Article · Wikipedia archive · Last revised Jun 12, 2026

Bell diagonal state

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

Last revised
Jun 12, 2026
Read time
≈ 3 min
Length
609 w
Citations
5
Source
Wikipedia ↗

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:

| ϕ + ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B + | 1 ⟩ A ⊗ | 1 ⟩ B ) {\displaystyle |\phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B})}
| ϕ − ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B − | 1 ⟩ A ⊗ | 1 ⟩ B ) {\displaystyle |\phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B})}
| ψ + ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B + | 1 ⟩ A ⊗ | 0 ⟩ B ) {\displaystyle |\psi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B})}
| ψ − ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B − | 1 ⟩ A ⊗ | 0 ⟩ 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

ϱ B e l l = p 1 | ϕ + ⟩ ⟨ ϕ + | + p 2 | ϕ − ⟩ ⟨ ϕ − | + p 3 | ψ + ⟩ ⟨ ψ + | + p 4 | ψ − ⟩ ⟨ ψ − | {\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 p 1 , p 2 , p 3 , p 4 {\displaystyle p_{1},p_{2},p_{3},p_{4}} is a probability distribution. Since p 1 + p 2 + p 3 + p 4 = 1 {\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 p m a x = max { p 1 , p 2 , p 3 , p 4 } {\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., p max ≤ 1 / 2 {\displaystyle p_{\text{max}}\leq 1/2} .2

2. Many entanglement measures have a simple formulas for entangled Bell-diagonal states:1

Relative entropy of entanglement: S r = 1 − h ( p max ) {\displaystyle S_{r}=1-h(p_{\text{max}})} ,3 where h {\displaystyle h} is the binary entropy function.

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

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

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

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

References

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.