Matrix Analysis

(Júlia Réffy, 2026)
  1. Finite-dimensional Hilbert spaces, Dirac formalism, bra and ket vectors. Trace and the Hilbert-Schmidt inner product.
  2. Special operators, spectral decomposition, functional calculus. Generators of B(H), self-adjoint subalgebras. Quantum states and measurements, Born rule.
  3. Absolute value, partial isometries, polar decomposition, singular values.
  4. Positive semi-definite order, minimum and maximum of self-adjoint operators. Trace minimum and maximum, optimal success and error probabilities of quantum state discrimination, max-relative entropy radius and center. Binary case, operational interpretation of the trace norm distance.
  5. Perspective function, classical f-divergences, convexity and monotonicity, variational distance, classical relative entropy and Rényi divergences.
  6. Monotonicity, convexity, and subadditivity of trace functions; Courant-Weyl-Fischer minimax theorem, Jensen inequality with operator weights. Neumann entropy, quantum Rényi entropies.
  7. Operator convex and operator monotone functions, basic examples, special integral representations.
  8. Tensor product of Hilbert spaces and operators. Asymptotic binary i.i.d. state discrimination problem, Audenaert’s inequality, attainability parts of the Stein, Chernoff and Hoeffding error exponents. Petz-type Rényi divergences, Umegaki relative entropy.
  9. Positive semi-definite block operators, Schur complement. Absolutely continuous part. Inequalities for positive and 2-positive super-operators.
  10. Operator perspective function, Kubo-Ando means, Petz-type and maximal quantum f-divergences, quantum Rényi divergences and relative entropies.
  11. Operator Jensen inequality, joint convexity/concavity of Kubo-Ando means and f-divergences.
  12. Discrete Weyl operators, partial trace via twirling. Completely positive maps in Kraus form, Stinespring dilation. Monotonicity of the Petz-type and the maximal f-divergences under CPTP maps.
  13. First and second derivatives of operator functions, characterization of operator monotonicity and convexity via derivatives.