ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

36 2. Gaussian states: structural properties
equilibrium at a temperature Tk, characterized by an average number of thermal
photons ¯nk which obeys Bose-Einstein statistics,
¯nk = νk − 1
2
=
1
. (2.32)
exp − 1
ωk
kBTk
The N quantities νk’s form the symplectic spectrum of the CM σ, and are
invariant under the action of global symplectic transformations on the matrix σ.
The symplectic eigenvalues can be computed as the orthogonal eigenvalues of the
matrix |iΩσ| [207] and are thus determined by N invariants of the characteristic
polynomial of such a matrix [208]. One global symplectic invariant is simply the
determinant of the CM (whose invariance is a consequence of the fact that Det S = 1
∀S ∈ Sp (2N,)), which once computed in the Williamson diagonal form reads
Det σ =
N
k=1
ν 2 k . (2.33)
Another important invariant under global symplectic operations is the so-called
seralian ∆ [GA6], defined as the sum of the determinants of all 2 × 2 submatrices
of a CM σ, Eq. (2.20), which can be readily computed in terms of its symplectic
eigenvalues as
N
∆(σ) = ν 2 k . (2.34)
The invariance of ∆σ in the multimode case [208] follows from its invariance in the
case of two-mode states, proved in Ref. [211], and from the fact that any symplectic
transformation can be decomposed as the product of two-mode transformations
[123].
2.2.2.2. Symplectic representation of the uncertainty principle. The symplectic eigenvalues
νk encode essential information on the Gaussian state σ and provide powerful,
simple ways to express its fundamental properties [211]. For instance, let us
consider the uncertainty relation (2.19). Since the inverse of a symplectic operation
is itself symplectic, one has from Eq. (2.23), S −1T ΩS −1 = Ω, so that Ineq. (2.19) is
equivalent to ν + iΩ ≥ 0. In terms of the symplectic eigenvalues νk the uncertainty
relation then simply reads
νk ≥ 1 . (2.35)
Inequality (2.35) is completely equivalent to the uncertainty relation (2.19) provided
that the CM σ satisfies σ ≥ 0.
We can, without loss of generality, rearrange the modes of a N-mode state such
that the corresponding symplectic eigenvalues are sorted in ascending order
k=1
ν− ≡ ν1 ≤ ν2 ≤ . . . ≤ νN−1 ≤ νN ≡ ν+ .
With this notation, the uncertainty relation reduces to ν− ≥ 1. We remark that the
full saturation of the uncertainty principle can only be achieved by pure N-mode
Gaussian states, for which
νi = 1 ∀i = 1, . . . , N ,
meaning that the Williamson normal form of any pure Gaussian state is the vacuum
|0〉 of the N-mode Hilbert space H . Instead, mixed states such that νi≤k = 1 and
νi>k > 1, with 1 ≤ k ≤ N, only partially saturate the uncertainty principle, with