ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

CHAPTER 10
Tripartite and four-partite state
engineering
In this Chapter, based mainly on Ref. [GA16], we provide the reader with a systematic
analysis of state engineering of the several classes of three-mode Gaussian
states characterized by peculiar structural and/or entanglement properties, introduced
in Chapter 7 (Secs. 7.3 and 7.4). We will also briefly discuss the instance
of those four-mode Gaussian states exhibiting unlimited promiscuous entanglement
[GA19], introduced in Chapter 8. For every family of Gaussian states, we will
outline practical schemes for their production with current optical technology.
General recipes to produce pure Gaussian states of an arbitrary number of
modes will be presented in the next Chapter.
10.1. Optical production of three-mode Gaussian states
10.1.1. The “allotment” box for engineering arbitrary three-mode pure states
The structural properties of generic pure three-mode Gaussian states, and their
standard form under local operations, have been discussed in Sec. 7.1.2. The computation
of the tripartite entanglement, quantified by the residual Gaussian contangle
of Eq. (7.36), for those states has been presented in full detail in Sec. 7.2.3. Here
we investigate how to produce pure Gaussian states of three modes with optical
means, allowing for any possible entanglement structure.
A viable scheme to produce all pure three-mode Gaussian states, as inspired
by the Euler decomposition [10] (see also Appendix A.1), would combine three independently
squeezed modes (with in principle all different squeezing factors) into
any conceivable combination of orthogonal (energy preserving) symplectic operations
(essentially, beam-splitters and phase-shifters, see Sec. 2.2.2). This procedure,
that is obviously legitimate and will surely be able to generate any pure state, is
however not, in general, the most economical one in terms of physical resources.
Moreover, this procedure is not particularly insightful because the degrees of bipartite
and tripartite entanglement of the resulting output states are not, in general,
easily related to the performed operations.
Here, we want instead to give a precise recipe providing the exact operations
to achieve an arbitrary three-mode pure Gaussian state with CM in the standard
form of Eq. (7.19). Therefore, we aim at producing states with any given triple
{a1, a2, a3} of local mixednesses, and so with any desired ‘physical’ [i.e. constrained
by Ineq. (7.17)] asymmetry among the three modes and any needed amount of tripartite
entanglement. Clearly, such a recipe is not unique. We provide here one
175