ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

Β
E KΒ 10K
5.2. Quantification and scaling of entanglement in fully symmetric states 103
3
2.5
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1.5
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0.5
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1 1.5 2 2.5 3 3.5 4
b 1ΜΒ
K5
K3
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Figure 5.4. Hierarchy of block entanglements of fully symmetric 2N-mode
Gaussian states of K × (2N − K) bipartitions (2N = 10) as a function of
the single-mode squeezing b. The block entanglements are depicted both for
pure states (solid lines) and for mixed states obtained from fully symmetric
(2N + 4)-mode pure Gaussian states by tracing out 4 modes (dashed lines).
resource for a given protocol. Let us further suppose that the protocol is optimally
implemented if the entanglement is concentrated between only two modes of the
global systems, as it is the case, e.g., in a CV teleportation protocol between two
single-mode parties [39]. Which choice of the bipartition between the modes allows
for the best entanglement concentration by a succession of local unitary operations?
In this framework, it turns out that assigning K = 1 mode at one party and all
the remaining modes to the other, as discussed in Sec. 5.2.1, constitutes the worst
localization strategy [GA5]. Conversely, for an even number of modes the best
option for localization is an equal K = N splitting of the 2N modes between the
two parties. The logarithmic negativity E βN |β N
N , concentrated into two modes by
local operations, represents the optimal localizable entanglement (OLE) of the state
σβ2N , where “optimal” refers to the choice of the bipartition. Clearly, the OLE of
a state with 2N + 1 modes is given by E βN+1 |β N
N . These results may be applied to
arbitrary, pure or mixed, fully symmetric Gaussian states.
5.2.2.2. Entanglement scaling with the number of modes. We now turn to the study
of the scaling behavior with N of the OLE of 2N-mode states, to understand how
the number of local cooperating parties can improve the maximal entanglement that
can be shared between two parties. For generic (mixed) fully symmetric 2N-mode
states of N × N bipartitions, the OLE can be quantified also by the entanglement
of formation EF , Eq. (4.17), as the equivalent two-mode state is symmetric, see
Sec. 5.1.2. It is then useful to compare, as a function of N, the 1 × 1 entanglement
of formation between a pair of modes (all pairs are equivalent due to the global
symmetry of the state) before the localization, and the N × N entanglement of
formation, which is equal to the optimal entanglement concentrated in a specific