ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

13.2. Entanglement distribution in Gaussian valence bond states 225
13.2.1. Short-range correlations
Let us consider a building block γ with s = smin ≡ (x + 1)/2. It is straightforward
to evaluate, as a function of x, the GVBS in Eq. (13.3) for an arbitrary number
of modes (we omit the CM here, as no particular insight can be drawn from the
explicit expressions of the covariances). By repeatedly applying the PPT criterion,
one can analytically check that each reduced two-mode block γout i,j is separable for
|i−j| > 1, which means that the output GVBS Γ out exhibits bipartite entanglement
only between nearest neighbor modes, for any value of x > 1 (for x = 1 we obtain
a product state).
While this certainly entails that Γ out is genuinely multiparty entangled, due to
the translational invariance, it is interesting to observe that, without feeding entanglement
in the input port γss of the original building block, the range of quantum
correlations in the output GVBS is minimum. The pairwise entanglement between
nearest neighbors will naturally decrease with increasing number of modes, being
frustrated by the overall symmetry and by the intrinsic limitations on entanglement
sharing (the so-called monogamy constraints [GA10], see Chapter 6). We can
study the asymptotic scaling of this entanglement in the limit x → ∞. One finds
that the corresponding partially-transposed symplectic eigenvalue ˜νi,i+1 is equal to
(N − 2)/N for even N, and [(N − 2)/N] 1/2 for odd N: neighboring sites are thus
considerably more entangled if the ring size is even-numbered. Such frustration
effect on entanglement in odd-sized rings, already devised in a similar context in
Ref. [272], is quite puzzling. An explanation may follow from counting arguments
applied to the number of parameters (which are related to the degree of pairwise
entanglement) characterizing a block-diagonal pure state on harmonic lattices (see
Sec. 11.2.1), as we will now show.
13.2.1.1. Valence bond representability and entanglement frustration. Let us make a
brief digression. It is conjectured that all pure N-mode Gaussian states can be
described as GVBS [202]. Here we provide a lower bound on the number M of
ancillary bonds required to accomplish this task, as a function of N. We restrict
to ground states of harmonic chains with spring-like interactions, i.e. to the blockdiagonal
Gaussian states of Sec. 11.2.1, which have been proven to rely on N(N −
1)/2 parameters [GA14], and which are GVBS with a CM of the form Eq. (13.4).
With a simple counting argument using Eq. (11.1), the total number of parameters
of the initial chain Γ of building blocks should be at least equal to that of the
target state, i.e.
N(2M + 1)(2M)/2 ≥ N(N − 1)/2 ,
which means M ≥ IntPart[( √ 4N − 3 − 1)/4]. This implies, for instance, that to
describe general pure states with at least N > 7 modes, a single EPR bond per
site is no more enough (even though the simplest case of M = 1 yields interesting
families of N-mode GVBS for any N, as shown in the following).
The minimum M scales as N 1/2 , diverging in the field limit N → ∞. As infinitely
many bonds would be necessary (and maybe not even sufficient) to describe
general infinite harmonic chains, the valence bond formalism is probably not helpful
to prove or disprove statements related to the entropic area scaling law [187] for