ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

110 6. Gaussian entanglement sharing
We will then lift the symmetry requirements and prove that CV entanglement,
once properly quantified, is monogamous for all Gaussian states [GA15]. This is
arguably the most relevant result of this Chapter, and one of the milestones of this
Dissertation.
The paradigmatic instance of tripartite CV entanglement, embodied by threemode
Gaussian states, will be treated independently and in full detail in the next
Chapter. In that case, let us anticipate that from the monogamy inequality a
measure of genuine tripartite entanglement emerges naturally (residual Gaussian
contangle), and we will prove it to be a full entanglement monotone under Gaussian
LOCC. Equipped with such a powerful tool to quantify tripartite entanglement, we
will proceed to investigate the entanglement sharing structure in three-mode Gaussian
states, unveiling the original feature named promiscuity: it essentially means
that bipartite and multipartite entanglement can enhance each other in Gaussian
states and be simultaneously maximized without violating the monogamy inequality
on entanglement sharing. In Chapter 8, the promiscuous sharing structure
of distributed CV entanglement will be shown to arise to an unlimited extent in
Gaussian states of at least four modes.
6.1. Distributed entanglement in multipartite continuous variable systems
Our primary aim, as in Ref. [GA10], is to analyze the distribution of entanglement
between different (partitions of) modes in Gaussian states of CV systems. The
reader is referred to Sec. 1.4 for a detailed, introductory discussion on the subject
of entanglement sharing.
6.1.1. The need for a new continuous-variable entanglement monotone
In Ref. [59] Coffman, Kundu and Wootters (CKW) proved for system of three
qubits, and conjectured for N qubits (this conjecture has now been proven by
Osborne and Verstraete [169]), that the bipartite entanglement E (properly quantified)
between, say, qubit A and the remaining two-qubits partition (BC) is never
smaller than the sum of the A|B and A|C bipartite entanglements in the reduced
states:
E A|(BC) ≥ E A|B + E A|C . (6.1)
This statement quantifies the so-called monogamy of quantum entanglement [230],
in opposition to the classical correlations, which are not constrained and can be
freely shared.
One would expect a similar inequality to hold for three-mode Gaussian states,
namely
E i|(jk) − E i|j − E i|k ≥ 0 , (6.2)
where E is a proper measure of bipartite CV entanglement and the indexes {i, j, k}
label the three modes. However, the demonstration of such a property is plagued
by subtle difficulties.
Let us for instance consider the simplest conceivable instance of a pure threemode
Gaussian state completely invariant under mode permutations. These pure
Gaussian states are named fully symmetric (see Sec. 2.4.3), and their standard
form CM [obtained by inserting Eq. (5.13) with L = 3 into Eq. (2.60)] is only
parametrized by the local mixedness b = (1/µβ) ≥ 1, an increasing function of the