Quantum Information Theory with Gaussian Systems

5.3 Result and outlook
5.3 Result and outlook
The calculations of the previous sections culminate in the following proposition and
provide its proof:
Proposition 5.1:
A private quantum channel with approximate security and discrete classical key
can be realized for coherent states by randomization with isotropic, uncorrelated
Gaussian noise. The protocol can be secured against all collective attacks, including
coherent schemes, involving a finite number of output states by considering
tensor products of input states. In particular, any two output states T(α), T(β) of
the randomization T for tensor products |α〉〈α|, |β〉〈β| of N coherent states with
f modes each are nearly indistinguishable in the sense of arbitrarily small trace
norm distance T(α) − T(β) 1 ≤ ǫ. This is accomplished by
⊲ addition of Gaussian noise with uniform covariance g ≥ g(ǫ, Emax, Nf),
⊲ restriction to a hypersphere of radius a ≥ a(g, Nf) in phase space and
⊲ discretization to K = K(a, δ, Nf) hypercubes with
⊲ diagonal δ ≤ δ(a, g, Nf),
where the exact values are established through Eqs.(5.18) and (5.20). The encryption
scheme requires (log 2 K)/N classical bits of the discrete key per input state
encrypted. Moreover, the phase space displacements determining the encryption
operations are defined deterministically and implicitly. Hence no preparatory communication
between sending and receiving parties is needed apart from exchange
of the global parameters and the classical key.
For the simplest case of single-mode coherent states without consideration of correlations,
the following corollary summarizes the more explicit results derived above:
Corollary 5.2:
For Nf = 1 the protocol guarantees security up to ǫ, i.e. T(α) − T(β) 1 ≤ ǫ,
with the following parameter values:
⊲ g ≥ 400 Emax/ǫ 2 ,
⊲ a ≥ 40 √ 2Emax ǫ−1 log(20/ǫ) 1/2 ,
⊲ δ ≤ ǫ
√
3 2 −1,
5 a /(4g ) + 2
⊲ K = 16 log(20/ǫ) 2 √
+ 80 Emax ǫ−1 log(20/ǫ) 1/2 2 .
The above parameter values have been derived for a specific protocol and with
the help of several estimations; this leaves plenty of space for optimization. A few
moretechnicalimprovements could be achieved by finding tighter estimations
for the various steps of the computation or by optimizing the contributions of the
terms in (5.18). A conceptual extension could include the application of correlated
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