Two-Qubit Bloch Sphere

Three unit spheres can represent the two-qubit pure states. One such model previously reported [1] is refined and presented. The three spheres are named the base sphere, entanglement sphere and fiber sphere. The base sphere and the entanglement sphere represent both the reduced density matrix of one qubit (the base qubit) as well as the non-local entanglement measure, concurrence, while the fiber sphere represents the other qubit (the fiber qubit) geometrically under a local unitary operation. When the bipartite state becomes separable, the base sphere and the fiber sphere seamlessly become the single-qubit Bloch sphere of each qubit. Since either qubit may be chosen to be the base qubit, two sets of such spheres can fully represent the reduced density matrices of both qubits as well as the concurrence where the concurrence value is the same in the two sets. The concurrence in this Bloch sphere is correctly related to the reduced density matrices. In particular, the entanglement (concurrence) and the imaginary part of coherence (off-diagonal element of the reduced density matrix) are related via an angle parameter and are represented together in the entanglement sphere. We illustrate partially entangled two-qubit states on the Bloch spheres. The significance of each sphere is discussed.