ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

60 4. Two-mode entanglement
where δô = 〈ô 2 〉 − 〈ô〉 2 for an operator ô. If ξ ≥ 1 then the state does not possess
non-local correlations [70]. The idealized EPR state [73] (simultaneous eigenstate
of the commuting observables ˆq1 − ˆq2 and ˆp1 + ˆp2) has ξ = 0. As for standard form
two-mode Gaussian states, Eq. (4.1), one has
δˆq1−ˆq2 = a + b − 2c+ , (4.20)
δˆp1+ˆp2 = a + b + 2c− , (4.21)
ξ = a + b − c+ + c− . (4.22)
Notice that ξ is not by itself a good measure of correlation because, as one can easily
verify, it is not invariant under local symplectic operations. In particular, applying
local squeezings with parameters ri = log vi and local rotations with angles ϕi to a
standard form state, we obtain
ξvi,ϑ = a
2
v 2 1 + 1
v 2 1
+ b
2
with ϑ = ϕ1 + ϕ2. Now, the quantity
¯ξ ≡ min
vi,ϑ ξvi,ϑ
v 2 2 + 1
v2
− c+v1v2 −
2
c−
cos ϑ , (4.23)
v1v2
has to be Sp (2,R) ⊕Sp (2,R) invariant. It corresponds to the maximal amount of EPR
correlations which can be distributed in a two-mode Gaussian state by means of
local operations. Minimization in terms of ϑ is immediate, yielding ¯ ξ = minvi ξvi,
with
ξvi = a
v
2
2 1 + 1
v2
+
1
b
v
2
2 2 + 1
v2
−
2
c+v1v2 − c−
v1v2
. (4.24)
The gradient of such a quantity is null if and only if
a v 2 1 − 1
v2
− |c+|v1v2 −
1
|c−|
= 0 , (4.25)
v1v2
b v 2 2 − 1
v2
− |c+|v1v2 −
2
|c−|
= 0 , (4.26)
v1v2
where we introduced the position c+c− < 0, necessary to have entanglement, see
Eq. (4.16). Eqs. (4.25, 4.26) can be combined to get
a v 2 1 − 1
v2
= b v
1
2 2 − 1
v2
. (4.27)
2
Restricting to the symmetric (a = b) entangled (⇒ c+c− < 0) case, Eq. (4.27)
and the fact that vi > 0 imply v1 = v2. Under such a constraint, minimizing ξvi
becomes a trivial matter and yields
¯ξ = 2 (a − |c+|)(a − |c−|) = 2˜ν− . (4.28)
We thus see that the smallest symplectic eigenvalue of the partially transposed state
is endowed with a direct physical interpretation: it quantifies the greatest amount
of EPR correlations which can be created in a Gaussian state by means of local
operations.
As can be easily verified by a numerical investigation, such a simple interpretation
is lost for nonsymmetric two-mode Gaussian states. This fact properly
exemplifies the difficulties of handling optimization problems in nonsymmetric instances,
encountered, e.g. in the computation of the entanglement of formation of
such states [95]. It also confirms that, in the special subset of two-mode (mixed)