Figure 1
The general scenario in five steps. (1) Quantum communication. Alice randomly picks signal states
from an ensemble
encoding a classical variable
. At the output of the channel, Bob detects the states via a quantum measurement. The corresponding outcomes define an output classical variable
correlated to
. (2) One-mode Gaussian channel. A one-mode Gaussian channel
corresponds to a canonical form
up to a pair of Gaussian unitaries
(at the input) and
(at the output). The central canonical form
can be dilated to a symplectic interaction
involving two ancillary modes
prepared in a TMSV state
. The dilation of the form is unique up to isometries acting on
. (3) Maximal dilation. By assuming Eve is in a finite box, the dilation can be extended (via an identity) to the remaining modes
of the environment (prepared in vacua). This maximal dilation of
is now unique up to unitaries
acting on
. (4) Collective Gaussian attack. All the output ancillas
provide an ensemble
, which Eve can detect to estimate
or
. By using an entropic bound for Eve’s accessible information, the extra ancillas and the extra unitary (dashed boxes in the figure) can be neglected. As a consequence, only the set
(solid boxes in the figure) is needed to characterize the attack. (5) Coherent-state protocol. Alice’s signal states
are coherent states
whose amplitudes encode a Gaussian variable (
). Bob’s measurement is a heterodyne detection retrieving the output amplitudes (
).
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