ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

11.2. Generic entanglement and state engineering of block-diagonal pure states 189
experimental facilities, one can thus choose either scheme when aiming to produce
pure three-mode Gaussian states.
11.2.4. Standard forms: generic-entangled ↔ block-diagonal
The special subset of pure N-mode Gaussian states emerging from our constructive
proof exhibits a distinct property: all correlations between “position” ˆqi and
“momentum” ˆpj operators are vanishing. Looking at Eq. (2.20), this means that
any such generic-entangled pure Gaussian state can be put in a standard form
where all the 2 × 2 submatrices of its CM are diagonal. The class of pure Gaussian
states exhibiting generic entanglement coincides thus with that formed by the
“block-diagonal” states discussed in Secs. 2.4.2 and 11.1.
The diagonal subblocks σi can be additionally made proportional to the identity
by local Williamson diagonalizations in the individual modes. This standard
form for generic-entangled N-mode Gaussian states, as already mentioned, can
be achieved by all pure Gaussian states for N = 2 [70] and N = 3 [GA11] (see
Sec. 7.1.2); for N ≥ 4, pure Gaussian states can exist whose number of independent
parameters scales as N(N − 2) and which cannot thus be brought in the
ˆq-ˆp block-diagonal form. Interestingly, all pure Gaussian states in our considered
block-diagonal standard form, are ground states of quadratic Hamiltonians with
spring-like interactions [11]. Let us now investigate the physical meaning of the
standard form.
Vanishing ˆq-ˆp covariances imply that the N-mode CM can be written as a
direct sum (see also Appendix A) σ = σq ⊕ σp, when the canonical operators
are arranged as (ˆq1, . . . , ˆqN, ˆp1, . . . , ˆpN ). Moreover, the global purity of σ imposes
σp = σ−1 q . Named (σq)ij = vqij and (σp)hk = vphk , this means that each vphk is
a function of the {vqij }’s. The additional N Williamson conditions vpii = vqii fix
the diagonal elements of σq. The standard form is thus completely specified by the
off-diagonal elements of the symmetric N × N matrix σq, which are, as expected,
N(N − 1)/2 ≡ ΞN from Eq. (11.1).
Proposition 1 of Sec. 11.2.2 acquires now the following remarkable physical
insight [GA14].
➢ Generic entanglement of pure Gaussian states. The structural properties
of pure block-diagonal N-mode Gaussian states, and in particular their
bipartite and multipartite entanglement, are completely specified (up to local
unitaries) by the ‘two-point correlations’ vqij = 〈ˆqiˆqj〉 between any pair of
modes, which amount to N(N − 1)/2 locally invariant degrees of freedom.
For instance, the entropy of entanglement between one mode (say i) and the
remaining N − 1 modes, which is monotonic in Det σi (see Sec. 2.3), is completely
specified by assigning all the pairwise correlations between mode i and any other
mode j = i, as Det σi = 1 −
j=i Det εij from Eq. (2.55). The rationale is that entanglement
in such states is basically reducible to a mode-to-mode one. This statement,
strictly speaking true only for the pure Gaussian states for which Proposition
1 holds, acquires a general validity in the context of the modewise decomposition of
arbitrary pure Gaussian states [116, 29, 92], as detailed in Sec. 2.4.2.1. We remark
that such an insightful correlation picture breaks down for mixed Gaussian states,
where also classical, statistical-like correlations arise.