Quantum Information Theory with Gaussian Systems

3.6 Teleportation criteria
can be truncated at n = 2 with f1 = f2 ≈ 0.6801 compared to f1 = f2 = 2
3 for the
best Gaussian cloner, which is obtained for n = 0.
While this scheme is conceptually clear, it relies on a nonlinear interaction in
the opa, which poses difficulties in the experimental realization. Recently, Leuchs
et al. [56] have proposed a scheme to realize the best symmetric Gaussian cloner
based on linear quantum optical elements alone, namely beam splitters and homodyne
detection. Their experiment implementing this scheme for 1-to-2 cloning was
reported to yield estimated fidelities of 0.643 ± 0.01 and 0.652 ± 0.01 for the two
clones. An implementation of the optimal cloner has not yet been reported in the
literature.
3.6 Teleportation criteria
The limits on cloning of coherent states constitute at the same time criteria which
allow to ascertain the successful conduction of a continuous-variable teleportation experiment.
In quantum information theory, teleportation is the task of transmitting an
arbitrary, unknown quantum state by sending only classical information [57,58,59].
This is not possible without the help of entangled states shared between sender and
receiver which provide sufficiently strong correlations. The process consists of three
steps, cf. Fig. 3.4: The sender, conventionally named Alice, performs a measurement
on the input system ρin and her part of the shared entangled resource ω. She communicates
the (classical) outcome c to the receiver, called Bob. Depending on this
result, he applies a suitable unitary transformation on his part of the entangled
state and ideally gets back the original input state in ρout. Note that the measurementdestroysthe
quantum information in the input state, i.e. the state of the
joint system on Alice’s side after the measurement does not convey any information
about the input state anymore. For continuous-variable systems, a common protocol
[59,60] uses a two-mode squeezed state as the entanglement resource. It consists
of measuring two commuting quadrature components of the joint system at Alice’s
side and applying the outcome as a phase space displacement on Bob’s system.
The fidelity of the output with respect to the original input state is determined
by thequalityof the entanglement, i.e. its amount quantified by a suitable entanglement
measure. 11 In the finite-dimensional case, perfect teleportation is in principle
possible with maximally entangled states as a resource. For continuous-variable
systems, the output only approximately resembles the input state, because a maximally
entangled state does not exist in this case. 12 If entanglement were not required,
the classical information could be stored and used to replicate the input state, i.e.
clone it. Reversing this argument shows that if the fidelity of the output state is
higher than the limit of classical cloning in Eq.(3.34), the process must indeed have
used entanglement. This turns the classical cloning limit into a success criterion for
11 The relevant entanglement measure is the entanglement of formation; see [62] for the relation
between fidelity and entanglement in continuous-variable teleportation.
12 Such states could be abstractly realized as infinitely entangled states [44]. However, these are
not normal states, i.e. they cannot be described by a density matrix.
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