Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory.
[1] The Bell diagonal state is defined as the probabilistic mixture of Bell states: In density operator form, a Bell diagonal state is defined as
ϕ
ϕ
is a probability distribution.
, a Bell diagonal state is determined by three real parameters.
The maximum probability of a Bell diagonal state is defined as
= max {
{\displaystyle p_{max}=\max\{p_{1},p_{2},p_{3},p_{4}\}}
A Bell-diagonal state is separable if all the probabilities are less or equal to 1/2, i.e.,
max
{\displaystyle p_{\text{max}}\leq 1/2}
Many entanglement measures have a simple formulas for entangled Bell-diagonal states:[1] Relative entropy of entanglement:
max
{\displaystyle S_{r}=1-h(p_{\text{max}})}
is the binary entropy function.
Entanglement of formation:
max
max
{\displaystyle E_{f}=h({\frac {1}{2}}+{\sqrt {p_{\text{max}}(1-p_{\text{max}})}})}
is the binary entropy function.
Negativity:
max
{\displaystyle N=p_{\text{max}}-1/2}
= log ( 2
max
{\displaystyle E_{N}=\log(2p_{\text{max}})}
Any 2-qubit state where the reduced density matrices are maximally mixed,
, is Bell-diagonal in some local basis.
Viz., there exist local unitaries