Role of Initial Entanglement and Non-Gaussianity in the Decoherence of Photon-Number Entangled States Evolving in a Noisy Channel

Abstract

We address the degradation of continuous variable (CV) entanglement in a noisy channel focusing on the set of photon-number entangled states. We exploit several separability criteria and compare the resulting separation times with the value of non-Gaussianity at any time, thus showing that in the low-temperature regime: (i) non-Gaussianity is a bound for the relative entropy of entanglement and (ii) Simon's criterion provides a reliable estimate of the separation time also for non-Gaussian states. We provide several evidences supporting the conjecture that Gaussian entanglement is the most robust against noise, i.e., it survives longer than a non-Gaussian one, and that this may be a general feature for CV systems in Markovian channels.

Simple proof of the robustness of Gaussian entanglement in bosonic noisy channels

“…the decoherence effects due to the latter types of channels on special families of Gaussian as well as non-Gaussian two-mode entangled states, all initially having the same mean energy [17]. Among the main findings of their work, they formulate a conjecture on the "robustness of Gaussian entanglement," according to which initial Gaussian pure states should be the ones whose entanglement is the last to vanish, compared to other classes of initial non-Gaussian pure states with equal energy, when they are all evolving through a noisy Markovian channel with loss and thermal hopping.…”

“…[17] by resorting to the extremality of Gaussian states [14], in particular to the fact that Gaussian states maximize entanglement among all CV states with fixed energy [18]. Our simple result establishes in full generality that no advantage can be gained by encoding information into entangled non-Gaussian states when they have to be transmitted through Gaussian noisy channels (under dissipation and/or amplification) and that Gaussian states are, hence, the most robust CV states against photon losses and thermal hopping.…”

“…Among all the possible input states, the two-mode squeezed state G 12 (0) = |ψ ψ|, where |ψ = n λ n √ 1 − λ 2 |n,n and n 0 = λ 2 /(1 − λ 2 ), is the only Gaussian instance (up to local unitary operations). We let each possible initial state undergo a dissipative evolution through the channel described by the following master equation [17],…”

“…It is important here that the energy of the evolving states 12 (t) at any time depends only on the initial energy and on the channel parameters; that is, if a set of states 12 (0) enter the channel all with the same energy, as it is the case in [17] and in the present discussion, their energies will remain equal to each other (yet globally affected by the open-system dynamics) throughout the whole evolution, regardless of the specific form of the states and of their Gaussian or non-Gaussian nature. Let us also remark once more that the channel in Eq.…”

“…Given that (as soon as N 1 ,N 2 = 0) the channel in Eq. (1) takes a finite "separation time" τ [ 12 (0)] to erase all the entanglement in any initial quantum state 12 (0) [17], the entanglement in the Gaussian state must be the last one to vanish, that is, τ [ G 12 (0)] τ [ 12 (0)] for all CV states 12 (0) with the same mean energy as G 12 (0). This proves the robustness of Gaussian entanglement conjecture of Ref.…”