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:

: $, \phi^+\rangle = \frac (, 0\rangle_A \otimes , 0\rangle_B + , 1\rangle_A \otimes , 1\rangle_B)$
: $, \phi^-\rangle = \frac (, 0\rangle_A \otimes , 0\rangle_B - , 1\rangle_A \otimes , 1\rangle_B)$
: $, \psi^+\rangle = \frac (, 0\rangle_A \otimes , 1\rangle_B + , 1\rangle_A \otimes , 0\rangle_B)$
: $, \psi^-\rangle = \frac (, 0\rangle_A \otimes , 1\rangle_B - , 1\rangle_A \otimes , 0\rangle_B)$

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

$\varrho^=p_1, \phi^+\rangle \langle \phi^+, +p_2, \phi^-\rangle\langle \phi^-, +p_3, \psi^+\rangle\langle \psi^+, +p_4, \psi^-\rangle\langle\psi^-,$

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\$.

Properties

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

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

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

Entanglement of formation: $E_f=h(\frac+\sqrt)$,where $h$ is the binary entropy function.

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

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

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.

References

{{DEFAULTSORT:Bell diagonal states
Quantum information science
Quantum states