ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

48 2. Gaussian states: structural properties
with ν +
αM = ν +
αM2 and ν− α = ν− α 2. Eq. (2.65) shows explicitly that the state with
CM σ ′′ , obtained from the original state with CM σ by exploiting local unitary
operations, is the tensor product of M + N − 2 uncorrelated single-mode states and
of a correlated two-mode Gaussian state.
According to this reduction, one immediately has that the amount of entanglement
(quantum correlations) present in any bisymmetric multimode Gaussian
state can be localized (concentrated) in an equivalent two-mode Gaussian state
(i.e. shared only by a single pair of modes), via local unitary operations [GA5].
These results and their consequences will be discussed in detail in Chapter 5. Let
us just note that fully symmetric Gaussian states, Eq. (2.60), are special instances
of bisymmetric states with respect to any global bipartition of the modes.