ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

4.4. Quantifying entanglement via purity measures: the average logarithmic negativity 77
privilege the role of S2. In fact, S2 can indeed, assuming some prior knowledge
about the state (essentially, its Gaussian character), be measured through conceivable
direct methods, in particular by means of single-photon detection schemes [87]
(of which preliminary experimental verifications are available [263]) or of the upcoming
quantum network architectures [80, 86, 165]. Very recently, a scheme to
locally measure all symplectic invariants (and hence the entanglement) of two-mode
Gaussian states has been proposed, based on number and purity measurements
[195]. Notice that no complete homodyne reconstruction [62] of the CM is needed
in all those schemes.
As already anticipated, for p = 2 we can provide analytical expressions for the
extremal entanglement in the space of global and marginal purities [GA2]
log −
EN max(µ1,2, µ) = −
1
µ +
EN min(µ1,2, µ) = −
log
µ1+µ2
2µ 2 1 µ2 2
1
µ 2 +
1
1
µ 2 −
2
1
2µ 2 − 1
2 −
µ1 + µ2 −
2
1
2
µ 2 1
(µ1 + µ2) 2 − 4µ2 1 µ2 2
µ
,
(4.58)
+ 1
µ 2 −
2
1
2µ 2 − 1
2 2 − 1
µ 2
.
(4.59)
Consequently, both the average logarithmic negativity δ ĒN , defined in Eq. (4.56),
and the relative error δ ĒN , given by Eq. (4.57), can be easily evaluated in terms
of the purities. The relative error is plotted in Fig. 4.6(b) for symmetric states as a
function of the ratio SLi /SL. Notice, as already pointed out in the general instance
of arbitrary p, how the error decays exponentially. In particular, it falls below 5% in
the range SL < SLi (µ > µi), which excludes at most very weakly entangled states
(states with EN 1). 9 Let us remark that the accuracy of estimating entanglement
by the average logarithmic negativity proves even better in the nonsymmetric case
µ1 = µ2, essentially because the maximal allowed entanglement decreases with the
difference between the marginals, as shown in Fig. 4.1(a).
The above analysis proves that the average logarithmic negativity ĒN is a reliable
estimate of the logarithmic negativity EN , improving as the entanglement
increases [GA2, GA3]. This allows for an accurate quantification of CV entanglement
by knowledge of the global and marginal purities. As we already mentioned,
the latter quantities may be in turn amenable to direct experimental determination
by exploiting recent single-photon-detection proposals [87] or in general interferometric
quantum-network setups. Let us stress, even though quite obvious, that
the estimate becomes indeed an exact quantification in the two crucial instances
of GMEMS (nonsymmetric thermal squeezed states) and GLEMS (mixed states of
partial minimum uncertainty), whose logarithmic negativity is completely determined
as a function of the three purities alone, see Eqs. (4.58, 4.59).
9 It is straightforward to verify that, in the instance of two-mode squeezed thermal (symmetric)
states, such a condition corresponds to cosh(2r) µ 1/4 . This constraint can be easily
satisfied with the present experimental technology: even for the quite unfavorable case µ = 0.5
the squeezing parameter needed is just r 0.3.