The quantized electromagnetic field is a well-known example of a continuous variable system. The partially reflecting mirror, termed the beam-splitter, is the primitive for all passive transformations . The beam-splitter transform has been used for the basis of generalizing the entropy power inequality to quantum mechanical setting [2,3]. The central formula is the map (ρX, ρY) ↦ ρX ⊞λ Y = Tr2(Uλ(ρX ⊗ ρY)U†λ), where Uλ is the beam-splitter transform. To achieve the same for the discrete or finite dimensional case [4,5] we need a generalization of the beam-splitter for finite dimensions. A promising approach is to use the discrete phase space distribution introduced by Wootters .
The aim of this assigment is to understand the phase space formalism for continuous and discrete quantum systems, and to define a discrete version of the usual beam-splitter in a way that is consistent in the limit of large dimensionality with the continuous variable case.
The assigment is suitable both for a bachelors and masters thesis. In case of a bachelor thesis the emphasis will be on the study of the phase space formalism and the formulation of the problem of the discrete beam-splitter, with the possibility to continue the research as master's thesis.
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- Wolfgang P. Schleich: "Quantum Optics in Phase Space," Wiley-VCH Verlag, Berlin (2001)
- Robert König and Graeme Smith: "The Entropy Power Inequality for Quantum Systems," IEEE Trans. Inf. Theory, 60, 1536-1548, (2014)
- Stefan Huber, Robert König, and Anna Vershynina: "Geometric inequalities from phase space translations," J. Math. Phys. 58, 012206 (2017)
- K. Audenaert, N. Datta, and M. Ozols: “Entropy power inequalities for qudits,” J. Math. Phys. 57, 052202 (2016)
- S. Guha et al.: “Thinning, photonic beam splitting, and a general discrete entropy power inequality,” in 2016 IEEE International Symposium on Information Theory (ISIT), IEEE, Barcelona, 2016. pp. 705–709.
- W. K. Wootters: "A Wigner-Function Formulation of Finite-State Quantum Mechanics," Ann. Phys. 176, 1-21 (1987)