From the author's point of view, all the basic concepts of quantum mechanics can be introduced using a quantum harmonic oscillator. Here are the spectral theorem (for the continuous and discrete case), quantum correlations (introduced using the von Neumann factor apparatus generated by the creation and annihilation operators) and quantum tomography (in which the symbols of observables become generalized functions on the space of basic functions - symbols of states). The course will be devoted to the consistent presentation of such a point of view.

PROGRAMME

1. Quantum measurements as generalized functions on the space of basic functions - quantum states. The case of a quantum harmonic oscillator.
2. The case of projector-valued measures. Spectral theorem. Quantum observables. Discrete and continuous spectra. Quantum oscillator. Coherent states. Function spaces with reproducing kernels.
3. Naimark's theorem on the dilation of positive operator-valued measures. Examples of positive operator-valued measures that are not projector-valued in finite-dimensional and infinite-dimensional spaces. Covariant positive operator- valued measures.
4. Probability distribution for the state-observable pair. Mathematical expectation, variance and covariance of observables. The Schrodinger- Robertson uncertainty relation.
5. The case of quantum observables that are linear combinations of the position and momentum operators. Weyl quantization. Fractional Fourier transform and its relation to the quantum oscillator.
6. Composite quantum systems. Separable and entangles states. Classical and quantum correlations. The Bell-Clauser-Horne-Shimony-Holt inequality. The Tsirelson border. Non-local games with classical and quantum strategies. The advantage of quantum strategies. Spatial and commuting correlations. Refutation of Tsirelson's conjecture about the coincidence of correlation classes.
7. Quantum channels. Kraus decomposition and its non-uniqueness. Quantum coding theorem. Examples of quantum channels: coding and measuring channels, classical-quantum and quantum-classical channels, entanglement breaking channels.
8. Projective unitary representations of finite groups. Quantum channels generated by projective unitary representations of groups. Majorization method for probability distributions. Output information characteristics of channels.
9. Various topographical representations of quantum mechanics. Wigner function, Hushimi-Kano function, optical quantum tomogram. Application to the study of quantum channels.

Program

Lecturer
Amosov Grigori Gennadievich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center