to GPTs.
For example, the concept of entanglement is strongly concerned with the tensor product of positive cones,
wwhich is shown in the topic about separability criterion or entanglement detection.

I also study standard quantum information theoretical topics.
I am especially interested in state discrimination, hypothesis testing, properties of entropic quantities, resource theory, separability criterion, computation complexity.
If you want to know such a theoretical aspect of quantum information theory, see some texts of quantum information theory, for example [7].

Mathematically, a model in GPTs is defined as a positive cone, which is a convex set closed under positive-valued multiplication.
Therefore, a study of GPTs has an aspect of analysis of positive cones.
At this point, we aim to characterize the positive cones corresponding to quantum and classical theory from all positive cones.
Actually, the positive cones corresponding to quantum and classical theory belong to a special class, called symmetric cones.
Symmetric cones heve been studied very well in mathematics.
For example, it is known that symmetric cones are 1-1 corresponding to Euclidean Jordan Algebras.
However, there are few studies to characterize symmetric cones from positive cones.
In particular, we want to give some physical-meaningful conditions to characterize symmetric cones.
Preceding mathematical studies do not have such a motivation.

If you want to know the above topic of positive and symmetric cones, see some texts of positive cones, for example [8].

- S. Popescu and D. Rohrlich''Quantum nonlocality as an axiom,''
*Found. Phys. l,***24**379 (1994).

*https://doi.org/10.1007/BF02058098* - L. Lami and C. Pa;azuelos and A. Winter''Ultimate Data Hiding in Quantum Mechanics and Beyond,''
*COMMUN MATH PHYS,***361**661?708(2018).

*https://doi.org/10.1007/s00220-018-3154-4* - H. Barnum, et.al.,''Entropy and information causality in general probabilistic theories,''
*Found. Phys.,***12**033024 (2010).

*https://iopscience.iop.org/article/10.1088/1367-2630/12/3/033024* - P. Janotta and H. Hinrichsen''Generalized probability theories: what determines the structure of quantum theory?,''
*J. Phys. A,***47**323001 (2014).

*https://iopscience.iop.org/article/10.1088/1751-8113/47/32/323001* - M. Pawłowski, et.al.,''Information causality as a physical principle,''
*Nature***461**, 1101–1104 (2009).

*https://www.nature.com/articles/nature08400* - Hayato Arai and Masahito Hayashi, ,
''Pseudo standard entanglement structure cannot be distinguished from standard entanglement structure.''
*New J. Phys.*(accepted manuscript).*https://doi.org/10.1088/1367-2630/acb565*. - M. Hayashi''Quantum Information Theory Mathematical Foundation,''
*Springer-Verlag Berlin Heidelberg,*2ed edition, (2017).

*https://doi.org/10.1007/978-3-662-49725-8* - J. Farautand A. Koranyi''Analysis on Symmetric Cones,''
*Oxford university press,*(1994).

*https://www.oupjapan.co.jp/en/node/18008?language=en*