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AIP conference proceedings

Euler angles parametrization of a two qubit system

Aleksander Aleksiev Stefanov, Dimitar M. Mladenov

Published Jan 1, 2019• 0 citations
Publisher

Abstract

We present a Cartan decomposition of SU(4), analogous to the Euler angles decomposition for SO(3). It can be used to parameterize the pure states of a two-qubit system and its relation to entanglement is briefly discussed. We also give a formula for the entanglement entropy in terms of this parametrization.
DOI
10.1063/1.5127498

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In quantum mechanics, sets of density matrices are important for numerous reasons. For example, their compact notation make them useful for describing decoherence and entanglement properties of multi-particle quantum systems. In particular, two two-state density matrices, otherwise known as two qubit density matrices, are important for their role in explaining quantum teleportation, dense coding, and computation theorems. The aim of this paper is to show an explicit parameterization for the Hilbert space of all two qubit density matrices. Such a parameterization would be extremely useful for numerical calculations concerning entanglement and other quantum information parameters. We would also like to know the properties of such parameterized two qubit density matrices; in particular, the representation of their convex sets, their sub-sets, and their set boundaries in terms of our parameterization. Here we present a generalized Euler angle parameterization for SU(4) and all possible two qubit density matrices as well as the corrected Haar Measures for SU(3) and SU(4) from this parameterization. The role of the parameterization in the Peres-Horodecki criteria will also be introduced as well as its usefulness in calculating entangled two qubit states.
In this paper we give an explicit parametrization for all two-qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU(4) and all possible two-qubit density matrices. The important group-theoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU(4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres–Horodecki criteria for separability and its corresponding usefulness in calculating entangled two-qubit states as represented through the parametrization. PACS numbers: 03.67.−a, 02.20.−a, 03.65.Ud 1.
In a previous paper [1] an Euler angle parametrization for SU (4) was given. Here we present the derivation of a generalized Euler angle parametrization for SU ( N ). The formula for the calculation of the Haar measure for SU ( N ) as well as its relation to Marinov's volume formula for SU ( N ) [2, 3] will also be derived. As an example of this parametrization's usefulness, the density matrix parametrization and invariant volume element for a qubit/qutrit, three qubit and two three-state systems, also known as two qutrit systems [4], will also be given.
In this paper we give an explicit parametrization for all two-qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU (4) and all possible two-qubit density matrices. The important group-theoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU (4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres–Horodecki criteria for separability and its corresponding usefulness in calculating entangled two-qubit states as represented through the parametrization.