ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

2.4. Standard forms of Gaussian covariance matrices 43
a pure state. A detailed analysis of the constraints imposed by Eq. (2.55), as obtained
in Ref. [GA18], is reported in Appendix A. Here, it suffices to say that by
proper counting arguments the CM of a pure N-mode Gaussian state is determined
by N 2 + N parameters in full generality. If one aims at evaluating entanglement,
one can then exploit the further freedom arising from local-unitary invariance and
reduce this minimal number of parameters to (see Appendix A.2.1)
#(σ p ) =
N(N − 1)/2, N ≤ 3 ;
N(N − 2), N > 3 .
(2.56)
An important subset of pure N-mode Gaussian states is constituted by those whose
CM which can be locally put in a standard form with zero direct correlations between
position ˆqi and momentum ˆpj operators, i.e. with all diagonal submatrices in
Eq. (2.20). This class encompasses basically all Gaussian states currently produced
in a multimode setting and employed in CV communication and computation processes,
and in general all Gaussian states of this form are ground states of some
harmonic Hamiltonian with spring-like interactions [11]. For these Gaussian states,
which we will refer to as block-diagonal — with respect to the vector of canonical
operators reordered as (ˆq1, ˆq2, . . . , ˆqN, ˆp1, ˆp2, . . . , ˆpN) — we have proven that the
minimal number of local-unitary-invariant parameters reduces to N(N − 1)/2 for
any N [GA14]. As they deserve a special attention, their structural properties will
be investigated in detail in Sec. 11.2.1, together with the closely related description
of their generic entanglement, and with the presentation of an efficient scheme for
their state engineering which involves exactly N(N − 1)/2 optical elements (singlemode
squeezers and beam-splitters) [GA14]. Notice from Eq. (2.56) that not only
single-mode and two-mode states, but also all pure three-mode Gaussian states are
of this block-diagonal form: an explicit standard form from them will be provided
in Sec. 7.1.2, and exploited to characterize their tripartite entanglement sharing as
in Ref. [GA11].
2.4.2.1. Phase-space Schmidt decomposition. In general, pure Gaussian states of a
bipartite CV system admit a physically insightful decomposition at the CM level
[116, 29, 92], which can be regarded as the direct analogue of the Schmidt decomposition
for pure discrete-variable states (see Sec. 1.3.1). Let us recall what happens
in the finite-dimensional case. With respect to a bipartition of a pure state |ψ〉 A|B
into two subsystems SA and SB, one can diagonalize (via an operation UA ⊗ UB
which is local unitary according to the considered bipartition) the two reduced
density matrices ϱA,B, to find that they have the same rank and exactly the same
nonzero eigenvalues {λk} (k = 1, . . . , min{dim HA, dim HB}). The reduced state
of the higher-dimensional subsystem (say SB) will accomodate (dim Hb − dim HA)
additional 0’s in its spectrum. The state |ψ〉 A|B takes thus the Schmidt form of
Eq. (1.20).
Looking at the mapping provided by Table 2.II, one can guess what happens for
Gaussian states. Given a Gaussian CM σ A|B of an arbitrary number N of modes,
where subsystem SA comprises NA modes and subsystem SB, NB modes (with
NA + NB = N), then one can perform the Williamson decomposition Eq. (2.29) in
both reduced CMs (via a local symplectic operation SA ⊕ SB), to find that they
have the same symplectic rank, and the same non-unit symplectic eigenvalues {νk}
(k = 1, . . . , min{NA, NB}). The reduced state of the higher-dimensional subsystem
(say SB) will accomodate (NB−NA) additional 1’s in its symplectic spectrum. With