Quantum Information Theory with Gaussian Systems

3.4 Optimization
wheren is the matrix fully occupied with 1, defined for Eq.(3.19). Decomposing
χout into the input part χin and the channel part t according to Eq.(3.16) yields
t(ξ) = exp −ξ T · ( n ⊗ 2 +n ⊗ 2) · ξ/4 . This is the characteristic function
(3.31) of the best symmetric Gaussian 1-to-n cloner considered in Section 3.4.2 for
the classical case, i.e. for a = b = 1. This cloner indeed yields equal single-copy
fidelities of fi = 1
2
, cf. Eq.(3.33a). It is covariant by design, cf. Section 3.3.2, and
also manifestly, because the output state inherits the displacement vector α from
the input state, see (3.35).
Remark: For a single clone, e.g. the first one, the output characteristic function
χout(ξ1, 0, . . .,0) = exp −ξ2 1 /4 + ξ2 1 /2 + iξT 1 · α corresponds to the input coherent
state |α〉〈α| plus two units of vacuum noise.
The characterization of classical 1-to-1 cloners or classical teleportation by time
reversal extends to cloners which are supplemented by ppt-bound entangled states
[45]:
Lemma 3.8:
Every classical teleportation protocol assisted by a ppt-bound entangled state ω
corresponds to a channel T which is completely positive under transposition of the
input density operator in the Schrödinger picture. That is, if Θ denotes matrix
transposition, then T∗ ◦ Θ is completely positive.
Remark: Note that in Schrödinger representation, time reversal of observables corresponds
to transposition of the Hermitian density operator, cf. [15].
Proof: Denote the Hilbert space of the input state ρ by HI and the Hilbert space of
the bipartite, ppt-entangled state ω by HA ⊗HB. Since the teleportation protocol is
classical, the corresponding channel T∗ in the Schrödinger picture can be represented
by a set of Kraus operators {Mi⊗Ri} in product form, cf. Section 2.3. The operators
Mi act on the input plus one part of the entangled state, i.e. on HI ⊗ HA, and play
the role of themeasurement. The Ri act on HB and turn the second part of ω
into the desired output state, thus corresponding to therepreparation. Hence T∗
is represented as
T∗(ρ) =
i
trI,A (Mi ⊗ Ri)(ρ ⊗ ω)(Mi ⊗ Ri) ∗ ,
where trI,A denotes the partial trace over subsystems I and A. Since the trace is
invariant under transposition of its argument, we can transpose the above expression
with respect to systems I and A to obtain for T∗ ◦ Θ:
(T∗ ◦ Θ)(ρ) = T∗(ρ T
) =
i
trI,A (Mi ⊗ Ri)(ρ ⊗ ω TA )(Mi ⊗ Ri) ∗ ,
where ω T A denotes the partial transposition of ω with respect to system A. If ω has
positive partial transpose, i.e. if ω T A ≥ 0, then T∗ ◦ Θ is completely positive, since it
is implemented by a set of Kraus operators {Mi ⊗ Ri}.
57