Quantum Information Theory with Gaussian Systems

f2(T)
1
0
(a)
1
f1(T)
f2(T)
1
0
(b)
λ = 1
2
3.2 Fidelities
1
f1(T)
Figure 3.1:
Schematic diagram of the convex set fsc of achievable worst-case single-copy fidelities
for 1-to-2 cloning. Any fidelity pair between the origin and the arc (e.g. on the
dotted line) can be realized by a classical mixture of an optimal cloner on the arc
and a fixed output state, represented by the origin. The shaded area of fidelities
is not accessible. The right diagram illustrates the interpretation of tangents. In
contrast to (a), the optimal cloners in (b) are the trivial cloners for small values
of λ or (1 − λ), indicated by the finite slope of the tangent in (0, 1) and (1, 0). See
text for further details.
Therefore, our task is to find the maximal worst-case joint fidelity with respect to
all cloners T,
inf
fjoint = sup
T
fjoint(T) = sup
T
ρ∈coh fjoint(T, ρ),
and the set fsc of all achievable n-tuples
f1, f2, . . . , fn of worst-case single-copy
fidelities.
This set is schematically depicted in Fig. 3.1 for the case of 1-to-2 cloning. Each
point in the diagram corresponds to a pair of worst-case single-copy fidelities for
the two clones in the output and thus to a cloner yielding these fidelities. The
achievable fidelities are of course restricted by the requirement that f1 ≤ 1 and
f2 ≤ 1. From two cloners one can construct a whole range of cloners by classical
mixing; the resulting fidelities lie on the line connecting the fidelity pairs of the two
initial cloners, indicated by the points on the dotted line. Consequently, the set fsc
is convex. The points with fidelities (f1, f2) = (1, 0) and (0, 1) represent the trivial
cloners which return the input state in one output subsystem and leave the other in
a fixed reference state. All fidelity pairs below and on the dashed line can be reached
by a classical mixture of these cloners and a fixed output state, represented by the
origin with (f1, f2) = (0, 0). Optimizing cloners has the effect of enlarging the convex
35