ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

4.5. Gaussian entanglement measures versus Negativities 83
saturation of the two sides of inequality (4.61). As a consequence of the above
discussion, this matrix would denote the optimal pure state σ p
opt. By solving the
system of equations Det (γq − Γ) = Det (Γ − γ−1 p ) = 0, where the matrices involved
are explicitly defined combining Eq. (4.60) and Eq. (4.68) with λ = 1, one finds the
following two solutions for the coordinates x0 and x1:
x ± 0 = (g + 1)s ± [(g − 1) 2 − 4d 2 ] (−d 2 + s 2 − g)
2 (d 2 + g)
x ± 1 = (g + 1) −d 2 + s 2 − g ± s (g − 1) 2 − 4d 2
2 (d 2 + g)
.
,
(4.75)
The corresponding pure state σp± = Γ ± ⊕ Γ ±−1 turns out to be, in both cases, a
two-mode squeezed state described by a CM of the form Eq. (2.22), with cosh(2r) =
x ± 0 . Because the single-mode determinant m2 = cosh 2 (2r) for these states, the
optimal m2 for GMEMS is simply equal to (x − 0 )2 . Summarizing,
⎧
1, g ≥ 2s − 1 [separable state] ;
m 2GMEMS
opt
⎪⎨
=
⎪⎩
(g+1)s− √ [(g−1) 2−4d2 ](−d2 +s2 2 −g)
4(d2 +g) 2
,
2|d| + 1 ≤ g < 2s − 1 .
(4.76)
Once again, also for the class of GMEMS the Gaussian EMs are not simple functions
of the symplectic eigenvalue ˜ν− alone. Consequently, they provide a quantification
of CV entanglement of GMEMS inequivalent to the one determined by the negativities.
Furthermore, we will now show how these results raise the problem of the
ordering of two-mode Gaussian states according to their degree of entanglement, as
quantified by different families of entanglement measures [GA7].
4.5.3. Entanglement-induced ordering of two-mode Gaussian states
We have more than once remarked that, in the context of CV systems, when one
restricts to symmetric, two-mode Gaussian states (which include all pure states)
the known computable measures of entanglement all correctly induce the same
ordering on the set of entangled states [GA7]. We will now show that, indeed, this
nice feature is not preserved moving to mixed, nonsymmetric two-mode Gaussian
states. We aim at comparing Gaussian EMs and negativities on the two extremal
classes of two-mode Gaussian states [GA3], introducing thus the concept of extremal
ordering. At fixed global and local purities, the negativity of GMEMS (which is the
maximal one) is obviously always greater than the negativity of GLEMS (which is
the minimal one). If for the same values of purities the Gaussian EMs of GMEMS
are larger than those of GLEMS, we will say that the extremal ordering is preserved.
Otherwise, the extremal ordering is inverted. In this latter case, which is clearly the
most intriguing, the states of minimal negativities are more entangled, with respect
to Gaussian EMs, than the states of maximal negativities, and the inequivalence
of the orderings, induced by the two different families of entanglement measures,
becomes manifest.
The problem can be easily stated. By comparing mGLEMS opt from Eq. (4.74) and
mGMEMS opt from Eq. (4.76), one has that in the range of global and local purities, or,
equivalently, of parameters {s, d, g}, such that
m GMEMS
opt
≥ m GLEMS
opt , (4.77)