ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

126 7. Tripartite entanglement in three-mode Gaussian states
We recall that the form of the CM of any Gaussian state can be simplified
through local (unitary) symplectic operations, that therefore do not affect the entanglement
or mixedness properties of the state, belonging to Sp (2,) ⊕N . Such
reductions of the CMs are called “standard forms”, as introduced in Sec. 2.4. For
the sake of clarity, let us write the explicit standard form CM of a generic pure
three-mode Gaussian state [GA11],
with
e ±
ij ≡
1
4 √ aiaj
σ p
sf =
⎛
⎜
⎝
a1 0 e + 12 0 e + 13 0
0 a1 0 e − 12 0 e − 13
e + 12 0 a2 0 e + 23 0
0 e − 12 0 a2 0 e − 23
e + 13 0 e + 23 0 a3 0
0 e − 13 0 e − 23 0 a3
(ai − aj) 2
2
− (ak − 1) (ai − aj) 2 − (ak + 1) 2
±
⎞
⎟ , (7.19)
⎟
⎠
(ai + aj) 2
2
− (ak − 1) (ai + aj) 2 − (ak + 1) 2
.
(7.20)
By direct comparison with Eq. (4.39), it is immediate to verify that each two-mode
reduced CM σij denotes a standard form GLEMS with local purities µi = a −1
i and
µj = a −1
j , and global purity µij ≡ µk = a −1
k . Notice also that the standard form of
any pure three-mode Gaussian state, Eq. (7.19), admits all 2×2 subblocks of the CM
simultaneously in diagonal form; this is no longer possible for completely general
pure Gaussian states of N ≥ 4 modes, as clarified in Appendix A.2.1. However,
pure Gaussian states which, for an arbitrary number of modes, are reducible to
such a “block-diagonal” standard form, are endowed with peculiar entanglement
properties [GA14], which will be investigated in Chapter 11.
Let us stress that, although useful in actual calculations, the use of CMs in
standard form does not entail any loss of generality, because all the results derived
in the present Chapter for N = 3 do not depend on the choice of the specific form of
the CMs, but only on invariant quantities, such as the global and local symplectic
invariants.
A first qualitative result which immediately follows from our study [GA11],
is that, regarding the classification of Sec. 7.1.1 [94], pure three-mode Gaussian
states may belong either to Class 5, in which case they reduce to the global threemode
vacuum, or to Class 2, reducing to the uncorrelated product of a single-mode
vacuum and of a two-mode squeezed state, or to Class 1 (fully inseparable state). No
two-mode or three-mode biseparable pure three-mode Gaussian states are allowed.
7.1.3. Mixed states
For the sake of completeness, let us briefly report that the most general standard
form σsf associated to the CM of any (generally mixed) three-mode Gaussian state