R Simon
Articles written in Pramana – Journal of Physics
Volume 65 Issue 6 December 2005 pp 981-993
Wigner distributions for finite dimensional quantum systems: An algebraic approach
S Chaturvedi E Ercolessi G Marmo G Morandi N Mukunda R Simon
We discuss questions pertaining to the definition of ‘momentum’, ‘momentum space’, ‘phase space’ and ‘Wigner distributions’; for finite dimensional quantum systems. For such systems, where traditional concepts of ‘momenta’ established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail
Volume 69 Issue 3 September 2007 pp 317-327 Research Articles
Ray space `Riccati' evolution and geometric phases for 𝑁-level quantum systems
S Chaturvedi E Ercolessi G Marmo G Morandi N Mukunda R Simon
We present a simple derivation of the matrix Riccati equations governing the reduced dynamics as one descends from the group $\mathbb{U}(N)$ describing the Schrõdinger evolution of an 𝑁-level quantum system to the various coset spaces and Grassmanian manifolds associated with it. The special case pertaining to the geometric phase in 𝑁-level systems is described in detail. Further, we show how the matrix Riccati equation thus obtained can be reformulated as an equation describing Hamiltonian evolution in a classical phase space and establish correspondences between the two descriptions.
Volume 73 Issue 3 September 2009 pp 471-483
The structure of states and maps in quantum theory
Sudhavathani Simon S P Rajagopalan R Simon
The structure of statistical state spaces in the classical and quantum theories are compared in an interesting and novel manner. Quantum state spaces and maps on them have rich convex structures arising from the superposition principle and consequent entanglement. Communication channels (physical processes) in the quantum scheme of things are in one-to-one correspondence with completely positive maps. Positive maps which are not completely positive do not correspond to physical processes. Nevertheless they prove to be invaluable mathematical tools in establishing or witnessing entanglement of mixed states. We consider some of the recent developments in our understanding of the convex structure of states and maps in quantum theory, particularly in the context of quantum information theory.
Volume 74 Issue 1 January 2010 pp 1-10 Research Articles
Hamilton’s theory of turns revisited
N Mukunda S Chaturvedi R Simon
We present a new approach to Hamilton’s theory of turns for the groups $SO(3)$ and $SU(2)$ which renders their properties, in particular their composition law, nearly trivial and immediately evident upon inspection. We show that the entire construction can be based on binary rotations rather than mirror reflections.
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