Bell diagonal state explained

Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory.^[1]

Definition

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

|\phi ^{+\rangle} = (1)/(\sqrt{ 2) } (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B)|\phi ^{-\rangle} = (1)/(\sqrt{ 2) } (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B)|\psi ^{+\rangle} = (1)/(\sqrt{ 2) } (|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B)|\psi ^{-\rangle} = (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 ^{+ \rangle} _{1 |\phi} \langle \phi ^{- \rangle\langle} _{2 |\phi} \phi ^{+ \rangle\langle} _{3 |\psi} \psi ^{- \rangle\langle\psi} _{4 |\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 _{max\leq} 1 / 2 .^[2]

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

Relative entropy of entanglement

S _{r=1-h(p} _{max)} ,^[3] where h is the binary entropy function.

Entanglement of formation

E _{f = h( (1)/(2)} + \sqrt{p _{max(1-p} _{max)})} ,where h is the binary entropy function.

Negativity

N = p _{max-1/2}

Log-negativity

E _{N=log(2} p _{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 ⊗} U _{2} such that U\rho U ^{\dagger} is Bell-diagonal.

Notes and References

1. Horodecki . Ryszard . Horodecki . Paweł . Horodecki . Michał . Horodecki . Karol . 2009-06-17 . Quantum entanglement . Reviews of Modern Physics . 81 . 2 . 865–942 . 10.1103/RevModPhys.81.865. quant-ph/0702225 . 2009RvMP...81..865H . 260606370 .
2. Horodecki . Ryszard . Horodecki . Michal/ . 1996-09-01 . Information-theoretic aspects of inseparability of mixed states . Physical Review A . 54 . 3 . 1838–1843 . 10.1103/PhysRevA.54.1838. 9913669 . quant-ph/9607007 . 1996PhRvA..54.1838H . 2340228 .
3. Vedral . V. . Plenio . M. B. . Rippin . M. A. . Knight . P. L. . 1997-03-24 . Quantifying Entanglement . Physical Review Letters . 78 . 12 . 2275–2279 . 10.1103/PhysRevLett.78.2275. quant-ph/9702027 . 1997PhRvL..78.2275V . 10044/1/300 . 16118336 .