Quantum Information Theory with Gaussian Systems

5 Gaussian private quantum channels
have shown in [94] that near-perfect encryption can be achieved with order of d log d
random unitary operations. 2 These results have been complemented in [95] by Ambainis
and Smith with a deterministic protocol. An investigation of private quantum
channels for continuous-variable systems has been started in [96]. Contrary to the
finite-dimensional case there is no ideal encryption due to the lack of a maximally
mixed state, so one has to rely on approximate encryption. Our aim is to rigorously
perform the related discussion for the encryption of coherent input states where the
unitary operations are shifts in phase space occurring with probabilities according to
a classical Gaussian weight function. On the one hand, the Gaussian weight function
renders the channel T between Alice and Eve quasi-free and the randomized states
are Gaussian, too. On the other hand, this weight function assures that the twirl 3
over the noncompact group of all phase space translations exists in the first place.
This randomization introduces classical noise which can be made large enough to
render two coherent input states arbitrarily indistinguishable by inducing a substantial
overlap between the resulting output states; see Fig. 5.1 for illustration. As a
measure of indistinguishability, we choose the trace norm 4 distance of the output
states at Eve’s end of the channel, T(ρ) − T(ρ ′ ) 1 . This quantity has the advantage
of an operational meaning since it equals the maximal difference in expectation
values of any measurement performed on these states [1].
However, this scheme has several inherent problems. First, the amount of noise
to be added depends on the input state; heuristically, the larger its amplitude, the
larger the variance of the Gaussian weight function has to be. To keep the protocol
as general as possible, this requires a bound on the amplitude of the input coherent
states |α〉〈α|, i.e. a bound on their occupation number expectation value and hence
on their energy 5 E = |α| 2 /2 ≤ Emax. Second, encryption with a continuous set
of phase space displacements would require an infinite key for each input state in
order to specify the phase space vector precisely. This problem can be overcome by
restricting the continuous integral for randomization to a finite area of phase space,
e.g. to a hypersphere with radius a, and approximating it with a finite sum over
a discrete set of displacements. Finally, since the encryption is only near-perfect,
the output state might be distinguishable up to a security parameter ǫ. A general
choice for the protocol with approximate security is whether it should be ablock
2 If output states are required to differ by at most ǫ > 0 in trace norm distance, approximately
(d log d)/ǫ 2 unitaries are needed. The operators can be chosen randomly, since the proof shows
that almost any such set of encryption operations yields the desired security.
3 A twirl is the averaging over all elements of a group, i.e. in our case the phase space displacements,
Z
dξ e −ξT ·G·ξ/4 Wξ ρ W ∗ ξ .
4 The trace norm X 1 of an operator X is defined as X 1 = tr|X| where |X| = √ X ∗ X is the
modulus of X.
5 The energy contained in a mode in a state ρ equals the occupation number expectation value in
that state scaled with the characteristic energy of ω of the associated harmonic oscillator with
frequency ω. We assume that the frequencies of all modes are the same. This would be the case
if all modes originate from the same laser mode, but states of this mode might be distinguished
e.g. by their temporal ordering.
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