ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

46 2. Gaussian states: structural properties
Standard form.— Let σβN be the CM of a fully symmetric N-mode Gaussian state.
The 2×2 blocks β and ζ of σβN , defined by Eq. (2.60), can be brought by means of
local, single-mode symplectic operations S ∈ Sp ⊕N
(2,) into the form β = diag (b, b)
and ζ = diag (z1, z2).
In other words, the standard form of fully symmetric N-mode states is such
that any reduced two-mode state is symmetric and in standard form, see Eq. (2.54).
Symplectic degeneracy.— The symplectic spectrum of σβN is (N − 1)-times degenerate.
The two symplectic eigenvalues of σβN , ν −
β and ν+
βN , read
where ν −
β
ν −
β = (b − z1)(b − z2) ,
ν +
β N = (b + (N − 1)z1)(b + (N − 1)z2) ,
is the (N − 1)-times degenerate eigenvalue.
(2.61)
Obviously, analogous results hold for the M-mode CM σ α M of Eq. (2.60), whose
2 × 2 submatrices can be brought to the form α = diag (a, a) and ε = diag (e1, e2)
and whose (M − 1)-times degenerate symplectic spectrum reads
ν − α = (a − e1)(a − e2) ,
ν +
α M = (a + (M − 1)e1)(a + (M − 1)e2) .
(2.62)
2.4.3.2. Bisymmetric M × N Gaussian states. Let us now generalize this analysis to
the (M + N)-mode Gaussian states, whose CM σ result from a correlated combi-
nation of the fully symmetric blocks σ α M and σ β N ,
σαM Γ
σ =
Γ T
σ β N
, (2.63)
where Γ is a 2M × 2N real matrix formed by identical 2 × 2 blocks γ. Clearly,
Γ is responsible of the correlations existing between the M-mode and the N-mode
parties. Once again, the identity of the submatrices γ is a consequence of the local
invariance under mode exchange, internal to the M-mode and N-mode parties.
States of the form of Eq. (2.63) will be henceforth referred to as “bisymmetric”
[GA4, GA5]. A significant insight into bisymmetric multimode Gaussian states can
be gained by studying the symplectic spectrum of σ and comparing it to the ones
of σ α M and σ β N .
Symplectic degeneracy.— The symplectic spectrum of the CM σ Eq. (2.63) of a
bisymmetric (M + N)-mode Gaussian state includes two degenerate eigenvalues,
with multiplicities M − 1 and N − 1. Such eigenvalues coincide, respectively, with
the degenerate eigenvalue ν − α of the reduced CM σ α M , and with the degenerate
eigenvalue ν −
β of the reduced CM σ β N .
Equipped with these results, we are now in a position to show the following
central result [GA5], which applies to all (generally mixed) bisymmetric Gaussian
states, and is somehow analogous to — but independent of — the phase-space
Schmidt decomposition of pure Gaussian states (and of mixed states with fully
degenerate symplectic spectrum).
Unitary localization of bisymmetric states.— The bisymmetric (M +N)-mode Gaussian
state with CM σ, Eq. (2.63) can be brought, by means of a local unitary (symplectic)
operation with respect to the M × N bipartition with reduced CMs σ α M
and σ β N , to a tensor product of M + N − 2 single-mode uncorrelated states, and of