Realistic continuous-variable quantum teleportation using a displaced Fock state channel

Abstract

We investigate ideal and non-ideal continuous-variable quantum teleportation protocols realized by using an entangled displaced Fock state resource. The characteristic function formulation is applied to measure the relative performance of displaced Fock state for teleporting squeezed and coherent states. It is found that for such single-mode input fields, the average fidelity remains at the classical threshold, suggesting that the displaced Fock states are not advantageous for teleportation. We also discuss the major decoherence effects, caused by the inaccuracy in Bell measurements and photon losses for the propagation of optical fields via fibre channels. The changes in the teleportation fidelity are described by adjusting the gain factor (g), reflectivity (R), mode damping ($\tau$), and the number of thermal photons ($n_\mathrm {th}$). The possibility of successful teleportation can be optimized by fixing these realistic parameters.

Appendix A. Details of Gaussian integration

Here, we outline the description of the Gaussian integration in detail.

$$\begin{aligned} I= & {} \int dx\,dy\,\exp (a_1x^2+a_2y^2+a_3xy+a_4x+a_5y)L_m[a(x^2+y^2)]\nonumber \\&\quad \times L_n[b(x^2+y^2)]\nonumber \\= & {} \sum _{p,q=0}^{m,n} \frac{(-a)^p(-b)^q m!n!}{p!^2q!^2(m-p)!(n-q)!}\nonumber \\&\times \int {dx\,dy\,\exp (a_1x^2+a_2y^2+a_3xy+a_4x+a_5y)(x^2+y^2)^{p+q}}\nonumber \\= & {} \sum _{p,q,r=0}^{m,n,p+q} \frac{(-a)^p(-b)^q m!n!(p+q)!}{p!^2q!^2(m-p)!(n-q)! r!(p+q-r)!}\nonumber \\&\times \int {dx\,dy\,\exp (a_1x^2+a_2y^2+a_3xy+a_4x+a_5y)x^{2r}y^{2p+2q-2r}} \end{aligned}$$

(A.1)

Now, rearranging the parameters as $\mu _x=\frac{2a_2a_4-a_3a_5}{4a_1a_2-a_3^2},~~\mu _y=\frac{2a_1a_5-a_3a_4}{4a_1a_2-a_3^2},\rho =-\frac{a_3}{2\sqrt{a_1a_2}},~~\sigma _x=\sqrt{-\frac{2a_2}{4a_1a_2-a_3^2}}, ~~\sigma _y=\sqrt{-\frac{2a_1}{4a_1a_2-a_3^2}},\,\,a_6=\frac{a_2a_4^2+a_1a_5^2-a_3a_4a_5}{4a_1a_2-a_3^2}$ and substituting $x=\sigma _xz_1+\mu _x$ and $y=\sigma _y(\rho z_1+\sqrt{1-\rho ^2}z_2) +\mu _y$, ($z_1$ and $z_2$ are standard normal variate) in (A.1), we obtain [41]

$$\begin{aligned} I e^{a_6}= & {} \sum _{p,q,r=0}^{m,n,p+q} \frac{(-a)^p(-b)^q m!n!(p+q)!}{p!^2q!^2(m-p)!(n-q)!r!(p+q-r)!}\\&\times \int {dx\,dy} \exp \bigg [-\frac{1}{2(1-\rho ^2)}\bigg \{\bigg (\frac{x-\mu _x}{\sigma _x}\bigg )^2+\bigg (\frac{y-\mu _y}{\sigma _y}\bigg )^2\\&- \frac{2\rho }{\sigma _x\sigma _y}\bigg (\frac{x-\mu _x}{\sigma _x}\bigg ) \bigg (\frac{y-\mu _y}{\sigma _y}\bigg )\bigg \}\bigg ]x^{2r}y^{2p+2q-2r}\\= & {} \sum _{p,q,r=0}^{m,n,p+q} \frac{(-a)^p(-b)^qm!n!(p+q)!}{p!^2q!^2(m-p)!(n-q)!r!(p+q-r)!}\sqrt{1-\rho ^2}\sigma _x\sigma _y\\&\times \sum _{s,t,u=0}^{2r,2p+2q-2r,t}{2r\atopwithdelims ()s}{2p+2q-2r\atopwithdelims ()t}{t\atopwithdelims ()u}\mu _x^{2r-s}\mu _y^{2p+2q-2r-t}\\&\times \sigma _x^{s}\sigma _y^{t}\rho ^{t-u}(1-\rho ^2)^{u/2}\int e^{-\frac{1}{2}(z_1^2+z_2^2)}z_1^{s+t-u}z_2^{u}dz_1dz_2\\= & {} \left\{ \begin{array}{lcl} \sum _{p,q,r=0}^{m,n,p+q} \frac{(-a)^p(-b)^qm!n!{{p+q}\atopwithdelims ()r}}{p!^2q!^2(m-p)!(n-q)!}\sqrt{1-\rho ^2}\sigma _x\sigma _y \sum _{s,t,u=0}^{2r,2p+2q-2r,t}{2r\atopwithdelims ()s}{2p+2q-2r\atopwithdelims ()t}\\ \times {t\atopwithdelims ()u}\mu _x^{2r-s}\mu _y^{2p+2q-2r-t}\sigma _x^{s} \sigma _y^{t}\rho ^{t-u}(1-\rho ^2)^{u/2}2^{\frac{s+t}{2}+1}\, \Gamma \left( \frac{s+t-u+1}{2}\right) \Gamma \left( \frac{u+1}{2}\right) ,\\ \text {if both } s+t\text { and } u\text { are even}\\ \\ 0,\,\,\,\,\, \text {otherwise} \end{array} \right. \end{aligned}$$

by using [42]

$$\begin{aligned} \int {dx\,e^{-ax^b}x^n} = \left\{ \begin{array}{lll} \frac{2\,\Gamma [(n+1)/b]}{b\,a^\frac{n+1}{b}},\,\,\,\text {if } n\text { is even}\\ \\ 0, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text {otherwise} \end{array} \right. \end{aligned}$$

(A.2)