# zbMATH — the first resource for mathematics

Parametrizations of degenerate density matrices. (English) Zbl 1380.15031
This paper develops a new parametrization for degenerate density matrices, i.e., for positive matrices in Hilbert spaces with trace equal to 1. The problem pertains to quantum systems and physics in general and is studied here for finite dimensions. The two linked conditions for density matrices introduce dependencies for their entries and therefore a parametrization is desirable that eliminates the redundancies effectively. A new continuous form is given for such a parametrization that does not rely solely on Lie algebras but in addition also uses the theory of homogeneous spaces. Two simple low dimensional examples complete the paper.
##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 81Q80 Special quantum systems, such as solvable systems 22E70 Applications of Lie groups to the sciences; explicit representations 15A18 Eigenvalues, singular values, and eigenvectors
Full Text:
##### References:
 [1] Akhtarshenas, S. J., Canonical coset parametrization and the bures metric of the three-level quantum systems, J. Math. Phys., 48, 1, 012102, (2007) · Zbl 1121.81059 [2] Akhtarshenas, S. J., Coset parametrization of density matrices, Opt. Spectrosc., 103, 3, 411-415, (2007) [3] Blanchard, Ph.; Brüning, E., Mathematical Methods in Physics Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics, 69, (2015), Birkhäuser · Zbl 1330.46001 [4] Bloch, F., Nuclear induction, Phys. Rev., 70, 7-8, 460-474, (1946) [5] Blum, K., Density Matrix Theory and Applications, 411, (2012), Springer · Zbl 1242.81002 [6] Brüning, E.; Chruściński, D.; Petruccione, F., Parametrizing density matrices for composite quantum systems, Open Syst. Inf. Dyn., 15, 4, 397-408, (2008) · Zbl 1188.81040 [7] Brüning, E.; Petruccione, F., Density matrices and their time evolution, Open Syst. Inf. Dyn., 15, 2, 109-121, (2008) · Zbl 1145.81015 [8] Byrd, M. S.; Khaneja, N., Characterization of the positivity of the density matrix in terms of the coherence vector representation, Phys. Rev. A, 68, 13, (2003) [9] Brüning, E.; Messina, A.; Mäkelä, H.; Petruccione, F., Topical review parametrizations of density matrices, J. Modern Opt., 59, 11, 1-20, (2012) · Zbl 1356.81244 [10] Harriman, J. E., Geometry of density matrices. I. definitions, $$N$$ matrices and 1 matrices, Phys. Rev. A, 17, 4, 1249-1256, (1978) [11] Jarlskog, C., A recursive parametrization of unitary matrices, J. Math. Phys., 46, 10, 4, (2005) · Zbl 1111.15027 [12] Jarlskog, C., Recursive parametrization and invariant phases of unitary matrices, J. Math. Phys., 47, 1, 013507, (2006) · Zbl 1111.81076 [13] Matsushima, Y., Differentiable Manifolds, (1972), Marcel Dekker, Inc. [14] Nielsen, M. A.; Chuang, I. L., Quantum Computation and Quantum Information, (2004), Cambridge University Press [15] Spengler, C.; Huber, M.; Hiesmayr, B. C., A composite parameterization of unitary groups, density matrices and subspaces, J. Phys. A, 43, 38, 385306, (2010) · Zbl 1198.81058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.