Enhancing quantum entanglement by photon addition and subtraction

The non-Gaussian operations effected by adding or subtracting a photon on entangled optical beams emerging from a parametric down-conversion process have been suggested to enhance entanglement. Heralded photon addition or subtraction is, as a matter of fact, at the heart of continuous-variable entanglement distillation. The use of such processes has recently been experimentally demonstrated in the context of the generation of optical coherent-state superpositions or the verification of canonical commutation relations. Here, we carry out a systematic study of the effect of local photon additions and subtractions on a two-mode squeezed vacuum state, showing that the entanglement generally increases with the number of such operations. This is analytically proven when additions or subtractions are restricted to one mode only, while we observe that the highest entanglement is achieved when these operations are equally shared between the two modes. We also note that adding photons typically provides a stronger entanglement enhancement than subtracting photons, while photon subtraction performs better in terms of energy efficiency. Furthermore, we analyze the interplay between entanglement and non-Gaussianity, showing that it is more subtle than previously expected.

Figure 1

Degaussification schemes analyzed in this work. Alice and Bob share a two-mode squeezed vacuum state, which results from applying a two-mode squeezing operation $\widehat{S}\left(r\right)$ on two vacuum modes ($A$ and $B$). They perform $k$ and $l$ photon additions (a) or subtractions (b) on their corresponding modes, generating the output state $|{\psi }_{\mathrm{add}}^{(k,l)}\rangle$ or $|{\psi }_{\mathrm{sub}}^{(k,l)}\rangle$, respectively.

Figure 2

Three equivalent degaussification schemes. Starting with the vacuum state, each of these three single-mode operations results in the same output state $|{\psi }^{\left(k\right)}\rangle$ up to a normalization factor (i.e., the probabilities of success are different): Alice's adding $k$ photons after the two-mode squeezer $\widehat{S}\left(r\right)$ [small solid-border (pink) box], Bob's subtracting $k$ photons after $\widehat{S}\left(r\right)$ [small dashed-border (green) box], or Alice's adding $k$ photons before $\widehat{S}\left(r\right)$ [small dashed-dotted border (yellow) box].

Figure 3

Entanglement entropy of the photon-added and -subtracted states, $|{\psi }_{\mathrm{add}}^{(k,l)}\rangle$ and $|{\psi }_{\mathrm{sub}}^{(k,l)}\rangle$, as a function of the number of operations $(k,l)$ for $\lambda =0.4$. Note that we have normalized it to the entanglement of the corresponding TMSV state, that is, $E_{\mathrm{add}}^{(0,0)}=E_{\mathrm{sub}}^{(0,0)}$ is set to 1. (a), (b) Density plots of $E_{\mathrm{add}}^{(k,l)}$ and $E_{\mathrm{sub}}^{(k,l)}$, respectively, in the $(k,l)$ space; darker regions correspond to lower entanglement. Thin lines are isolines (lines of equal entanglement), while thick straight lines correspond to $k=l$ (dashed), $k+l=10$ (dashed-dotted), and $l=4$ (dotted). The entanglement of the latter three lines is plotted in (c), (d), and (e), respectively, with dark (blue) and light (red) curves denoting the photon-added and photon-subtracted states, respectively. Even though $k$ and $l$ are physically discrete variables, we have taken them to be continuous variables by using the $\Gamma$ function as an analytic extension of the factorial function (this holds for the rest of the figures in the article).

Figure 4

Same as Fig. 3, but for $\lambda =0.22$. For conciseness, we only show $E_{\mathrm{sub}}^{(k,l)}$ as a density plot (a) as well as the entanglement along the lines $k+l=10$ (b; dashed-dotted line) and $l=4$ (c; dotted line), for both the photon-added [dark (blue) line] and the photon-subtracted [light (red) line] states. Note that while the photon-added states have the same behavior as for $\lambda =0.4$, this is not the case for the photon-subtracted states; see details in the text.

Figure 5

Entanglement energy-efficiency ${\eta }^{(k,l)}$ of the photon-added [dark (blue) curves] and photon-subtracted [light (red) curves] states as a function of the number of operations $k$ and $l$ for $\lambda =0.4$ (a)–(c) and $\lambda =0.22$ (d)–(f). As we did for the entanglement, we show the evolution of the efficiency along the $k=l$ (a), (d), $k+l=10$ (b), (e), and $l=4$ (c), (f) lines.

Figure 6

Non-Gaussianity ${\overline{G}}^{(k,l)}$ of the photon-added [dark (blue) curves] and photon-subtracted [light (red) curves] states as a function of the number of operations $k$ and $l$ for $\lambda =0.4$ (a)–(c) and $\lambda =0.22$ (d)–(f). As we did for the entanglement and the entanglement energy efficiency, we show the evolution of the non-Gaussianity along the $k=l$ (a), (d), $k+l=10$ (b), (e), and $l=4$ (c), (f) lines. Note that, in contrast to the entanglement and the entanglement energy efficiency, the non-Gaussianity has the same qualitative behavior for both $\lambda$ values.