Quantum states of two qubits
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 :
|
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}\))
.
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]
^ a b 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 .
^ a b 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 .
^ 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 .