ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

202 12. Multiparty quantum communication with Gaussian resources
quadrature operators (see Refs. [236, 235] for further details):
with
ˆqtel = ˆqin − ˆqrel ,
ˆptel = ˆpin + ˆptot ,
ˆqrel = ˆqk − ˆql ,
ˆptot = ˆpk + ˆpl + gN
j=k,l
ˆpj ,
(12.20)
(12.21)
where gN is an experimentally adjustable gain. To compute the teleportation fidelity
from Eq. (12.10), we need the variances of the operators ˆqrel and ˆptot of
Eq. (12.21). From the action of the N-splitter, Eq. (12.19), we find
〈(ˆqrel) 2 〉 = 2n2e −2(¯r−d) ,
〈(ˆptot) 2 〉 = [2 + (N − 2)gN ] 2 n1e −2(¯r+d)
+ 2[gN − 1] 2 (N − 2)n2e 2(¯r−d) /4 .
(12.22)
The optimal fidelity can be now found in two straightforward steps: 1) minimizing
〈(ˆptot) 2 〉 with respect to gN (i.e. finding the optimal gain g opt
N ); 2) minimizing
the resulting φ with respect to d (i.e. finding the optimal bias d opt
N ). The results
are
g opt
N = 1 − N/ (N − 2) + 2e 4¯r
n2/n1 , (12.23)
d opt 1
N = ¯r +
4 log
N
(N − 2) + 2e4¯r
. (12.24)
n2/n1
Inserting Eqs. (12.22–12.24) in Eq. (12.10), we find the optimal teleportationnetwork
fidelity, which can be put in the following general form for N modes
F opt
N =
1
, ˜ν (N)
Nn1n2
− ≡
2e4¯r . (12.25)
+ (N − 2)n1/n2
1 + ˜ν (N)
−
For N = 2, ˜ν (2)
− ≡ ˜ν− from Eq. (12.9), showing that the general multipartite
protocol comprises the standard bipartite one as a special case.
By comparison with Eq. (12.15), we observe that, for any N > 2, the quantity
˜ν (N)
− plays the role of a “generalized symplectic eigenvalue”, whose physical meaning
will be clear soon. Before that, it is worth commenting on the form of the optimal
resources, focusing for simplicity on the pure-state setting (n1,2 = 1). The optimal
form of the shared N-mode symmetric Gaussian states, for N > 2, is neither
unbiased in the qi and pi quadratures (like the states discussed in Ref. [32] for
three modes), nor constructed by N equal squeezers (r1 = r2 = ¯r). This latter case,
which has been implemented experimentally for N = 3 [277], is clearly not optimal,
yielding fidelities lower than 1/2 for N ≥ 30 and ¯r falling in a certain interval
[236] [see Fig. 12.3(b)]. The explanation of this paradoxical behavior, provided by
the authors of Ref. [236], is that their teleportation scheme might not be optimal.
Our analysis [GA9] shows instead that the problem does not lie in the choice of
the protocol, but rather in the form of the employed states. If the shared Nmode
resources are prepared by suitable pre-processing — or transformed by local
unitary (symplectic on the CM) operations — into the optimal form of Eq. (12.24),
the teleportation fidelity is guaranteed to be nonclassical [see Fig. 12.3(a)] as soon