ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

42 2. Gaussian states: structural properties
the non-local blocks, while leaving the local ones unaffected (as they are proportional
to the identity). Different sets of N entries in the non-diagonal sub-matrices
can be thus set to zero. For an even total number of modes, all the non-diagonal
blocks σ12, σ34,. . . ,σ (N−1)N describing the correlations between disjoint pairs of
quadratures can be diagonalized (leading to the singular-value diagonal form of each
block), with no conditions on all the other blocks. For an odd number of modes,
after the diagonalization of the blocks relating disjoint quadratures, a further nondiagonal
block involving the last mode (say, σ1N) can be put in triangular form by
a rotation on the last mode.
Notice finally that the locally invariant degrees of freedom of a generic Gaussian
state of N modes are (2N +1)N −3N = 2N 2 −2N, as follows from the subtraction of
the number of free parameters of the local symplectics from the one of a generic state
— with an obvious exception for N = 1, for which the number of free parameters
is 1, due to the rotational invariance of single-mode Williamson forms (see the
discussion about the vacuum state in Appendix A.2.1).
2.4.1.1. Standard form of two-mode Gaussian states. According to the above counting
argument, an arbitrary (mixed) Gaussian state of two modes can be described, up
to local unitary operations, by 4 parameters. Let us briefly discuss this instance
explicitly, to acquaint the reader with the symplectic playground, and since twomode
Gaussian states are the paradigmatic examples of bipartite entangled states
of CV systems.
The expression of the two-mode CM σ in terms of the three 2 × 2 matrices α,
β, γ, that will be useful in the following, takes the form [see Eq. (2.20)]
α γ
σ =
γT
. (2.53)
β
For any two-mode CM σ there is a local symplectic operation Sl = S1 ⊕ S2 which
brings σ in the standard form σsf [218, 70]
S T ⎛
a
⎜
l σSl = σsf ≡ ⎜ 0
⎝ c+
0
a
0
c+
0
b
0
c−
0
⎞
⎟
⎠ . (2.54)
0 c− 0 b
States whose standard form fulfills a = b are said to be symmetric. The covariances
a, b, c+, and c− are determined by the four local symplectic invariants Detσ =
(ab − c 2 +)(ab − c 2 −), Detα = a 2 , Detβ = b 2 , Detγ = c+c−. Therefore, the standard
form corresponding to any CM is unique (up to a common sign flip in c− and c+).
Entanglement of two-mode Gaussian states is the topic of Chapter 4.
2.4.2. Pure states
The CM σ p of a generic N-mode pure Gaussian state satisfies the condition
− Ω σ p Ω σ p = . (2.55)
This follows from the Williamson normal-mode decomposition of the CM, σ p =
SS T , where S is a symplectic transformation. Namely,
−Ωσ p Ωσ p = −ΩSS T ΩSS T = −ΩSΩS T = −ΩΩ = .
The matrix identity Eq. (2.55) provides a set of (not mutually independent) polynomial
equations that the elements of a generic CM have to fulfill in order to represent