ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

270 A. Standard forms of pure Gaussian states under local operations
the antisymmetric condition (A.6): this number turns out to match the number of
free parameters in σqp for N = 3, but it is no longer enough to make σqp null for
pure states with N ≥ 4. This is further discussed in Chapter 11.
Summing up, we have rigorously determined the number of “locally irreducible”
free parameters of pure Gaussian states [GA18], unambiguously showing that the
quantification and qualification of the entanglement (which, by definition, is preserved
under local unitary operations) in such states of N modes is completely
determined by 1 parameter for N = 2 and (N 2 − 2N) parameters for N > 2, as
reported in Eq. (2.56).