Bell diagonal state

Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory.

Definition
The Bell diagonal state is defined as the probabilistic mixture of Bell states:


 * $$|\phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B)$$
 * $$|\phi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B)$$
 * $$|\psi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B)$$
 * $$|\psi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B)$$

In density operator form, a Bell diagonal state is defined as

$$\varrho^{Bell}=p_1|\phi^+\rangle \langle \phi^+|+p_2|\phi^-\rangle\langle \phi^-|+p_3|\psi^+\rangle\langle \psi^+|+p_4|\psi^-\rangle\langle\psi^-|$$

where $$p_1,p_2,p_3,p_4$$is a probability distribution. Since $$p_1+p_2+p_3+p_4=1$$, a Bell diagonal state is determined by three real parameters. The maximum probability of a Bell diagonal state is defined as $$ p_{max}=\max\{p_1,p_2,p_3,p_4\}$$.

Properties
1. A Bell-diagonal state is separable if all the probabilities are less or equal to 1/2, i.e., $$p_\text{max}\leq 1/2$$.

2. Many entanglement measures have a simple formulas for entangled Bell-diagonal states:

Relative entropy of entanglement: $$S_r=1-h(p_\text{max})$$, where $$h$$ is the binary entropy function.

Entanglement of formation: $$E_f=h(\frac{1}{2}+\sqrt{p_\text{max}(1-p_\text{max})})$$,where $$h$$ is the binary entropy function.

Negativity: $$N=p_\text{max}-1/2$$

Log-negativity: $$E_N=\log(2 p_\text{max} )$$

3. Any 2-qubit state where the reduced density matrices are maximally mixed, $$\rho_A=\rho_B=I/2$$, is Bell-diagonal in some local basis. Viz., there exist local unitaries $$U=U_1\otimes U_2$$ such that $$U\rho U^{\dagger} $$is Bell-diagonal.