Entangled Bloch spheres: Bloch matrix and two-qubit state space

We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.

Figure 1

Cross sections of the qutrit Bloch vector space. Allowed regions in the hypersphere are shaded.

Figure 2

The Bloch spheres of the two subsystems for (a) generic state, (b) pure state, (c) product state, and (d) singlet state. In each Bloch sphere, the local Cartesian axes vectors are in black, the subsystem's Bloch vector is in red, and the scaled correlation axes ($x_i{\widehat{m}}_i$ or $x_i{\widehat{n}}_i$) dashed in blue. The scaled correlation axes are mutually orthogonal in each Bloch sphere, and are labeled with their index $i$ to indicate the correlation pairing between the two Bloch spheres.

Figure 3

The effect of the irreducible nonlocal unitary transformation due to ${\mathring{U}}_1\left({\theta }_1\right)$ on a generic quantum state. The three singular values $x_i$ are plotted against the parameter angle ${\theta }_1$ in the domain $[-\pi ,\pi ]$.

Figure 4

Singular value diagrams. The regions in singular value space $x_1,x_2,x_3$ where positivity is satisfied, for fixed values of $\vec{g}$ and $\vec{h}$. Bounded by the unit cube, with the origin in the rear bottom left. Regions with $d=1$ in blue and $d=-1$ in red.

Figure 5

Relative Bloch vector diagrams. The coordinates of $\vec{g}=(g_1,g_2,g_3)$ where positivity is satisfied, for fixed values of $\mathrm{\Sigma }$ and $\vec{h}$. Axes in the range $[-1,1]$, with the origin at the center of the cube. Regions with $d=1$ in blue and $d=-1$ in red.