ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

8.2. Entanglement in partially symmetric four-mode Gaussian states 149
by reversing time in the subspace of any chosen single subsystem [218, 265] (PPT
criterion, see Sec. 3.1.1). This inseparability condition is readily verified for the
family of states in Eq. (8.1) yielding that, for all nonzero values of the squeezings
s and a, σ is entangled with respect to any global bipartition of the modes. This
follows from the global purity of the state, together with the observation that the
determinant of each reduced one- and two-mode CM obtainable from Eq. (8.1) is
strictly bigger than 1 for any nonzero squeezings. The state is thus said to be fully
inseparable [111], i.e. it contains genuine four-partite entanglement.
Following our previous studies on CV entanglement sharing (see Chapters 6 and
7) we choose to measure bipartite entanglement by means of the Gaussian contangle
Gτ , an entanglement monotone under Gaussian LOCC, computable according to
Eq. (6.13).
In the four-mode state with CM σ, we can evaluate the bipartite Gaussian
contangle in closed form for all pairwise reduced (mixed) states of two modes i
and j, described by a CM σ i|j. By applying again PPT criterion (see Sec. 3.1.1),
one finds that the two-mode states indexed by the partitions 1|3, 2|4, and 1|4, are
separable. For the remaining two-mode states the computation is possible thanks to
the results of Sec. 4.5.2. Namely, the reduced state of modes 2 and 3, σ23, belongs
to the class of GMEMS (defined in Sec. 4.3.3.1); for it Eq. (4.76) yields 21
m 2|3 =
−1+2 cosh 2 (2a) cosh 2 s+3 cosh(2s)−4 sinh 2 a sinh(2s)
4[cosh 2 a+e 2s sinh 2 a] , a < arcsinh[ √ tanh s] ;
1, otherwise.
(8.2)
On the other hand, the states σ 1|2 and σ 3|4 are GMEMMS (defined in Sec. 4.3.2),
i.e. simultaneous GMEMS and GLEMS, for which either Eq. (4.74) or Eq. (4.76)
give
m 1|2 = m 3|4 = cosh 2a . (8.3)
Accordingly, one can compute the pure-state entanglements between one probe
mode and the remaining three modes. In this case one has simply m i|(jkl) = Det σi.
One finds from Eq. (8.1),
m 1|(234) = m 4|(123) = cosh 2 a + cosh(2s) sinh 2 a ,
m 2|(134) = m 3|(124) = sinh 2 a + cosh(2s) cosh 2 a .
(8.4)
Concerning the structure of bipartite entanglement, Eqs. (6.13, 8.3) imply that
the Gaussian contangle in the mixed two-mode states σ1|2 and σ3|4 is 4a2 , irrespective
of the value of s. This quantity is exactly equal to the degree of entanglement
in a pure two-mode squeezed state Si,j(a)S T i,j
(a) of modes i and j generated with
the same squeezing a. In fact, the two-mode mixed state σ 1|2 (and, equivalently,
σ 3|4) serves as a proper resource for CV teleportation [39, 89], realizing a perfect
transfer (unit fidelity 22 ) in the limit of infinite squeezing a.
21 We refer to the notation of Eq. (6.13) and write, for each partition i|j, the corresponding
parameter m i|j involved in the optimization problem which defines the Gaussian contangle.
22 The fidelity F ≡ 〈ψ in |ϱ out |ψ in 〉 (“in” and “out” being input and output state, respectively)
quantifies the teleportation success, as detailed in Chapter 12. For single-mode coherent input
states and σ 1|2 or σ 3|4 employed as entangled resources, F = (1+e −2a cosh 2 s) −1 . It reaches unity
for a ≫ 0 even in presence of high interpair entanglement (s ≫ 0), provided that a approaches
infinity much faster than s.