Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and Education" (GRID'2021), Dubna, Russia, July 5-9, 2021 SOLVING THE UHLMANN EQUATION FOR THE BURES- FISHER METRIC ON THE SUBSET OF RANK-DEFICIENT QUDIT STATES M. Bures1,2,a, A. Khvedelidze2,3,4, D. Mladenov5 1 Institute of Experimental and Applied Physics Czech Technical University in Prague, Husova 240/5 110 00 Prague 1. Czech Republic 2 Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia 3 A.Razmadze Mathematical Institute, Iv.Javakhishvili Tbilisi State University, Tbilisi, Georgia 4 Institute of Quantum Physics and Engineering Technologies, Georgian Technical University, Tbilisi, Georgia 5 Faculty of Physics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd, 1164 Sofia, Bulgaria E-mail: a bures@physics.muni.cz The Bures-Fisher metric on the subset of the state space of an N-level quantum system, consisting of rank-k density matrices is given by a solution to the Uhlmann equation. Solving the Uhlmann equation on, we use its decomposition into a finite union of strata of different SU(N) orbit types with all admissible isotropy groups Hα . Solution to the Uhlmann equation on the corresponding orbits stratum defines uniquely the Bures-Fisher metric for a rank deficient states. Keywords: quantum computing, quantum information, quantum Fisher information, Bures metric, qubit, qutrit, qudit Martin Bures, Arsen Khvedelidze, Dimitar Mladenov Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). 358 Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and Education" (GRID'2021), Dubna, Russia, July 5-9, 2021 1. Introduction Modern developments in theoretical quantum metrology have given rise to a fresh interest in the quantum Fisher information and the corresponding Riemannian geometry structure, the Bures metrics on the state space of a finite dimensional quantum system (see e.g. the review in [1] and references therein). Currently, it is a common view that: • the Bures metric is locally equivalent to a Riemannian metric determined by the quantum analog of the Fisher information matrix [2, p.262]; • the quantum Fisher information matrix and the Bures metric are equivalent to each other, except at the points where the rank of the density matrix changes [3-6]; To clarify these interrelations, the knowledge of generic topological features and differential- geometrical properties of the convex body of quantum states is very useful. More precisely, aiming to determine the Bures-Fisher metric on the subset of fixed rank-k states, , it is helpful to decompose it into components of strata of orbits of adjoint action of the unitary group. This decomposition follows by combining two partitions of the state space into different topological subspaces. The first one is a well-defined partition, the stratification of into unitary orbit types: (1) where is the stratum associated to the isotropy group . The second partition is the decomposition of the state space into subsets , consisting of density matrices of rank , (2) Comparing (1) and (2), we arrive at the decomposition of each component of a fixed rank-k as a union of orbits of certain types, (3) In the next section, we briefly summarize the results of studying the Uhlmann equation (4) on each component of (3). It will be outlined that the knowledge of the topological decomposition (3) and the unique Bures-Fisher metric on unitary strata allows to define the corresponding metric on rank-k subset and analyze the singularities of this metric in the neighbourhood of states with different ranks. 359 Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and Education" (GRID'2021), Dubna, Russia, July 5-9, 2021 2. Bures-Fisher metric from the Uhlmann equation Let be an element of , a set of unit trace semi-positive density matrices of rank . For a given , consider the equation (4) for an unknown 1-form . According to A.Uhlmann [7], the solution to (4) determines the Bures- Fisher metric on as (5) In order to treat as a Riemannian manifold endowed with the Bures-Fisher metric, we need to analyze the existence and uniqueness of the solution to (4) for all k=1,2,..., N. Following the standard theory of systems of linear equations, one can easily formulate the corresponding conditions on the density matrix, which guarantee the existence and uniqueness of the Bures-Fisher metric (5). The Uhlmann equation, being a system of linear equations, has its solution represented as (6) where is some particular solution and stands for a general solution of the corresponding homogeneous equation. If then is trivial, while for singular density matrices, i.e. rank deficient states with , the number of linearly independent solutions of the homogeneous Uhlmann equation is . However, it can be shown that the metric form (5), evaluated at the vectors tangent to turns out to be independent of all parameters in the solution to the Uhlmann equation. To verify these statements, we note that equation (4) is a special form of the famous Sylvester matrix equation with matrices 1 for an unknown matrix X: AX + XB = C. (7) In 1884, Sylvester (cf.[8]) considered the homogeneous version of this equation and thereby showed that the condition for (7) to have a unique solution is that A and −B have no eigenvalues in common. Following these propositions for equal Hermitian matrices A=B=A†, one can prove that a) equation (4) admits the Bures-Fisher metric on states of all possible ranks; b) for rank deficient states, rank (ρ)=k