Abstract
Let
H
1
,
H
2
be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose
ρ
i
is a state in
H
i
,i=1,2.
Let
C(
ρ
1
,
ρ
2
)
be the convex set of all states
ρ
in
H=
H
1
⊗
H
2
whose marginal states in
H
1
and
H
2
are
ρ
1
and
ρ
2
respectively. Here we present a necessary and sufficient criterion for a
ρ
in
C(
ρ
1
,
ρ
2
)
to be an extreme point. Such a condition implies, in particular, that for a state
ρ
to be an extreme point of
C(
ρ
1
,
ρ
2
)
it is necessary that the rank of
ρ
does not exceed
(
d
2
1
+
d
2
2
−1
)
1/2
,
where
d
i
=dim
H
i
,i=1,2.
When
H
1
and
H
2
coincide with the 1-qubit Hilbert space
C
2
with its standard orthonormal basis
{|0>,|1>}
and
ρ
1
=
ρ
2
=1/2I
it turns out that a state
ρ∈C(1/2I,1/2I)
is extremal if and only if
ρ
is of the form
|Ω><Ω|
where
|Ω>=
1
2
√
(|0>|
ψ
0
>+|1>|
ψ
1
>),
{|
ψ
0
>,|
ψ
1
>}
being an arbitrary orthonormal basis of
C
2
.
In particular, the extremal states are the maximally entangled states.