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: $E_{N}=\log(2p_{\text{max}})$

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.