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 state]]s. It can be written as :$p\_I\; |\backslash Phi^+\; \backslash rangle\; \backslash langle\; \backslash Phi^+|\; +\; p\_x\; |\backslash Psi^+\; \backslash rangle\; \backslash langle\; \backslash Psi^+|\; +\; p\_y\; |\backslash Psi^-\; \backslash rangle\; \backslash langle\; \backslash Psi^-|\; +\; p\_z\; |\backslash Phi^-\; \backslash rangle\; \backslash langle\; \backslash Phi^-|$. In matrix form it looks like : 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 $(p\_1,\; p\_2,\; p\_3,\; p\_4)$ can be permuted to any other order by local unitaries. Unilateral $\backslash pi$ rotation around the x-, y- and z-axes and bilateral $\backslash pi/2$ 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 measure | entanglement measures]] have a simple formulas for entangled Bell-diagonal states ** [[Relative entropy of entanglement]]: $E\_r\; =\; 1\; -\; h(p\_\{max\})$, where $h$ is the binary entropy function $h(x)\; =\; -\; x\; \backslash log\_2(x)\; -\; (1-x)\; \backslash log\_2(1-x)$quant-ph/9702027 ** [[Entanglement of formation]]: $E\_f\; =\; h\backslash left(1/2\; +\; \backslash sqrt\{p\_\{max\}(1-p\_\{max\})\}\backslash right)$quant-ph/9604024 ** [[Negativity]]: $N\; =\; p\_\{max\}\; -\; 1/2$ ** [[Log-negativity]]: $E\_N\; =\; \backslash log(2\; p\_\{max\})$ * Any 2-qubit state where both qubits are maximally mixed, $\backslash rho\_A\; =\; \backslash rho\_B\; =\; I/2$, is bell-diagonal in some local basis. I.e. there exist local unitaries $U\_1,\; U\_2$ such that $U\_1\; \backslash otimes\; U\_2\; \backslash rho\_\{AB\}\; U\_1^\backslash dagger\; \backslash otimes\; U\_2^\backslash dagger$ is bell-diagonal.quant-ph/9607007 == Visualization == [[Image:Belldiagstates_beta.png|thumb|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: :$\backslash beta\_0\; =\; \backslash frac\{1\}\{2\}\; (\; p\_I\; +\; p\_x\; +\; p\_y\; +\; p\_z\; )$ :$\backslash beta\_1\; =\; \backslash frac\{1\}\{2\}\; (\; p\_I\; -\; p\_x\; -\; p\_y\; +\; p\_z\; )$ :$\backslash beta\_2\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (\; p\_I\; -\; p\_z\; )$ :$\backslash beta\_3\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (\; p\_x\; -\; p\_y\; )$ The coordinate $\backslash beta\_0$ will always be equal to 1/2, and $\backslash beta\_1\; \backslash ldots\; \backslash beta\_3$ can be plotted in 3D. In these coordinates the Bell states are located at :$|\backslash Phi+\; \backslash rangle:\; \backslash left(\backslash frac\{1\}\{2\},\backslash frac\{1\}\{\backslash sqrt\{2\}\},\; 0\; \backslash right)$, $|\backslash Psi+\; \backslash rangle:\; \backslash left(-\backslash frac\{1\}\{2\},0,\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash right)$, $|\backslash Psi-\; \backslash rangle:\; \backslash left(-\backslash frac\{1\}\{2\},0,-\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash right)$, $|\backslash Phi-\backslash rangle:\; \backslash left(\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{\backslash sqrt\{2\}\},0\backslash right)$ [[Image:Belldiagstates_gamma.png|thumb|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. :$\backslash gamma\_0\; =\; \backslash frac\{1\}\{2\}(p\_I\; +\; p\_x\; +\; p\_y\; +\; p\_z)$ :$\backslash gamma\_1\; =\; \backslash frac\{1\}\{2\}(p\_I\; -\; p\_x\; -\; p\_y\; +\; p\_z)$ :$\backslash gamma\_2\; =\; \backslash frac\{1\}\{2\}(p\_I\; -\; p\_x\; +\; p\_y\; -\; p\_z)$ :$\backslash gamma\_3\; =\; \backslash frac\{1\}\{2\}(p\_I\; +\; p\_x\; -\; p\_y\; -\; p\_z)$ In these coordinates, the Bell states are situated at :$|\backslash Phi+\; \backslash rangle:\; (\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\})$, $|\backslash Psi+\; \backslash rangle:\; (-\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\})$, $|\backslash Psi-\; \backslash rangle:\; (-\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\})$, $|\backslash Phi-\backslash rangle:\; (\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\})$ The $\backslash beta$-coordinate system has the advantage that two of the edges are parallel to axes of the coordinate system. The $\backslash gamma$-coordinate system on the other hand inherits more of the symmetry from the cube. Both coordinate transformations are orthogonal, and the transformation from $p\_i$ to $\backslash gamma\_i$ is its own inverse. [[Category:Quantum States]]