Quaternion representation and symplectic spin tomography

TL;DR

This paper explores advanced mathematical frameworks for quantum state tomography, introducing quaternion-based representations for discrete variables and analyzing symplectic and spin tomography methods.

Contribution

It proposes a novel quaternion-based representation for discrete variable tomograms and discusses an analog of symplectic tomography in this context.

Findings

Quaternion representation for discrete variables introduced

Application to qubit-state tomograms demonstrated

Connection between symplectic and spin tomography analyzed

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 related 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 representation 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.