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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 R½ðtGފ 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  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

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