ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

148 8. Unlimited promiscuity of multipartite Gaussian entanglement
We prove that multimode Gaussian states exist, that can possess simultaneously
arbitrarily large pairwise bipartite entanglement between some pairs of modes and
arbitrarily large genuine multipartite entanglement among all modes without violating
the monogamy inequality (6.17) on entanglement sharing. The class of
states exhibiting such unconstrained simultaneous distribution of quantum correlations
are producible with standard optical means (as we will detail in Sec. 10.2),
the achievable amount of entanglement being technologically limited only by the
attainable degree of squeezing. This unexpected feature of entanglement sheds new
light on the role of the fundamental laws of quantum mechanics in curtailing the
distribution of information. On a more applicative ground, it serves as a prelude
to implementations of quantum information processing in the infinite-dimensional
scenario that cannot be achieved with qubit resources.
To illustrate the existence of such phenomenon, we consider the simplest nontrivial
instance of a family of four-mode Gaussian states, endowed with a partial
symmetry under mode exchange.
8.2. Entanglement in partially symmetric four-mode Gaussian states
8.2.1. State definition
We start with an uncorrelated state of four modes, each one initially in the vacuum
of the respective Fock space, whose corresponding CM is the identity. We apply a
two-mode squeezing transformation S2,3(s), Eq. (2.24), with squeezing s to modes
2 and 3, then two further two-mode squeezing transformations S1,2(a) and S3,4(a),
with squeezing a, to the pairs of modes {1, 2} and {3, 4}. The two last transformations
serve the purpose of redistributing the original bipartite entanglement,
created between modes 2 and 3 by the first two-mode squeezing operations, among
all the four modes. For any value of the parameters s and a, the output is a pure
four-mode Gaussian state with CM σ,
σ = S3,4(a)S1,2(a)S2,3(s)S T 2,3(s)S T 1,2(a)S T 3,4(a) . (8.1)
Explicitly, σ is of the form Eq. (2.20) where
σ1 = σ4 = [cosh 2 (a) + cosh(2s) sinh 2 (a)]2 ,
σ2 = σ3 = [cosh(2s) cosh 2 (a) + sinh 2 (a)]2 ,
ε1,2 = ε3,4 = [cosh 2 (s) sinh(2a)]Z2 ,
ε1,3 = ε2,4 = [cosh(a) sinh(a) sinh(2s)]2 ,
ε1,4 = [sinh 2 (a) sinh(2s)]Z2 ,
ε2,3 = [cosh 2 (a) sinh(2s)]Z2 ,
with Z2 = 1 0
0 −1 . It is immediate to see that a state of this form is invariant under
the double exchange of modes 1 ↔ 4 and 2 ↔ 3, as Si,j = Sj,i and operations on
disjoint pairs of modes commute. Therefore, there is only a partial symmetry under
mode permutations, not a full one like in the case of the three-mode GHZ/W states
and in general the states of Sec. 2.4.3.
8.2.2. Structure of bipartite entanglement
Let us recall that in a pure four-mode Gaussian state and in its reductions, bipartite
entanglement is equivalent to negativity of the partially transposed CM, obtained