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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:

${\displaystyle |\phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B})}$

${\displaystyle |\phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B})}$

${\displaystyle |\psi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B})}$

${\displaystyle |\psi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B})}$

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

${\displaystyle \varrho ^{Bell}=p_{1}|\phi ^{+}\rangle \langle \phi ^{+}|+p_{2}|\phi ^{-}\rangle \langle \phi ^{-}|+p_{3}|\psi ^{+}\rangle \langle \psi ^{+}|+p_{4}|\psi ^{-}\rangle \langle \psi ^{-}|}$

where ${\displaystyle p_{1},p_{2},p_{3},p_{4}}$ is a probability distribution. Since ${\displaystyle p_{1}+p_{2}+p_{3}+p_{4}=1}$, a Bell diagonal state is determined by three real parameters. The maximum probability of a Bell diagonal state is defined as ${\displaystyle p_{max}=\max\{p_{1},p_{2},p_{3},p_{4}\}}$.

Properties

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

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

Relative entropy of entanglement: ${\displaystyle S_{r}=1-h(p_{\text{max}})}$, where ${\displaystyle h}$ is the binary entropy function.

Entanglement of formation: ${\displaystyle E_{f}=h({\frac {1}{2}}+{\sqrt {p_{\text{max}}(1-p_{\text{max}})}})}$,where ${\displaystyle h}$ is the binary entropy function.

Negativity: ${\displaystyle N=p_{\text{max}}-1/2}$

Log-negativity: ${\displaystyle E_{N}=\log(2p_{\text{max}})}$

3. Any 2-qubit state where the reduced density matrices are maximally mixed, ${\displaystyle \rho _{A}=\rho _{B}=I/2}$, is Bell-diagonal in some local basis. Viz., there exist local unitaries ${\displaystyle U=U_{1}\otimes U_{2}}$ such that ${\displaystyle U\rho U^{\dagger }}$ is Bell-diagonal.

References

1. ^ Horodecki, Ryszard; Horodecki, Paweł; Horodecki, Michał; Horodecki, Karol (2009-06-17). "Quantum entanglement". Reviews of Modern Physics. 81 (2): 865–942. arXiv:quant-ph/0702225. Bibcode:2009RvMP...81..865H. doi:10.1103/RevModPhys.81.865. S2CID 260606370.
2. ^ Horodecki, Ryszard; Horodecki, Michal/ (1996-09-01). "Information-theoretic aspects of inseparability of mixed states". Physical Review A. 54 (3): 1838–1843. arXiv:quant-ph/9607007. Bibcode:1996PhRvA..54.1838H. doi:10.1103/PhysRevA.54.1838. PMID 9913669. S2CID 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. arXiv:quant-ph/9702027. Bibcode:1997PhRvL..78.2275V. doi:10.1103/PhysRevLett.78.2275. hdl:10044/1/300. S2CID 16118336.

• Quantum information science
• Quantum states