ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

5.2. Quantification and scaling of entanglement in fully symmetric states 99
5.1.3.1. The case of the basset hound. To give an example, we can consider a bisymmetric
1 × 2 three-mode Gaussian state, 12 where the CM of the last two modes
(constituting subsystem SB) is assumed in standard form, Eq. (2.54). Because of
the symmetry, the local symplectic transformation responsible for entanglement
concentration in this simple case is the identity on the first mode (constituting
subsystem SA) and just a 50:50 beam-splitter transformation B2,3(1/2), Eq. (2.26),
on the last two modes [268] (see also Sec. 9.2.1). The entire procedure of unitary
localization and delocalization of entanglement [GA5] is depicted in Fig. 5.1. Interestingly,
it may be referred to as “cut-off and regrowth of the head of a basset
hound”, where in our example the basset hound pictorially represents a bisymmetric
three-mode state. However, the breed of the dog reflects the fact that the unitary
localizability is a property that extends to all 1×N [GA4] and M ×N [GA5] bisymmetric
Gaussian states (in which case, the basset hound’s body would be longer
and longer with increasing N). We can therefore address bisymmetric Gaussian
states as basset hound states, if desired.
In this canine analogy, let us take the freedom to remark that fully symmetric
states of the form Eq. (2.60), as a special case, are of course bisymmetric under
any bipartition of the modes; this, in brief, means that any conceivable multimode,
bipartite entanglement is locally equivalent to the minimal two-mode, bipartite
entanglement (consequences of this will be deeply investigated in the following).
Pictorially, remaining in the context of three-mode Gaussian states, this special
type of basset hound state resembles a Cerberus state, in which any one of the
three heads can be cut and can be reversibly regrown.
5.2. Quantification and scaling of entanglement in fully symmetric states
In this Section we will explicitly compute the block entanglement (i.e. the entanglement
between different blocks of modes) for some instances of multimode Gaussian
states. We will study its scaling behavior as a function of the number of modes and
explore in deeper detail the localizability of the multimode entanglement. We focus
our attention on fully symmetric L-mode Gaussian states (the number of modes
is denoted by L in general to avoid confusion), endowed with complete permutation
invariance under mode exchange, and described by a 2L × 2L CM σ β L given
by Eq. (2.60). These states are trivially bisymmetric under any bipartition of the
modes, so that their block entanglement is always localizable by means of local symplectic
operations. Let us recall that concerning the covariances in normal forms
of fully symmetric states (see Sec. 2.4.3), pure L-mode states are characterized by
ν −
β
= ν+
β L = 1 in Eq. (2.61), which yields
z1 = (L − 2)(b2 − 1) + (b 2 − 1) [L ((b 2 − 1) L + 4) − 4]
2b(L − 1)
z2 = (L − 2)(b2 − 1) − (b 2 − 1) [L ((b 2 − 1) L + 4) − 4]
2b(L − 1)
,
.
(5.13)
12 The bipartite and genuinely tripartite entanglement structure of three-mode Gaussian
states will be extensively investigated in Chapter 7, based on Ref. [GA11]. The bisymmetric
three-mode Gaussian states will be also reconsidered as efficient resources for 1 → 2 telecloning
of coherent states in Sec. 12.3, based on Ref. [GA16].