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Allegra, Giorda, and Paris Reply: In our Letter [we
addressed the evolution of photon-number entangled states
state of a harmonic oscillator in noisy channels described
A ¼ 12 Àð1 þ NTÞ, B ¼ 12 ÀNT, where À is the damping
factor of the channel and NT the average number of thermal
excitations of the channel. Upon exploiting several non-
equivalent separability criteria we found evidence that
entanglement of Gaussian PNES survives longer, and
thus we drew a conjecture about the generality of this
result. General arguments supporting the conjecture have
been then put forward in On the other hand, the
conjecture need not to be true if one enlarges the set of
time tG for Gaussian states with same initial energy (open
separability criteria, e.g., to include the negativity of the
symbols) or entanglement (full symbols). (Left) PSSV states:
density matrix under partial transposition (NDPT). Indeed,
black circles, 0 ¼ 0:1; E0 ¼ 0:013 (notice that the curves over-
in ] it has been shown that the conjecture is too strong to
lap); red squares, 0 ¼ 1:0; E0 ¼ 0:3. (Right) jc 01i states: black
be maintainable in the general case.

circles, jc1j2 ¼ 0:5; red squares, jc1j2 ¼ 0:25; blue triangles,
The preceding Comment [presents two examples
where the conjecture is violated. The first is the states
jc 01i ¼ c0j00i þ c1j11i with 0 c21 1=2, whereas the
one includes the NDPT criterion. Nevertheless, for the
second example concerns photon subtracted squeezed vac-
uum (PSSV), i.e., the PNES with c n ¼/ ðn þ 1Þxnþ1 with
Comment [which violate the conjecture, the residual
initial entanglement 0 2 f0:1; 1g and initial energy E0 2
entanglement is extremely low. Therefore, it remains an
f0:013; 0:3g. Not surprisingly, these examples confirm that
open question whether these, as well as other kinds of
by enlarging the set of separability criteria the conjecture is
states, could represent a useful resource in quantum com-
not maintainable. However, these examples are worth ana-
munication protocols. The issue of degradation of continu-
lyzing in order to assess whether their ‘‘robustness’’ com-
ous variable entanglement in noisy channels is indeed an
pared to Gaussian states has some relevant physical
extremely relevant one, and it is worth analyzing in detail
consequences. With this aim we have evaluated the resid-
ual NDPT N R½ðtGÞ displayed by the above states at theSimon (Gaussian) separation time tG (i.e., at the time inwhich Gaussian states with the same initial entanglement
Michele Allegra,1,2 Paolo Giorda,1 and Matteo G. A. Paris3
(note that one photon at optical frequency $ 1015 Hz
Dipartimento di Fisica TeoricaUniversita` degli Studi di Torino
corresponds to a temperature T $ 5 Â 104 K).

The results are shown in Fig. where, in order to give
an estimate of the level of residual entanglement, we report
maximal entanglement at the energy of ðtGÞ, i.e., theentanglement displayed by a pure Gaussian state with
Received 17 October 2011; published 30 November 2011DOI:
same energy, and N 0 is the initial entanglement. For
PACS numbers: 03.67.Mn, 03.65.Yz, 42.50.Dv
PSSV states (left-hand panels), we show that although theirseparation time can be larger than tG, their residual entan-
[1] M. Allegra, P. Giorda, and M. G. A. Paris,
glement is very small for all values of B=A and, in par-
ticular, for realistic values of NT, i.e., B=A ( 1. Similar
results are obtained for jc 01i (right-hand panels).

[3] K. K. Sabapathy, J. S. Ivan, and R. Simon,
In summary, the conjecture that we put forward in our
Letter, based on a correct comparison of several different
[4] J. Lee, M. S. Kim, and H. Nha, preceding Comment, Phys.

separability criteria, is not maintainable in a strict sense if

Source: http://www.isi.it/wp-content/uploads/publication/document/physrevlett_107_238902_1384178139.pdf

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