Two-Qubit Bloch Sphere

Chu-Ryang Wie

doi:10.3390/physics2030021

Abstract

Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base and fiber spheres seamlessly become the single-qubit Bloch spheres of each qubit. Since either qubit can be chosen as the base qubit, two alternative sets of these three spheres are available, where each set fully represents the bipartite pure state, and each set has information of the reduced density matrix of its base qubit. Comparing this model to the two Bloch balls representing the reduced density matrices of the two qubits, each Bloch ball corresponds to two unit spheres in our model, namely, the base and entanglement spheres. The concurrence–coherence complementarity is explicitly shown on the entanglement sphere via a single angle.

Highlights

The Bloch sphere is a geometrical representation of pure single-qubit states as a point on the unit sphere [1]
For two-qubit pure states, for example, a practical Bloch sphere model may try to depict the states using a point on the unit spheres where the linear superposition of basis states and the entanglement may be perceivable from the angle coordinates of the points, represent the local single-qubit gates as rotation operators on the local Bloch spheres, parameterize the entanglement property [8] using a small number of angle coordinates, and represent the joint two-qubit gates as some kind of controlled rotations on the Bloch spheres
This may be a reasonable option because the fiber spheres do not provide any measurable local information about the fiber qubit except that, with the fiber sphere, each 3-sphere set is equal to the bipartite state vector of Equation (1) and that the effect of some single-qubit unitary operations on the fiber qubit may be depicted as a simple rotation on the fiber sphere

Summary

Introduction

The Bloch sphere is a geometrical representation of pure single-qubit states as a point on the unit sphere [1]. Applied the geometric algebra of Doran, Lasenby, and Gull [11] to analyze the space of two-qubit and three-qubit pure states, separating the local and non-local degrees of freedom He used a single angle to represent the entanglement measure in terms of the von Neuman entropy, which at an angle of π/2 was 1 (i.e., maximal entanglement) [6]. Is first given and a more complete version of the two-qubit pure state Bloch sphere model is presented, as well as an improved version of the entanglement sphere that includes an inner sphere of a radius pertinent to the value of concurrence, namely, the entanglement measure This Bloch sphere model is based on the Hopf fibration, as reported in Wie [5], and can have two alternative versions because, in mapping the two-qubit state space S7 using a Hopf fibration, the assignment of the two physical qubits to the Hopf base and fiber spaces is arbitrary. Possible applications and implications of the model are discussed and summarized

Summary of a Two-Qubit Bloch Sphere Model

Two-qubit

Improved Entanglement Sphere and All Three Bloch Spheres

Bloch Sphere Examples with a Two-Qubit Quantum Circuit

Discussion

Summary