ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

6.2. Monogamy of distributed entanglement in N-mode Gaussian states 117
systems with an arbitrary number N of modes and parties S1, . . . , SN , thus generalizing
the results of the previous subsection.
As a measure of bipartite entanglement, we employ the Gaussian tangle τG
defined via the square of negativity, Eqs. (6.14, 6.15), in direct analogy with the
case of N-qubit systems [169]. Our proof is based on the symplectic analysis of
CMs (see Chapter 2) and on the properties of Gaussian entanglement measures (see
Sec. 3.2.2). The monogamy constraint has important implications on the structural
characterization of entanglement sharing in CV systems [GA10, GA11, GA16, GA15],
in the context of entanglement frustration in harmonic lattices [272], and for practical
applications such as secure key distribution and communication networks with
continuous variables (see Part V).
Given an arbitrary N-mode Gaussian state ϱS1|S2...SN , we now prove the general
monogamy inequality
τG(ϱS1|S2...SN ) ≥
N
l=2
τG(ϱS1|Sl ) , (6.21)
where we have in general renamed the modes so that the probe subsystem in
Eq. (6.17) is S1, for mere convenience.
To this end, we can assume without loss of generality that the reduced two-mode
states ϱS1|Sl = TrS2...Sl−1Sl+1...SN ϱS1|S2...SN of subsystems (S1Sl) (l = 2, . . . , N)
are all entangled. In fact, if for instance ϱS1|S2 is separable, then τG(ϱS1|S3...SN ) ≤
τG(ϱS1|S2...SN ) because the partial trace over the subsystem S2 is a local Gaussian
operation that does not increase the Gaussian entanglement. Furthermore, by
the convex roof construction of the Gaussian tangle, it is sufficient to prove the
monogamy inequality for any pure Gaussian state ϱ p
(see also Refs. [59,
S1|S2...SN
169]). Therefore, in the following we can always assume that ϱS1|S2...SN is a pure
Gaussian state for which the reduced states ϱS1|Sl (l = 2, . . . , N) are all entangled.
We start by computing the left-hand side of Eq. (6.21). Since ϱS1|S2...SN is
a 1 × (N − 1) pure Gaussian state, its CM σ is characterized by the condition
Eq. (2.55), which implies
Det α +
N
Det γl = 1 , (6.22)
l=2
where γl is the matrix encoding intermodal correlations between mode 1 and mode
l in the reduced state ϱS1|Sl (l = 2, . . . , N), described by a CM [see Eq. (4.1)]
σ S1|Sl =
⎛
⎜
⎝
σ1,1 σ1,2 σ1,2l−1 σ1,2l
σ2,1 σ2,2 σ2,2l−1 σ2,2l
σ2l−1,1 σ2l−1,2 σ2l−1,2l−1 σ2l−1,2l
σ2l,1 σ2l,2 σ2l,2l−1 σ2l,2l
⎞
⎟
⎠ =
α γl
γ T l
β l
. (6.23)
As ϱ S1|Sl is entangled, Det γ l is negative [218], see Eq. (4.16). It is useful to
introduce the auxiliary quantities
such that one has Det α = 1 +
l Υl/4 .
Υl = −4Det γl > 0 , (6.24)