Bell diagonal state

Short description: Quantum states of two qubits

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:

$|{\varphi }^{+}⟩=\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |0{⟩}_B+|1{⟩}_A\otimes |1{⟩}_B)$

$|{\varphi }^{-}⟩=\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |0{⟩}_B-|1{⟩}_A\otimes |1{⟩}_B)$

$|{\psi }^{+}⟩=\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |1{⟩}_B+|1{⟩}_A\otimes |0{⟩}_B)$

$|{\psi }^{-}⟩=\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |1{⟩}_B-|1{⟩}_A\otimes |0{⟩}_B)$

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

${\varrho }^{Bell}=p_1|{\varphi }^{+}⟩⟨{\varphi }^{+}|+p_2|{\varphi }^{-}⟩⟨{\varphi }^{-}|+p_3|{\psi }^{+}⟩⟨{\psi }^{+}|+p_4|{\psi }^{-}⟩⟨{\psi }^{-}|$

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., $p_{\text{max}}\le 1/2$.

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 $U\rho U^{†}$is Bell-diagonal.