Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
istic function, χout(ξ1, . . . , ξn) = χT(Ω ξ)χin(
i ξi), we can compute the fidelity of
such cloners. Since there is only one output subsystem, the distinction between singlecopy
and joint fidelity is not necessary and we write fclassical(T) = fjoint(T) = fi(T).
For covariant T, we can evaluate the fidelity for the fixed input state |0〉〈0|:
T, |0〉〈0| = tr T |0〉〈0| |0〉〈0|
fclassical(T) = fclassical
dξ
=
2π χT
√
2ξ χin(ξ)
2 1 dξ
=
2 2π χT(ξ)χin(ξ)
= 1
2 trρT |0〉〈0| ≤ 1
2 , (3.34)
where χin(ξ) = exp(−ξ 2 /4) is the characteristic function of the input state |0〉〈0|.
This bound is tight, which has been proven in [63]. However, for completeness and
further investigation we provide
Lemma 3.7:
The fidelity bound (3.34), fclassical(T) ≤ 1
2 , can be reached by a heterodyne measurement
and preparation of coherent states according to the measurement result.
Moreover, this scheme can be extended to a 1-to-n cloner which yields the same
fidelity and is Gaussian as well as covariant.
Proof: A heterodyne measurement is modeled as a povm9 |µ〉〈µ|/(2π) based on
coherent states µ ∈ S ccr(Ξin, σin) . The probability pα(µ) of finding the measurement
outcome µ for a coherent input state α ∈ S ccr(Ξin, σin) is
pα(µ) = tr |α〉〈α| |µ〉〈µ|/(2π) = 1
dξ
2π 2π χα(ξ) χµ(ξ)
= 1
dξ
2π 2π exp−ξ 2 /2 + iξ T
· (µ − α) = e −(µ−α)2 /2
/(2π),
where χα(ξ) = exp(−ξ2 /2 + iξ T · α) is the characteristic function of the coherent
state α (likewise for µ) and the bar denotes complex conjugation. In order to produce
n clones of the input state, the output ρout is a classical mixture of n-fold tensor
products |µ〉〈µ| ⊗n of coherent states µ, weighted with the probabilities pα(µ):
ρout = dµ pα(µ) |µ〉〈µ| ⊗n ,
resulting in a characteristic function
χout(ξ) = tr
ρout Wξ = dµ pα(µ) tr |µ〉〈µ| ⊗n
Wξ
= 1
dµ exp
2π
−(µ − α) 2 /2 − ξ 2 /4 + i n j=1 ξT j · µ
= exp −ξ 2 /4 + ( n j=1 ξj )2 /2 + i n j=1 ξT j · α
= exp −ξ T
· ( n ⊗ 2 + 2n ⊗ 2) · ξ/4 + i n j=1 ξT j · α , (3.35)
9 A positive-operator-valued measurement (povm) [1], also called generalized measurement, is a
discrete or continuous set of positive operators {Mj} which resolve unity, i.e. R dj Mj = where
the symbol R dj denotes a discrete summation or a continuous integration.
56