det [ α ] + det [ β ] + 2 det [ γ ] det[ γ ]=0 = det [ α ] + det [ β ] ≤ 4 det [ σ ] + 1 / 4 . (16) det [ C ] + det [ f ] $^{2}$det [ C + S − 1 B f − $^{1}$[ g ] ] − 2 det [ f ] det [ S ] ≤ 4 det [ f ] $^{2}$det [ γ$_{TMSS}$$_{,}$$_{r}$ + ( O ⊕ S 1 B f − $^{1}$[ g ]) ] + 1 / 4 . (17) 4 det [ g ] ≥ det [ f ] 2 + 2 det [ f ] + 1 = (det [ f ] + 1) $^{2}$, (18) σ$_{AB}$ = ( S$_{A}$ ⊕ S$_{B}$ )[ γ$_{TMSS}$$_{,}$$_{r}$ ] , (12) σ$_{A}$$_{1}$$_{,...,A}$$_{N}$$_{−}$$_{1}$$_{,B}$ = (19) ( S$_{A,N}$$_{−}$$_{1}$ ⊕ S$_{B}$ ) [ ( γ$_{vac}$ )$_{A}$$_{1}$$_{,...,A}$$_{N}$ − 2 ⊕ ( γ$_{TMSS}$$_{,r}$ )$_{A}$$_{N}$ − $_{1}$,B ] . ( 1 ⊕ fS$_{B}$ )[ γ$_{TMSS}$$_{,}$$_{r}$ ] + ( O ⊕ g ) = ( C S ( fS$_{B}$ ) T fS$_{B}$S fS$_{B}$ [ C ] + g ) ≡ ( α γ γ T β ) , (13) det [ C ] + det [ fS$_{B}$ [ C ] + g ] − 2 det [ fS$_{B}$S ] ≤ 4 det [( 1 ⊕ fS$_{B}$ )[ γ$_{TMSS}$$_{,}$$_{r}$ ] + ( O ⊕ g )] + 1 / 4 . (14) det [ α ] + det [ β ] − 2 det [ γ ] det[ γ ]=0 = det [ α ] + det [ β ] ≤ 4 det [ σ ] + 1 / 4 (15) Summing up, qualitatively , the entanglement of all pure Gaussian states evolves in the same way under a fixed one-sided Gaussian channel: Just as in the qubit case, the separability of the output state depends only on the channel, not on the initial state. The question whether or not it also evolves in the same way quantita- tively , hence whether or not a full Gaussian entanglement evolution equation similar to (2) exists, is still open. The answer to this question depends crucially on the choice of the right entanglement measure for a Gaussian version of (2). Numerical calculations ruled out the existence of a Gaussian entanglement evolution equation for the two best-known entanglement measures in the context of For det [ f ] = 0, we have det [ γ ] = det [ fS$_{B}$S ] = det [ f ] det [ S$_{B}$S ] = 0 and the separability condition (7) reduces to Now there are two cases. Assume first that det [ f ] = 0. In this case, the channel separability condition (11) is automatically fulfilled because any Gaussian channel also fulfills (10). Therefore, it has to be shown that, in this case, the output state is indeed separable, as implied by (11) together with (10). Again, the unitary S$_{A,N}$$_{−}$$_{1}$ can be interchanged with the channel (which only acts on mode B ) and undone. From here on, the proof for the case of an N -mode input state works exactly like the two-mode case since the modes in the vacuum state are irrelevant for the entanglement of the two modes in the two-mode squeezed state. where C ≡ 1 / 2 cosh( r ) 1 2 and S ≡ 1 / 2 sinh( r ) σ$_{z}$ . Ac- cording to (7), the separability of (13) is equivalent to where again the abbreviation A [ B ] ≡ ABA T has been used. The transformation S$_{A}$ of mode A commutes with the channel which acts on mode B and can be undone after the channel since local unitary operations do not change the entanglement of the output state. There- fore, using (12), the separability of the output state ( 1 ⊕ f )[ σ$_{AB}$ ] + ( O ⊕ g ) is equivalent to the separability of the state The generalization to the case of a Gaussian channel acting on one mode of a pure N -mode Gaussian state is straightforward: Any pure N -mode Gaussian input state can be transformed via a local ( N − 1)-mode unitary S$_{A,N}$$_{−}$$_{1}$ and a local one-mode unitary S$_{B}$ into a two- mode squeezed state on the mode A$_{N}$$_{−}$$_{1}$ and the mode B as well as N − 2 unrelated modes ( A$_{1}$, . . . , A$_{N}$$_{−}$$_{2}$ ) in the vacuum state with CM γ$_{vac}$ = 1 2 1 [7]: which establishes (11). Using that S − 1 B f $^{−}$1[ g ] is a symmetric matrix, this in- equality can be evaluated (using computer algebra) and reduces to Proof of Proposition 1. Every pure, entangled two-mode Gaussian state σ$_{AB}$ is related to a two-mode squeezed state γ$_{TMSS}$$_{,}$$_{r}$ via local unitary operations which act on the CM as symplectic matrices S$_{A}$ and S$_{B}$ [9] Note also the close similarity of the separability condi- tion (11) and the physicality condition (10) for the chan- nel matrices. For any two real matrices f, g , there is a quantum channel such that f, g describe its action on CMs according to (9) if the determinant of the noise ma- trix g is large enough, i.e. if 4 det [ g ] ≥ (det [ f ] − 1) $^{2}$. If g also fulfills 4 det [ g ] ≥ (det [ f ] + 1) $^{2}$, this channel will only output separable states. Assume now that det [ f ] = 0, hence f is invertible. Using that det [ S$_{B}$ ] = 1 for every symplectic matrix S$_{B}$ , equation (14) can be rewritten as Therefore, if det [ f ] = 0, the output state is indeed sepa- rable. Holevo [8] provided general conditions under which Gaussian channels disentangle all input states, which are in agreement with (11). His results, however, do not im- ply that any Gaussian channel which leaves some pure entangled Gaussian input state entangled must leave all such states entangled, as required for the existence of a Gaussian entanglement evolution equation. entanglement of X and leaves some of the entanglement of Y intact [16]. But, under the same conditions (for det [ f ] = 0), the separability condition (15) is implied by condition (6) which expresses the physicality of the output CM: 3