ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

266 A. Standard forms of pure Gaussian states under local operations
A.2. Degrees of freedom of pure Gaussian states
Pure Gaussian states are characterized by CMs with Williamson form equal to the
identity. As we have seen (Sec. 2.2.2.1), the Williamson decomposition provides a
mapping from any Gaussian state into the uncorrelated product of thermal (generally
mixed) states: such states are pure (corresponding to the vacuum), if and only
if all the symplectic eigenvalues are equal to 1.
The symplectic eigenvalues of a generic CM σ are determined as the eigenvalues
of the matrix |iΩσ|, where Ω stands for the symplectic form. Therefore, a Gaussian
state of N modes with CM σ is pure if and only if
− σΩσΩ = 2N . (A.2)
It will be convenient here to reorder the CM, and to decompose it in the three
sub-matrices σq, σp and σqp, whose entries are defined as
(σq)jk = Tr [ϱˆqj ˆqk], (σp)jk = Tr [ϱˆpj ˆpk], (σqp)jk = Tr [ϱ{ˆqj, ˆpk}/2], (A.3)
such that the complete CM σ is given in block form by
σq σqp
σ =
. (A.4)
σ T qp
Let us notice that the matrices σq and σp are always symmetric and strictly positive,
while the matrix σqp does not obey any general constraint.
Eqs. (A.2) and (A.4) straightforwardly lead to the following set of conditions
σp
σqσp = N + σ 2 qp , (A.5)
σqpσq − σqσ T qp = 0 , (A.6)
T 2
σpσq = N + σqp , (A.7)
σ T qpσp − σpσqp = 0 . (A.8)
Now, the last two equations are obviously obtained by transposition of the first
two. Moreover, from (A.5) one gets
while Eq. (A.6) is equivalent to
σp = σ −1
q (N + σ 2 qp) , (A.9)
σ −1
q σqp − σ T qpσ −1
q = 0 (A.10)
(the latter equations hold generally, as σq is strictly positive and thus invertible).
Eq. (A.10) allows one to show that any σp determined by Eq. (A.9) satisfies the
condition (A.8). Therefore, only Eqs. (A.5) and (A.6) constitute independent constraints
and fully characterize the CM of pure Gaussian states.
Given any (strictly positive) matrix σq and (generic) matrix σqp, the fulfillment
of condition (A.6) allows to specify the second moments of any pure Gaussian state,
whose sub-matrix σp is determined by Eq. (A.9) and does not involve any additional
degree of freedom.
A straightforward counting argument thus yields the number of degrees of freedom
of a generic pure Gaussian state, by adding the entries of a generic and symmetric
N × N matrix and subtracting the equations of the antisymmetric condition
(A.6): N 2 + N(N + 1)/2 − N(N − 1)/2 = N 2 + N, in compliance with the number
dictated by the Euler decomposition of a symplectic operation:
σ = S T 2N S = O T Z 2 O . (A.11)