ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

A.2. Degrees of freedom of pure Gaussian states 269
obviously wrong, again due to a loss of parameters in the transformations of particular
invariant states. We have already inspected this very case and pointed out
such invariances in our treatment of the Schmidt decomposition (previous Section):
we know that the number of locally irreducible free parameters is just min(M, N)
in this case, corresponding to the tensor product of two-mode squeezed states and
uncorrelated vacua.
For N ≥ 3, local single-mode operations can fully reduce the number of degrees
of freedom of pure Gaussian states by their total number of parameters. The
issue encountered for two-mode states does not occur here, as the first single-mode
rotations can act on different non-diagonal blocks of the CM (i.e., pertaining to
the correlations between different pairs of modes). The number of such blocks is
clearly equal to (N 2 − N)/2 while the number of local rotations is just N. Only
for N = 1, 2 is the latter value larger than the former: this is, ultimately, why the
simple subtraction of degrees of freedom only holds for N ≥ 3. To better clarify
this point, let us consider a CM σ 3m in the limiting instance N = 3. The general
standard form for (mixed) three-mode states implies the conditions (see Sec. 2.4.1)
and
diag (σ 3m
q ) = diag (σ 3m
p ) (A.12)
σ 3m
qp =
⎛
⎝
0 0 0
0 0 u
s t 0
⎞
⎠ . (A.13)
The diagonal of σ3m q coincides with that of σ3m p (which always results from the local
single-mode Williamson reductions) while six entries of σ3m qp can be set to zero. Let
us now specialize to pure states, imposing the conditions (A.5) and (A.6). Eq. (A.6)
results into a linear system of three equations for the non-null entries of σ3m qp , with
coefficients given by the entries of σ3m q . The definite positivity of σ3m q implies that
the sub-system on s and t is determinate and thus imposes s = t = 0. This fact
already implies (σ3m qp ) 2 = 0 and thus σ3m p = (σ3m q ) −1 . As for u, the system entails
that, if a = 0, then the entry (σ3m q )13 = 0. But, as is apparent from Eq. (A.2) (and
from σ > 0), the determinant of the CM of any pure state has to be equal to 1. Now,
working out the determinant of the global CM σ 3m under the assumptions (A.12),
(A.13) and s = t = (σ 3m
q )13 = 0 one gets Det σ 3m = (α + (u 2 − σ)β)/(α − σβ)
with β > 0 (again from the strict positivity of σ 3m ), which is equal to 1 if and only
if u = 0. Therefore, for pure three-mode Gaussian states, the matrix σ3m qp can be
made null by local symplectic operations alone on the individual modes. The entries
of the symmetric positive definite matrix σ3m q are constrained by the necessity of
Eqs. (A.5) — which just determines σ3m p — and (A.12), which is comprised of three
independent conditions and further reduces the degrees of freedom of the state to
the predicted value of three. An alternative proof of this is presented in Sec. 7.1.2
[GA11].
Let us also incidentally remark that the possibility of reducing the sub-matrix
σqp to zero by local single-mode operations is exclusive to two-mode (pure and
mixed) and to three-mode pure states. This is because, for general Gaussian states,
the number of parameters of σqp after the local Williamson diagonalizations is given
by N(N − 1) (two per pair of modes) and only N of these can be canceled out by
the final local rotations, so that only for N < 3 can local operations render σqp null.
For pure states and N > 2 then, further N(N − 1)/2 constraints on σqp ensue from