ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

Eq. (13.3) thus takes the form
13.1. Gaussian valence bond states 223
Γ out = C −1 ⊕ C , (13.4)
where C is a circulant N ×N matrix [18] and the phase space operators are assumed
here to be ordered as (ˆq1, ˆq2, . . . , ˆqN, ˆp1, ˆp2, . . . , ˆpN). It can be shown that a CM
of the form Eq. (13.4) corresponds to the ground state of the quadratic Hamiltonian
ˆH = 1
2
i
ˆp 2 i +
i,j
ˆqiVij ˆqj ,
with the potential matrix given by V = C 2 [11]. The GVBS we are going to investigate,
therefore, belong exactly to the class of block-diagonal pure N-mode Gaussian
states which, in Sec. 11.2.1, have been shown to achieve “generic entanglement”. We
will now interpret the entanglement and in general the distribution of correlations
in GVBS in terms of the structural and entanglement properties of the building
block γ.
13.1.1. Properties of the building block
In the Jamiolkowski picture of Gaussian operations [202, 205, 90], different valence
bond projectors correspond to differently entangled Gaussian building blocks. Let
us recall some results on the characterization of bipartite entanglement from Part II
of this Dissertation.
According to the PPT criterion, a Gaussian state is separable (with respect
to a 1 × N bipartition) if and only if the partially transposed CM satisfies the
uncertainty principle, see Sec. 3.1.1. As a measure of entanglement, for two-mode
symmetric Gaussian states γ i,j we can adopt either the logarithmic negativity EN ,
Eq. (3.8), or the entanglement of formation EF , computable in this case [95] via the
formula Eq. (4.17). Both measures are equivalent being monotonically decreasing
functions of the positive parameter ˜νi,j, which is the smallest symplectic eigenvalue
of the partial transpose ˜γ i,j of γ i,j. For a two-mode state, ˜νi,j can be computed
from the symplectic invariants of the state [GA3] (see Sec. 4.2.1) , and the PPT
criterion Eq. (3.6) simply yields γ i,j entangled as soon as ˜νi,j < 1, while infinite
entanglement (accompanied by infinite energy in the state) is reached for ˜νi,j → 0 + .
We are interested in studying the quantum correlations of GVBS of the form
Eq. (13.3), and in relating them to the entanglement properties of the building block
γ, Eq. (13.1). The building block is a pure three-mode Gaussian state. As discussed
in Sec. 7.1.2, its standard form covariances Eq. (13.2) have to vary constrained to
the triangle inequality (7.17) for γ to describe a physical state [GA11]. This results
in the following constraints on the parameters x and s,
x + 1
x ≥ 1 , s ≥ smin ≡ . (13.5)
2
Let us keep the parameter x fixed: this corresponds to assigning the CM of
mode 3 (output port). Straightforward applications of the PPT separability conditions,
and consequent calculations of the logarithmic negativity Eq. (3.8), reveal
that the entanglement between the first two modes in the CM γss (input port) is
monotonically increasing as a function of s, ranging from the case s = smin when
γss is separable to the limit s → ∞ when the block γss is infinitely entangled.
Accordingly, the entanglement between each of the first two modes γs of γ and the