ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

proving that
8.2. Entanglement in partially symmetric four-mode Gaussian states 151
min{g[m 2 1|(234) ] − g[m2 1|2 ], g[m2 2|(134) ] − g[m2 1|2 ] − g[m2 2|3 ]}
is nonnegative. The first quantity always achieves the minimum yielding
Gτ res (σ) ≡ Gτ (σ1|(234)) − Gτ (σ1|2) (8.5)
= arcsinh 2
[cosh 2 a + cosh(2s) sinh 2 a] 2
− 1 − 4a 2 .
Since cosh(2s) > 1 for s > 0, it follows that Gτ res > 0 and Ineq. (6.17) is satisfied.
The entanglement in the global Gaussian state is therefore distributed according
to the laws of quantum mechanics, in such a way that the residual Gaussian
contangle Gτ res quantifies the multipartite entanglement not stored in couplewise
form. Those quantum correlations, however, can be either tripartite involving three
of the four modes, and/or genuinely four-partite among all of them. We can now
quantitatively estimate to what extent such correlations are encoded in some tripartite
form: as an anticipation, we will find them negligible in the limit of high
squeezing a.
8.2.3.2. Tripartite entanglement estimation. Let us first observe that in the tripartitions
1|2|4 and 1|3|4 the tripartite entanglement is zero, as mode 4 is not entangled
with the block of modes 1,2, and mode 1 is not entangled with the block of modes
3,4 (the corresponding three-mode states are then said to be biseparable [94], see
Sec. 7.1.1). The only tripartite entanglement present, if any, is equal in content
(due to the symmetry of the state σ) for the tripartitions 1|2|3 and 2|3|4, and can
be quantified via the residual Gaussian contangle determined by the corresponding
three-mode monogamy inequality (6.17).
The residual Gaussian contangle of a Gaussian state σ of the three modes i,
j, and k (which is an entanglement monotone under tripartite Gaussian LOCC for
pure states [GA10], see Sec. 7.2.2.1), takes the form as in Eq. (7.36),
Gτ (σ i|j|k) ≡ min
(i,j,k)
Gτ (σ i|(jk)) − Gτ (σ i|j) − Gτ (σ i|k) , (8.6)
where the symbol (i, j, k) denotes all the permutations of the three mode indexes.
We are interested here in computing the residual tripartite Gaussian contangle,
Eq. (8.6), shared among modes 1, 2 and 3 in the reduced mixed three-mode state
σ123 obtained from Eq. (8.1) by tracing over the degrees of freedom of mode 4. To
quantify such tripartite entanglement exactly, it is necessary to compute the mixedstate
1 × 2 Gaussian contangle between one mode and the block of the two other
modes. This requires solving the nontrivial optimization problem of Eq. (6.13) over
all possible pure three-mode Gaussian states — not necessarily in standard form,
Eq. (7.19). However, from the definition itself, Eq. (6.13), the bipartite Gaussian
contangle Gτ (σi|(jk)) (with i, j, k a permutation of 1, 2, 3) is bounded from above
by the corresponding bipartite Gaussian contangle Gτ (σ p
i|(jk) ) in any pure, threemode
Gaussian state with CM σ p
i|(jk) ≤ σi|(jk). As an ansatz we can choose pure
three-mode Gaussian states whose CM σ p
123 has the same matrix structure of our
mixed state σ123 (in particular, zero correlations between position and momentum
operators, and diagonal subblocks proportional to the identity), and restrict the
optimization to such a class of states. This task is accomplished by choosing a pure