XXXIX Workshop on Geometric Methods in Physics 19.06–25.06.2022 XI School on Geometry and Physics 27.06–1.07.2022
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In order to secure the right conditions for our Workshop we have decided to exceptionally move (only for this year's meeting) the site of our Workshop to the campus of our University in Białystok.

# Martin Bures

## On basic stratified structures in quantum information geometry

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\title[]{On basic stratified structures in quantum information geometry}

%----------Author1
\author{M.Bure\v s}
% 1
Institute of Experimental and Applied Physics\\
Czech Technical University in Prague\\
Husova 240/5, 110 00 Prague 1, Czech Republic\\
\& \\
% 2
Laboratory of Information Technologies \\
Joint Institute for Nuclear Research \\
141980 Dubna, Russia
}
\email{bures@physics.muni.cz}
%----------Author2
\author{A.Khvedelidze}
% 1
Iv. Javakhishvili, Tbilisi State University \\
Tbilisi, Georgia \\
\& \\
% 2
Institute of Quantum Physics and Engineering Technologies \\
Georgian Technical University \\
Tbilisi, Georgia \\
\& \\
%
Laboratory of Information Technologies \\
Joint Institute for Nuclear Research \\
141980 Dubna, Russia
}
\email{akhved@jinr.ru}
%----------Author3
Theoretical Physics Department, Faculty of Physics \\
Sofia University "St Kliment Ohridski" \\
5 James Bourchier Blvd, \\
1164 Sofia, Bulgaria
}
%----------Author4
\author{S.Velkov}
Theoretical Physics Department, Faculty of Physics \\
Sofia University "St Kliment Ohridski" \\
5 James Bourchier Blvd, \\
1164 Sofia, Bulgaria
}
\email{stanislavbg@yahoo.co.uk}
\begin{document}

\maketitle
Searching for non-trivial physical consequences of quantum theory, the knowledge of the mathematical structure of the set of quantum states can be a reliable guide.
The state space $\mathfrak{P}_N$ of an $N$-level quantum system consists of $N\times N$ Hermitian, normalized semi-positive density matrices,
\begin{equation*}
\mathfrak{P}_N= \{\,X \in M_N(\mathbb{C})\, | \,
X = X^\dagger\,, \ X\geq 0\,, \ \mbox{Tr}\, X = 1\,
\}\,.
\end{equation*}
During the last two decades, following the request coming from advanced quantum technologies and quantum information science, the state space $\mathfrak{P}_N$ has been studied in various contexts, among them convex-geometric,
topological, differential-geometric, etc.
(see, e.g. reviews \cite{BengtssonZyczkowski2017,Amari2016,DAndrea2021} and references therein.)

In the present report, we discuss some features of the underlying stratified structure of $\mathfrak{P}_N$. It will be outlined
that among three admissible partitions of $\mathfrak{P}_N$, namely by the adjoint
$SU(N)$ orbits, by the corresponding orbit types, or by the subsets of density matrices with fixed ranks, only the last decomposition determines the Whitney stratification. Based on this observation, we expand some results of our recent paper~\cite{Bures:2021fzy}, devoted to the study of the Bures-Fisher metric for rank deficient states, which are non-maximal dimensional strata of the Whitney stratification.
We comment on the existence of a generalized stratified metric on the whole state space.

The first author was supported from EU Regional Development Fund-Project No.CZ.02.1.01/0.0/0.0/16\_019/0000766.
The research of A.K and D.M was supported in part by the Collaborative grant "Bulgaria-JINR".

\begin{thebibliography}{2}
% 1
\bibitem{BengtssonZyczkowski2017}
I.~Bengtsson and K.~Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition. Cambridge University Press,
(2017)
% 2
\bibitem{Amari2016}
S. Amari,
Information Geometry and Its Applications, Springer, Japan, 2016.
% 3
\bibitem{DAndrea2021}
F. D'Andrea and D. Franco,
On the pseudo-manifold of quantum states,
Differential Geometry and its Applications 78,
101800 (2021).
\bibitem{Bures:2021fzy}
Solving the Uhlmann Equation for the Bures-Fisher Metric on the Subset of Rank-Deficient Qudit States,
9th International Conference on Distributed Computing and Grid Technologies in Science and Education, 358-362 (2021).

\end{thebibliography}
\end{document}

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