Research, publications

Research interests: Matrix analysis, Foundations of Quantum mechanics, Quantum information theory

Papers:

  • D. Petz, A. Szántó: Complementary subalgebras in finite quantum systems, QP--PQ: Quantum Probab. White Noise Anal., vol. 27. (Eds: R. Rebolledo and M. Orsz\'ag), World Scientific, 2011, 282--287.
  • D. Petz, A. Szántó, M. Weiner: Complementarity and the algebraic structure of four-level quantum systems, J. Infin. Dim. Analysis Quantum Prob., 12(2009), 1–18.
  • H. Ohno, D. Petz, A. Szántó: Quasi-orthogonal subalgebras of 4×4 matrices, Linear Alg. Appl. 425(2007), 109–118.
  • K.M. Hangos, D. Petz, A. Szántó, F. Szöllősi: State tomography for two qubits using reduced densities, J. Phys. A: Math. Gen. 39(2006), 10901–10907.

Lectures, Seminars

Mátrixanalízis

Szerda, 12:15-13:45

Házifeladatok

Eredmények

Irodalom:

  • D. Petz: Matrix analysis with some applications
  • Rajendra Bhatia: Matrix Analysis, Springer, 1997
  • Rajendra Bhatia: Positive Definite Matrices, Princeton Univ. Press, 2007
  • Roger A. Horn, Charles R. Johnson: Topics in Matrix Analysis, Cambridge Univ. Press, 1991
  • Nicholas J. Higham: Functions of Matrices: Theory and Computation, SIAM, 2008

2010 Matrix Analysis

R507, Wednesday, 12:15-13:45

Literature:

  • Lecture notes of D. Petz
  • Rajendra Bhatia: Matrix Analysis, Springer, 1997
  • Roger A. Horn, Charles R. Johnson: Topics in Matrix Analysis, Cambridge Univ. Press, 1991
  • Nicholas J. Higham: Functions of Matrices: Theory and Computation, SIAM, 2008

Lectures:

  • Sept. 8.: Rehearsal of basic linear algebra and functional analysis
  • Sept. 15.: Positive matrices, characterizations. Square root, polar decomposition. Minimax principle for eigenvalues, Weyl monotonicity. Hadamard product, Schur theorem. Homework
  • Sept. 22: Norms on matrices, convergence. Neumann series. Von Neumann's mean ergodic theorem. Convergence of monotonic bounded sequences, iteration for square root. Orthogonal projections, intersection. Homework
  • Oct. 6.: Functional calculus for matrices, function of arbitrary rank 1 operator. Exponential function, derivative, exp(A+B) and commutativity. Pauli matrices. Suzuki-Lie-Trotter formula. Golden-Thompson-Lieb inequality. Homework
  • Oct. 13.: Proof of Golden-Thompson inequality. Square root, logarithm, log(aX), log(A+B). Baker-Campbell-Hausdorff formula, Hadamard lemma. Sine, cosine, identites. Derivation. Matrix monotonity. Homework
  • Oct. 20.: Convex functionals, von Neumann entropy, Legendre transform. Hadamard inequality, Hadamard conjecture. Homework
  • Oct. 27.: Matrix convex functions, Hansen-Pedersen theorem. Löwner-Heinz inequality. Connection of matrix monotonity and martix convexity Homework
  • Nov. 3.: Löwner's theorems. Pick functions, Nevanlinna's theorem. Gaussian distributions, Boltzmann entropy, information geometry. Geometric mean, properties. Inequality of the harmonic, geometric and arithmetic means Homework
  • Nov. 10.: Geometric mean for three matrices. Connections, Kubo-Ando theorem. Symmetric means. Logarithmic mean Homework
  • Nov. 24.: Positive maps, Examples: linear functionals, Lyapunov equation. Jordan decomposition. Kadison-Choi inequalities. Russo-Dye theorem, Krein extension theorem. Completely positive maps. Choi-Kraus theorem. Homework
  • Dec. 1.: Stinespring dilation theorem. Partial trace, Pauli Channel. Arveson extension theorem. Schwarz inequality for completely positive maps