Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
ρin
c
ω
Alice Bob
Figure 3.4:
Teleportation scheme: Alice and Bob share a bipartite entangled state ω. Alice
performs a measurement on the input state ρin and her part of ω. She sends the
classical outcome c to Bob, who adjusts his part of ω accordingly. This yields the
output ρout.
continuous-variable teleportation. Until recently, the value of this bound was only assumed
to be 1
2 (though various papers provided ample evidence [60,63,64,65,66]). By
the calculations in Section 3.4.3, published in [a], we could ascertain this value and
thus prove the criterion. Moreover, since the derivation included procedures assisted
by ppt-bound entanglement, we could even extend the criterion to this case:
Corollary 3.12:
A process that replicates a single input coherent state by measuring the input,
forwarding classical information only and repreparing output states with a fidelity
must have necessarily been assisted by non-ppt entanglement.
exceeding 1
2
In [46] it has been proven in a more general context that the bound fclassical ≤ 1
2
is valid and tight for classical measure-and-prepare schemes where the fidelity is
averaged over a flat distribution of input coherent states. For the standard teleportation
protocol [59] involving only measurements of the quadrature components, i.e.
the field operators Qi and Pi, the findings of [60] imply that the maximum fidelity
for teleportation of coherent states without supplemental entanglement is 1
2 . Our
above result is more general as it does not make additional assumptions about the
measure-and-prepare scheme and, moreover, distinguishes between ppt and non-ppt
entanglement. Note, however, that teleportation of coherent states with the standard
protocol can be described by a local-realistic model, cf. [60].
Another connection between cloning and teleportation concerns the distribution
of quantum information. It is not necessarily clear that the output state of the
teleportation process is the best remaining approximation to the input state. In
fact, if the fidelity of the teleportation output is low, the input state might have
not been used efficiently and still retain most of the information. However, if the
fidelity of the output with respect to the original input state exceeds the single-copy
fidelity of the optimal (non-Gaussian) 1-to-2 cloner, then the output system must
64
ρout