ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

APPENDIX A
Standard forms of pure Gaussian states
under local operations
In this Appendix, based on Ref. [GA18], we study the action of local unitary operations
on a general CM of a pure N-mode Gaussian state and compute the minimal
number of parameters which completely characterize pure Gaussian states up to
local unitaries.
A.1. Euler decomposition of symplectic operations
Central to our analysis will be the following general decomposition of a symplectic
transformation S (referred to as the “Euler” or “Bloch-Messiah” decomposition [10,
37]):
S = O ′ ZO, (A.1)
where O, O ′ ∈ K(N) = Sp (2N,) ∩ SO(2N) are orthogonal symplectic transformations,
while
Z = ⊕ N
zj 0
j=1
1 ,
0 zj
with zj ≥ 1 ∀ j. The set of such Z’s forms a non-compact subgroup of Sp (2N,)
comprised of local (single-mode) squeezing operations (borrowing the terminology of
quantum optics, where such transformations arise in degenerate parametric downconversion
processes). Moreover, let us also mention that the compact subgroup
K(N) is isomorphic to the unitary group U(N), and is therefore characterized
by N 2 independent parameters. To acquaint the reader with the flavor of the
counting arguments which will accompany us through this Appendix (and with the
nontrivial aspects contained therein), let us combine the Williamson and the Euler
decompositions to determine the number of degrees of freedom of a generic N-mode
Gaussian state (up to first moments), given by N + 2N 2 + N − N = 2N 2 + N.
The N subtracted from the sum of the numbers of symplectic eigenvalues and of
degrees of freedom of a symplectic operation takes into account the invariance under
single-mode rotations of the local Williamson forms — which ‘absorbs’ one degree
of freedom per mode of the symplectic operation describing the state according to
Eq. (2.29). Actually, the previous result is just the number of degrees of freedom of a
2N × 2N symmetric matrix (in fact, the only constraint σ has to fulfill to represent
a physical state is the semidefinite σ + iΩ ≥ 0, which compactly expresses the
uncertainty relation for many modes [208]).
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