MAGIC048: Quantum Statistics

Contents: The recent advances in Quantum Information and Quantum Computation have brought a paradigm shift in the way we think about encoding and manipulating information. Atoms and photons are carriers of a new type of information and thanks to the modern technology we have reached the point where we can manipulate and measure individual quantum systems. A fundamental implication of these developments is that statistical inference based on data obtained by measuring a limited number of individual systems, will play a much greater role in quantum theory.
These lectures give an short overview of the current status in quantum statistics starting from the first methods developed in the 70's, and up to the latest theoretical and experimental results. The guiding principle is to adapt and extend well established `classical' statistical inference techniques to the quantum set-up, and to identify the `purely quantum' features that need to be explored. In parallel, some recent practical applications will be discussed.
Literature:

• Artiles, L, Gill, R., Guta, M., An invitation to quantum tomography, J. Royal Statist. Soc. B, 67, (2005), 109-134.
• Barndorff-Nielsen O.E., Gill, R., Jupp, P. E., On quantum statistical inference (with discussion), J. R. Statist. Soc. B, 65, (2003), 775-816.
• Guta M., Janssens B., Kahn J., Optimal estimation of qubit states with continuous time measurements, Commun. Math. Phys., 277, (2008), 127-160.
• Helstrom C.W., Quantum Detection and Estimation Theory, Academic Press, New York (1976).
• Holevo A.S., Probabilistic and Statistical Aspects of Quantum Theory, North-Holland (1982).
• Nielsen, M. A. and Chuang, I. L., Quantum Computation and Quantum Information, Cambridge University Press, (2000)

Basic courses on: Quantum Mechanics and/or Hilbert space theory, statistics and probablity.

1. Quantum mechanics revisited:
   Hilbert space, selfadjoint and positive operators, states, measurements;
2. Notions of statistical inference:
   statistical decision problems, Cramer-Rao bound, bias estimation for coin toss;
3. Quantum state estimation preliminaries:
   Quantum Fisher information, quantum Cramer-Rao bound, Holevo bound;
4. Estimation for covariant families of states:
   Covariant measurements, seed of measurement, optimality, examples;
5. Quantum state discrimination:
   Helstrom measurement, classical and quantum Chernoff bound, square-root measurement;
6. Quantum Homodyne Tomography:
   quantum harmonic oscillator, homodne measurements, Radon transform, pattern functions, consistent estimators;
7. Estimation of Gaussian states:
   definition of Gaussian states, heterodyne measurements, optimality;
8. Optimal estimation for qubit states (I):
   spin coherent states, irreducible representations of SU(d), quantum central limit theorem;
9. Optimal estimation for qubit states (I):
   local asymptotic normality, adaptive measurements, asymptotic optimality;
10. Further topics:
    Estimation of unitary channels, quantum cloning.