ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

2.4. Standard forms of Gaussian covariance matrices 45
accordance with Eq. (2.56), and as explained by alternative arguments in Appendix
A.2.1. In other words, according to the notation of Eq. (2.54), any pure twomode
Gaussian state is symmetric (b = a) and its standard form elements fulfill
c± = ± √ a2 − 1.
Notice also that the phase-space decomposition discussed here is special to
Gaussian states and is independent from the general Schmidt decomposition at the
Hilbert space level, Eq. (1.20), which can be obtained for any pure state. For CV
systems, it will contain in principle infinite terms, as the local bases are infinitedimensional.
To give an example, the two-mode squeezed state, whose CM in its
“phase-space Schmidt decomposition” is of the form Eq. (2.22), admits the following
Hilbert-space Schmidt decomposition [16]
|ψ sq 〉i,j = 1
cosh r
∞
tanh n r |n〉 i |n〉 j , (2.59)
n=0
with local Schmidt bases given by the number states in the Fock space of each
mode.
We will now show that for (generally mixed) Gaussian states with some local
symmetry constraints, a similar phase-space reduction is available, such that multimode
properties (like entanglement) can be unitarily reduced to two-mode ones
[GA4, GA5].
2.4.3. Symmetric states
Very often in quantum information, and in particular in the theory of entanglement,
peculiar roles are played by symmetric states, that is, states that are either invariant
under a particular group of transformations — like Werner states of qudits [264] —
or under permutation of two or more parties in a multipartite system, like ground
and thermal states of some translationally invariant Hamiltonians (e.g. of harmonic
lattices) [11]. Here we will introduce classes of Gaussian states invariant under all
permutation of the modes (fully symmetric states) or exhibiting such permutationinvariance
locally in each of the two subsystems across a global bipartition of the
modes (bisymmetric states). For both we will provide a standard form based on
the special properties of their symplectic spectrum. We will limit ourself to a
collection of results, which will be useful for the computation and exploitation of
entanglement in the corresponding states. All the proofs can be found in Ref. [GA5].
Unless explicitly stated, we are dealing with generally mixed states.
2.4.3.1. Fully symmetric Gaussian states. We shall say that a multimode Gaussian
state ϱ is “fully symmetric” if it is invariant under the exchange of any two modes. In
the following, we will consider the fully symmetric M-mode and N-mode Gaussian
states ϱ α M and ϱ β N , with CMs σ α M and σ β N . Due to symmetry, we have that
the CM, Eq. (2.20), of such states reduces to
⎛
⎞
α ε · · · ε
⎜
σαM = ⎜
⎝
ε
.
.
α
ε
ε
. ..
.
.
ε
⎟
⎠
ε · · · ε α
, σ ⎛
β ζ · · · ζ
⎞
⎜
βN = ⎜
⎝
ζ
.
.
β
ζ
ζ
. ..
.
.
ζ
⎟ ,
⎠
(2.60)
ζ · · · ζ β
where α, ε, β and ζ are 2 × 2 real symmetric submatrices (the symmetry of ε and
ζ stems again from the symmetry under the exchange of any two modes).