7. Quantum formalism on the plane: POVM-Toeplitz quantization, Naimark theorem and linear polarization of the light

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Context: Measurements, i.e. distributions around a given value, are described by positive operator- valued measures (POVM) in quantum mechanics. It is now admitted that POVM play a crucial role in quantum formalism, specially in quantum measurement and quantum information. With Roberto Beneduci, Jean-Pierre Gazeau and Amedeo Perri, I studied various interesting aspects of POVM on the Euclidean plane, the simplest Hilbert space, which is already rich in illustrations of non-trivial aspects of quantum formalism.

Method: I constructed generalised coherent states and compared the quantisation of a function using integral quantisation and Toeplitz quantisation. I used our finding to determine whether Naimark’s theorem is valid with integral quantisation. These general results were studied in depth in two dimensions by restricting the analysis to functions defined on the circle. I illustrated these results with an example of quantum measurements on the plane, which can be viewed as the linear polarisation of the light through Stokes parameters. Finally, I defined the notion of incompatibility between two observables in this framework.

Results:

• I found that Toeplitz quantisation is a kind of integral quantisation, and that Naimark’s theorem is valid. Naimark’s dilations were also given.
• In 2d, the quantisation of functions on the unit circle shows that the quantization map gives a non-commutative version of R3 identified as a Fourier subspace.
• Applying Naimark’s theorem made possible the identification between the quantum operator of a POVM and its density matrix.
• Density matrices take the form of a polarisation tensor when circular polarisation is neglected.
• I showed that, after a sequential measurement, we can identify the degree of mixing of a density matrix to the fuzziness of a quantum observable. Therefore, the compatibility conditions of two POVMs can be expressed in terms of Stokes parameters.