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QUATERNION REPRESENTATION AND SYMPLECTIC SPIN TOMOGRAPHY

Aleksey K. Fedorov, Evgeniy O. Kiktenko

Published Jan 1, 2014• 3 citations
OpenAlex

Abstract

Quantum tomography for continuous variables is based on the symplectic transformation group acting in the phase space. A particular case of symplectic tomography is optical tomography re-lated to the action of a special orthogonal group. In the tomographic description of spin states, the connection between special unitary and special orthogonal groups is used. We analyze the represen-tation for spin tomography using the Cayley–Klein parameters and discuss an analog of symplectic tomography for discrete variables. We propose a representation for tomograms of discrete variables through quaternions and employ the qubit-state tomogram to illustrate the method elaborated.

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Quantum tomography for continuous variables is based on the symplectic transformation group acting in the phase space. A particular case of symplectic tomography is optical tomography re-lated to the action of a special orthogonal group. In the tomographic description of spin states, the connection between special unitary and special orthogonal groups is used. We analyze the represen-tation for spin tomography using the Cayley–Klein parameters and discuss an analog of symplectic tomography for discrete variables. We propose a representation for tomograms of discrete variables through quaternions and employ the qubit-state tomogram to illustrate the method elaborated.
The tomographic-probability description of quantum states is reviewed. The symplectic tomography of quantum states with continuous variables is studied. The symplectic entropy of the states with continuous variables is discussed and its relation to Shannon entropy and information is elucidated. The known entropic uncertainty relations of the probability distribution in position and momentum of a particle are extended and new uncertainty relations for symplectic entropy are obtained. The partial case of symplectic entropy, which is optical entropy of quantum states, is considered. The entropy associated to optical tomogram is shown to satisfy the new entropic uncertainty relation. The example of Gaussian states of harmonic oscillator is studied and the entropic uncertainty relations for optical tomograms of the Gaussian state are shown to minimize the uncertainty relation.
We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU (3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.
The existing relation between the tomographic description of quantum states and the convolution algebra of certain discrete groupoids represented on Hilbert spaces will be discussed. The realizations of groupoid algebras based on qudit, photon-number (Fock) states and symplectic tomography quantizers and dequantizers will be constructed. Conditions for identifying the convolution product of groupoid functions and the star--product arising from a quantization--dequantization scheme will be given. A tomographic approach to construct quasi--distributions out of suitable immersions of groupoids into Hilbert spaces will be formulated and, finally, intertwining kernels for such generalized symplectic tomograms will be evaluated explicitly.
We adopt methods of symplectic tomography and discrete phase space for the description of states of discrete variable quantum systems (qudits). The proposed tomographic functions are constructed as generalized analogs of the center-of-mass and the cluster tomograms and associated with finite linear manifolds in the discrete phase space. Hilbert spaces of considered qudits must have the power of a prime dimension, so the corresponding phase spaces are the vector spaces over finite fields. We find conditions for the nonnegativity of the constructed functions and obtain formulae for the density matrix restoration.
The intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and some of well-known quantum tomographies, such as tomography of rotation group, spinor tomography and tomography of unitary group, is established. Also both the atomic decomposition and Banach frame nature of these quantum tomographic examples are revealed in detail. Finally, we consider separability criteria for any state with group theoretical wavelet transform on Banach spaces.