Non-Gaussian swapping of entangled resources

Abstract

We investigate the continuous-variable entanglement swapping protocol in a non-Gaussian setting, with non-Gaussian states employed either as entangled inputs and/or as swapping resources. The quality of the swapping protocol is assessed in terms of the teleportation fidelity achievable when using the swapped states as shared entangled resources in a teleportation protocol. We thus introduce a two-step cascaded quantum communication scheme that includes a swapping protocol followed by a teleportation protocol. The swapping protocol is fed by a general class of tunable non-Gaussian states, the Squeezed Bell states, which, by means of controllable free parameters, allows to pass in a continuous way from Gaussian twin beams up to maximally non-Gaussian squeezed number states. In the realistic instance, taking into account the effects of losses and imperfections, we show that as the input two-mode squeezing increases, optimized non-Gaussian swapping resources allow for a monotonically increasing enhancement of the fidelity compared to the corresponding Gaussian setting. This result suggests that the use of non-Gaussian resources can be useful to guarantee the success of continuous-variable entanglement swapping in the presence of decoherence.

Appendices

A CV entanglement swapping protocol in the characteristic function representation

The characteristic function representation provides a most elegant and compact description of the CV teleportation protocol [82]. Such description proves to be particularly convenient in the instance of non-Gaussian resources [14,15,16,17] and has been generalized to include the non-ideal case [15]. In this appendix, we apply this formalism to the description of the realistic CV entanglement swapping protocol, schematically illustrated in Fig. 1.

Let $\rho _{12}^{A}$ and $\rho _{34}^{B}$ denote the density matrices associated with the two-mode entangled input pure state of modes 1 and 2, and the two-mode entangled pure resource of modes 3 and 4, respectively. The global four-mode initial state is the biseparable state $\rho _{0}= \rho _{12}^{A} \otimes \rho _{34}^{B}$ and the corresponding initial global characteristic function $\chi _{0}$ associated with $\rho _{0}$ reads:

$$\begin{aligned} \chi _{0}(\alpha _1;\alpha _2;\alpha _3;\alpha _4)= Tr\left[ \prod _{j=1}^{4} D_j (\alpha _j) \rho _{0}\right] = \chi _{12}(\alpha _1;\alpha _2) \; \chi _{34}(\alpha _3;\alpha _4) \,, \end{aligned}$$

(10)

where Tr denotes the trace operation, $D_j (\alpha _j)$ denotes the displacement operator of mode j $(j=1,\ldots ,4)$, $\chi _{12}$ is the characteristic function of the two-mode input state, and $\chi _{34}$ is the characteristic function of the two-mode resource. By introducing the quadrature operators $X_{j}=\frac{1}{\sqrt{2}}(a_{j}+a_{j}^{\dag })$ and $P_{j}=\frac{i}{\sqrt{2}}(a_{j}^{\dag }-a_{j})$, and the corresponding phase space variables $x_{j}= \frac{1}{\sqrt{2}}(\alpha _{j}+\alpha _{j}^{*})$ and $p_{j}=\frac{i}{\sqrt{2}}( \alpha _{j}^{*}-\alpha _{j})$, the characteristic function can be written in terms of $x_j$, $p_j$, i.e., $\chi _{0}(\alpha _1;\alpha _2;\alpha _3;\alpha _4)\equiv \chi _{0}(x_1,p_1;x_2,p_2;x_3,p_3;x_4,p_4)$.

The first step of the protocol consists of a Bell measurement at the first user’s location. The modes 2 and 3 are mixed at a balanced beam splitter; the effects of photon losses and the inefficiencies of the photodetectors are simulated by two additional fictitious beam splitters placed in front of the detectors, characterized by the transmissivities $T_{j}^2$ (reflectivity $R_{j}^2=1-T_{j}^2$), $j=2,3$. Let us denote by ${\tilde{x}}$ and ${\tilde{p}}$ the homodyne measurements of the first quadrature of the mode 3 and of the second quadrature of the mode 2, respectively. The description of realistic Bell measurements using the formalism of the characteristic function is discussed in full detail in Ref. [15]. Here we just provide the final expression of the characteristic function $\chi _{Bm}(x_1,p_1;x_4,p_4)$ associated with the entire measurement process:

$$\begin{aligned} \chi _{Bm}(x_1,p_1;x_4,p_4)= & {} \frac{{\mathcal {P}}^{-1}({\tilde{p}},{\tilde{x}}) }{(2\pi )^{2}} \int \hbox {d}\xi \hbox {d}\upsilon \, e^{i\xi {\tilde{p}} - i {\tilde{x}} \upsilon } \exp \left[ -\frac{R_2^2}{4} \xi ^2 -\frac{R_3^2}{4} \upsilon ^2 \right] \nonumber \\&\times \chi _{12} \left( x_1,p_1; \frac{T_2 \xi }{\sqrt{2}},\frac{T_3 \upsilon }{\sqrt{2}} \right) \chi _{34} \left( \frac{T_2 \xi }{\sqrt{2}},-\frac{T_3 \upsilon }{\sqrt{2}}; x_4,p_4\right) \,, \nonumber \\ \end{aligned}$$

(11)

where the function ${\mathcal {P}}({\tilde{p}},{\tilde{x}})$ is the distribution of the measurement outcomes ${\tilde{p}}$ and ${\tilde{x}}$, that is:

$$\begin{aligned} {\mathcal {P}}({\tilde{p}},{\tilde{x}})= & {} \frac{1 }{(2\pi )^{2}} \int \hbox {d}\xi \hbox {d}\upsilon \, e^{i\xi {\tilde{p}} - i {\tilde{x}} \upsilon } \exp \left[ -\frac{R_2^2}{4} \xi ^2 -\frac{R_3^2}{4} \upsilon ^2 \right] \nonumber \\&\times \chi _{12} \left( 0,0; \frac{T_2 \xi }{\sqrt{2}},\frac{T_3 \upsilon }{\sqrt{2}} \right) \chi _{34} \left( \frac{T_2 \xi }{\sqrt{2}},-\frac{T_3 \upsilon }{\sqrt{2}};0,0\right) . \end{aligned}$$

(12)

After measurement, modes 1 and 4 propagate in noisy channels (e.g., optical fibers) toward Alice’s and Bob’s locations, respectively. The dynamics of a multimode system subject to decoherence is described, in the interaction picture, by the following master equation for the density operator $\rho$ [91, 92]:

$$\begin{aligned} \partial _{t} \rho \,=\, \sum _{i=1,4} \frac{\Upsilon _i}{2} \left\{ n_{th,i} L[a_{i}^{\dag }] \rho + (n_{th,i}+1) L[a_{i}] \rho \right\} \,, \end{aligned}$$

(13)

where the Lindblad superoperators are defined as $L[{\mathcal {O}}] \rho \equiv 2 {\mathcal {O}} \rho \mathcal {O^{\dag }} - \mathcal {O^{\dag }} {\mathcal {O}} \rho - \rho \mathcal {O^{\dag }} {\mathcal {O}}$, $\Upsilon _i$ is the mode damping rate, and $n_{th,i}$ is the number of thermal photons in mode i. Because of decoherence due to propagation in the noisy channels, the characteristic function (11) can be rewritten in the following form:

$$\begin{aligned} \chi _{t}(x_{1},p_{1};x_{4},p_{4}) \,=\,&\chi _{Bm}(e^{-\frac{1}{2}\Upsilon _1 t}x_{1},e^{-\frac{1}{2}\Upsilon _1 t}p_{1};e^{-\frac{1}{2}\Upsilon _4 t}x_{4},e^{-\frac{1}{2}\Upsilon _4 t}p_{4}) \nonumber \\&\times \; e^{-\frac{1}{2} \sum _{i=1,4} (1-e^{-\Upsilon _i t})\left( \frac{1}{2}+n_{th,i}\right) (x_{i}^{2}+p_{i}^{2})}. \end{aligned}$$

(14)

The description of the technical features of the experimental apparatus, e.g., the efficiency of the photodetectors, and characteristics as the length of the channels (fibers), and the temperature of the environment, is complete once the quantities $T_{j}$ (equivalently $R_{j}$, $j=2,3$), $\Upsilon _i$, and $n_{th,i}$ ($i=1,4$) are specified and fixed at certain given values.

In the last step of the protocol, two local unitary displacements $\lambda _1$ and $\lambda _4$ are performed at Alice’s and Bob’s locations; a local unitary displacement $\lambda _1 = - g_1 ( {\tilde{x}} - i{\tilde{p}})$ is performed on mode 1, and a local unitary displacement $\lambda _4 = g_4 ( {\tilde{x}} + i{\tilde{p}})$ is performed on mode 4. The real parameters $g_1$ and $g_4$ denote the gain factors of the displacement transformations [93]. After such local unitary operations, the characteristic function reads:

$$\begin{aligned} \chi _D(x_1,p_1;x_4,p_4) = e^{-i\sqrt{2}{\tilde{x}}(g_1 p_1 - g_4 p_4)-i\sqrt{2}{\tilde{p}}(g_1 x_1 + g_4 x_4)} \; \chi _{t}(x_1,p_1;x_4,p_4) . \end{aligned}$$

(15)

Finally, in order to obtain the output characteristic function $\chi _{out}(x_1,p_1;x_4,p_4)$, describing the output two-mode entangled state of the entanglement swapping protocol, one must take the average over all the possible outcomes ${\tilde{p}}$ and ${\tilde{x}}$ of the Bell measurements:

$$\begin{aligned} \chi _\mathrm{out}^{(\mathrm swapp)}(x_1,p_1;x_4,p_4) = \int \hbox {d}{\tilde{x}} \hbox {d}{\tilde{p}} {\mathcal {P}}({\tilde{p}},{\tilde{x}})\chi _D(x_1,p_1;x_4,p_4), \end{aligned}$$

(16)

where $\tau _i = \Upsilon _i t$. The above integral yields the final expression (3) for the characteristic function associated with the swapped resource.

The core mathematical task is thus the explicit evaluation of the characteristic function $\chi _\mathrm{out}^{(\mathrm swapp)}(x_1,p_1;x_4,p_4)$ associated with the output of the realistic swapping protocol, as expressed by Eq. (3). We have determined its analytical expression for the most general non-Gaussian setting, that is the entanglement swapping of SB input states using SB states as resources. As the class of SB states contains as special cases both the Gaussian TB states and the non-Gaussian PS states, the general expression of the output characteristic function reduces to the explicit expression for these special Gaussian and non-Gaussian cases as well. We do not report here the explicit analytical expression of $\chi _\mathrm{out}^{(\mathrm swapp)}(x_1,p_1;x_4,p_4)$, as it is exceedingly long and cumbersome and does not yield any particularly useful physical insight. On the other hand, having obtained the explicit expression of the output characteristic function of the swapping protocol, it is straightforward to compute the output characteristic function $\chi _\mathrm{out}^{(\mathrm telep)}(x_{4},p_{4})$ of the subsequent ideal teleportation protocol, Eq. (5).

Finally, we have derived the analytical expression for the teleportation fidelity ${\mathcal {F}}_{X^{sw}Y}$, Eq. (6), which in the most general instance is ${\mathcal {F}}_{\mathrm{SB}^{sw}\mathrm{SB}}$. Such a fidelity depends on the following parameters: the squeezing amplitudes and phases $r_{12}$, $\phi _{12}$, $r_{34}$, $\phi _{34}$ and the angles and phases $\delta _{12}$, $\theta _{12}$, $\delta _{34}$, $\theta _{34}$ of the input states and of the resources; the parameters associated with the experimental apparatus are listed in Table 2. Without loss of generality, as already verified in Refs. [14, 15], one can obtain some significant simplifications by fixing the phases of the SB states. Specifically, we set the non-Gaussian phases $\theta _{12}=\theta _{34}=0$ and the squeezing phases $\phi _{12}=\phi _{34}=\pi$ in Eq. (1). With such choice, the teleportation fidelity depends on the two gains $g_i$ $(i=1,4)$ through the total gain parameter ${\tilde{g}}=g_1+g_4$, both in the ideal and in the realistic protocols.

B Performance criteria for the swapping protocol

In the paper, to check efficiency a specific criterion have been exploited, based on a cascaded scheme, i.e., a swapping protocol followed by a teleportation protocol. We mentioned in the introduction, in fact, that, unlike the Gaussian case where only the entanglement content is relevant [94], when non-Gaussian resources are exploited the intrinsic nonlinearity of interaction leads to different results depending on the specific quantity chosen to be optimized.

We now show this aspect by optimizing three different objects:

• (a) the squeezing content of the output swapped state,

• (b) the fidelity between the input and the output two-mode states,

• (c) the fidelity of teleportation of a coherent state, following the cascade scheme adopted in this paper.

For a better understanding, here we resort to a simplified experimental setup: we consider an ideal swapping protocol, with a Gaussian twin beam as the input two-mode entangled state, and a Bell-like state as the two-mode entangled resource, which are described, respectively, by the following characteristic functions:

$$\begin{aligned}&\chi _{12}^{(TwB)}(x_1,p_1;x_2,p_2)= e^{-\frac{1}{4}\cosh 2r_{12}(x_1^2+p_1^2+x_2^2+p_2^2)+\frac{1}{2}\sinh 2r_{12}(x_1 x_2-p_1 p_2)} \,, \end{aligned}$$

(17)

$$\begin{aligned}&\chi _{34}^{(Bell)}(x_3,p_3;x_4,p_4)= e^{-\frac{1}{4}(x_3^2+p_3^2+x_4^2+p_4^2)} \Big \{\cos ^2 \delta _{34} +(x_3 x_4 -p_3 p_4) \sin \delta _{34} \cos \delta _{34} \nonumber \\&\quad +\left( \frac{x_3^2+p_3^2}{2}-1\right) \left( \frac{x_4^2+p_4^2}{2}-1\right) \sin ^2 \delta _{34} \Big \} \,. \end{aligned}$$

(18)

With the above simplified choice of the resource, the non-Gaussian character is preserved together with the presence of parameters (angles) that can be optimized, but it is possible to explicitly compute all the quantities one is interested in.

According to Eq. (4) for the swapping output, in this case we have a swapped state described by the characteristic function:

$$\begin{aligned} \chi _{14}^{(sw)}(x_1,p_1;x_4,p_4)= & {} e^{-\frac{1}{4}\cosh 2r_{12}(x_1^2+p_1^2+x_4^2+p_4^2)-\frac{1}{2}(x_4^2+p_4^2) +\frac{1}{2}\sinh 2r_{12}(x_1x_4-p_1p_4)} \nonumber \\&\times \left\{ 1+ (\cos 2\delta _{34}-1) \left( \frac{x_4^2+p_4^2}{2}-\frac{x_4^4+p_4^4}{8}- \frac{x_4^2 p_4^2}{4} \right) \right. \nonumber \\&\quad \left. +\sin 2\delta _{34}\frac{x_4^2- p_4^2}{2} \right\} . \end{aligned}$$

(19)

Now we perform the optimization procedure in the three different cases.

• (a) Optimization of the squeezing content of the output swapped state. Given the generalized two-mode quadratures $X_{hk}=\frac{1}{\sqrt{2}}(a_h+a_k +h.c.)$ and $P_{hk}=\frac{i}{\sqrt{2}}(a_h^{\dag }+a_k^{\dag } - h.c.)$, it is straightforward to compute the variances $\Delta X_{hk}^2$ and $\Delta P_{hk}^2$ by exploiting the above analytical expressions for characteristic functions. The squeezed quadrature for the input twin beam Eq. (17) is $P_{12}$, i.e., $\Delta P_{12}^2 = e^{-2r_{12}}$, and the optimization procedure implies minimization over the non-Gaussian parameter $\delta _{34}$ of the difference between the second quadrature variances associated with the swapped state and the input state:

  $$\begin{aligned} \min _{\delta _{34}} \left\{ (\Delta P_{14}^2)^{(sw)}-(\Delta P_{12}^2)^{(in)} \right\} \,. \end{aligned}$$

  (20)

  Minimization leads to the optimal value (independent of $r_{12}$) for the angle $\delta _{34}^{(\mathrm opt)}$ that is reported in the first row of Table 3.

• (b) Optimization of the fidelity between the input and the output two-mode states. In this case we must compute:

  $$\begin{aligned} \max _{\delta _{34}} {\mathcal {F}}(r_{12},\delta _{34}), \end{aligned}$$

  (21)

  with

  $$\begin{aligned}&{\mathcal {F}}(r_{12},\delta _{34}) \end{aligned}$$

  (22)
  $$\begin{aligned}&= \frac{1}{(2\pi )^2} \int \hbox {d}x_1 \hbox {d}p_1 \hbox {d}x_4 \hbox {d}p_4 \chi _{12}^{(\mathrm TB)}(x_1,p_1;x_4,p_4) \chi _{14}^{(sw)}(-x_1,-p_1;-x_4,-p_4). \end{aligned}$$

  (23)

  Maximization leads to the optimal value for the angle $\delta _{34}^{(\mathrm opt)}$ that is reported in the second row of Table 3.

• (c) Optimization of the fidelity of teleportation of a coherent state, following the cascade scheme adopted in this paper.

The cascaded scheme here described provides in this simplified case the optimal angle $\delta _{34}^{(\mathrm opt)}$ reported in the third row of Table 3.

Thus, the optimized angles do not coincide, identifying different optimized resources for the three different cases.

Table 3 Optimized angles