Bell-diagonal state

A Bell diagonal state is a 2-qubit state that is diagonal in the Bell basis. In other words, it is a mixture of the four Bell states. It can be written as

p_I |\Phi^+ \rangle \langle \Phi^+| + p_x |\Psi^+ \rangle \langle \Psi^+| + p_y |\Psi^- \rangle \langle \Psi^-| + p_z |\Phi^- \rangle \langle \Phi^-|.

In matrix form it looks like

\frac{1}{2} \begin{bmatrix} p_I + p_z & 0 & 0 & p_I - p_z \\ 0 & p_x + p_y & p_x - p_y & 0 \\ 0 & p_x - p_y & p_x + p_y & 0 \\ p_I - p_z & 0 & 0 & p_I + p_z \\ \end{bmatrix}

where the matrix is in the computational basis.

Because of the simple structure, many questions that are difficult to answer for general 2-qubit states simplify when they are restricted to Bell-diagonal states.

Properties

  • The weights (p1,p2,p3,p4) can be permuted to any other order by local unitaries. Unilateral π rotation around the x-, y- and z-axes and bilateral π / 2 rotations around the same axes are sufficient for this.
  • A Bell-diagonal state is separable if all the probabilities are less or equal to 1/2.
  • Many entanglement measures have a simple formulas for entangled Bell-diagonal states
  • Any 2-qubit state where both qubits are maximally mixed, ρA = ρB = I / 2, is bell-diagonal in some local basis. I.e. there exist local unitaries U1,U2 such that U_1 \otimes U_2 \rho_{AB} U_1^\dagger \otimes U_2^\dagger is bell-diagonal.<ref>quant-ph/9607007</ref>

Visualization

The Bell-diagonal states visualized in the β-coordinate system. The separable states are indicated in the centre of the tetraherdon.

The set of Bell-diagonal states can be visualized as a tetrahedron where the four Bell states are the corners. The following change of coordinate system makes the plotting of states easy:

\beta_0 = \frac{1}{2} ( p_I + p_x + p_y + p_z )
\beta_1 = \frac{1}{2} ( p_I - p_x - p_y + p_z )
\beta_2 = \frac{1}{\sqrt{2}} ( p_I - p_z )
\beta_3 = \frac{1}{\sqrt{2}} ( p_x - p_y )

The coordinate β0 will always be equal to 1/2, and \beta_1 \ldots \beta_3 can be plotted in 3D. In these coordinates the Bell states are located at

|\Phi+ \rangle: \left(\frac{1}{2},\frac{1}{\sqrt{2}}, 0 \right), |\Psi+ \rangle: \left(-\frac{1}{2},0,\frac{1}{\sqrt{2}}\right), |\Psi- \rangle: \left(-\frac{1}{2},0,-\frac{1}{\sqrt{2}}\right), |\Phi-\rangle: \left(\frac{1}{2},-\frac{1}{\sqrt{2}},0\right)
The Bell-diagonal states visualized in the γ-coordinate system.


Another useful coordinate system is the one where the corners of the tetrahedron lie in four of the corners of a cube, with the edges going along the diagonals of the cube's faces.

\gamma_0 = \frac{1}{2}(p_I + p_x + p_y + p_z)
\gamma_1 = \frac{1}{2}(p_I - p_x - p_y + p_z)
\gamma_2 = \frac{1}{2}(p_I - p_x + p_y - p_z)
\gamma_3 = \frac{1}{2}(p_I + p_x - p_y - p_z)

In these coordinates, the Bell states are situated at

|\Phi+ \rangle: (\frac{1}{2},\frac{1}{2},\frac{1}{2}), |\Psi+ \rangle: (-\frac{1}{2},-\frac{1}{2},\frac{1}{2}), |\Psi- \rangle: (-\frac{1}{2},\frac{1}{2},-\frac{1}{2}), |\Phi-\rangle: (\frac{1}{2},-\frac{1}{2},-\frac{1}{2})

The β-coordinate system has the advantage that two of the edges are parallel to axes of the coordinate system. The γ-coordinate system on the other hand inherits more of the symmetry from the cube. Both coordinate transformations are orthogonal, and the transformation from pi to γi is its own inverse.


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