1. Introduction

Quantum teleportation [1–3] is arguably one of the most important elementary protocols in quantum communication. In particular, the entanglement swapping [4,5] is a key component of quantum repeaters [6–8]. In optics, deterministic quantum teleportation of qubits encoded into states of single photons [9] is hindered by the impossibility to perform complete Bell measurement with linear optics [10,11]. Using ancilla photons, the probability of successful teleportation can be made arbitrarily close to $1$, but at the expense of a greatly increased setup complexity. An interesting alternative is offered by the continuous variable teleportation [12–14], where the quantum channel is formed by a two-mode squeezed vacuum state and the Bell measurement is implemented with a balanced beam splitter and two homodyne detectors. This protocol enables deterministic teleportation, as both the Bell measurement and the required corrective coherent displacement operations on the receiver’s side are experimentally feasible. However, the quantum state transfer is imperfect, because perfect teleportation would require a non-physical infinitely squeezed state.

The continuous-variable teleportation can be subject to various modifications to optimize its performance. The ordinary unity gain teleportation faithfully transfers mean values of the field quadratures but adds thermal noise to the teleported state. With a suitable choice of the gain one can get rid of the added noise and the teleportation becomes equivalent to a purely lossy channel [15,16]. Especially challenging is the teleportation of highly non-classical non-Gaussian states such as Fock states. Noiseless continuous-variable teleportation of Fock states becomes possible at the cost of making the teleportation protocol probabilistic, accepting only the cases when the outcomes of homodyne detections in the Bell measurement are sufficiently close to zero [17].

Very recently, a different probabilistic modification of the ordinary continuous-variable quantum teleportation protocol has been proposed and experimentally demonstrated [18]. This modified teleportation protocol utilizes a measurement-based, or emulated, noiseless amplification. Ideal noiseless amplification is an (unphysical) operation described by operator $G^{\hat {n}}$ with $G>1$, where $\hat {n}$ is the photon number operator [19]. If applied to a coherent state $|\alpha \rangle$, the noiseless amplification increases the state amplitude, $G^{\hat {n}}|\alpha \rangle =e^{(G^2-1)|\alpha |^2/2} |G\alpha \rangle$. When applied to a squeezed state, noiseless amplification can increase its squeezing. More generally, noiseless amplification can be useful for suppressing losses in quantum communication [20,21]. In physical implementations, the noiseless amplification is always realized only approximately, and only the amplitudes of several lowest Fock states are modulated as desired, since the transformation must always be described by a bounded operator. Ancilla single-photon states [22–25] or conditional photon addition and subtraction [26–29] are required to physically implement the approximate noiseless amplification, which makes it experimentally challenging. Approximate physical noiseless quantum amplification is a non-Gaussian operation that, when operated outside the regime of high-fidelity amplification, can generate highly non-classical and non-Gaussian states [30].

In scenarios, where the noiseless amplification is directly followed by heterodyne or homodyne detection, the noiseless amplification can instead be emulated by suitable data processing [31–33], which can greatly simplify the experiment. The data processing involves a filtering with an inverse Gaussian filter, which is unbounded, hence a cut-off needs to be introduced. This reflects the unphysicality of the ideal noiseless amplifier $G^{\hat {n}}$ that cannot be perfectly emulated with nonzero success probability. The emulated, or measurement-based, noiseless amplification has been successfully tested experimentally and utilized for enhanced quantum cloning of coherent states [34,35].

In Ref. [18], the emulation of noiseless amplification is applied to the outcomes of continuous-variable Bell measurement. The authors show that this procedure can improve the fidelity of coherent state teleportation, at the cost of making the teleportation probabilistic. The authors also observe and analyze other effects such as purification of teleported displaced thermal states, where the teleported state can exhibit smaller thermal noise than the input state while maintaining the unity gain teleportation. In the present work, we perform a detailed theoretical analysis of the teleportation protocol proposed and demonstrated in Ref. [18]. Our aim is to provide additional insights into this protocol and clarify its benefits and limitations. Our study is focused on teleportation of Gaussian states, where a simple and unifying picture can be established.

The rest of the paper is organized as follows. In Section 2 we briefly recapitulate the ordinary continuous-variable teleportation and then analyze the improvement of teleportation of coherent states via emulation of noiseless amplification. In Section 3 we generalize the findings from Section 2 to arbitrary Gaussian states. As additional examples, we consider in Section 4 teleportation of displaced thermal states and displaced squeezed states. Finally, Section 5 contains a short discussion and conclusions.

2. Emulation of noiseless amplification and teleportation of coherent states

Consider teleportation of coherent states $|\alpha \rangle$ with the Braunstein-Kimble teleportation scheme [13,14], as illustrated in Fig. 1. Let us assume that the quantum channel is formed by pure two-mode squeezed vacuum state with squeezing constant $r$, $|\Psi _{\mathrm {EPR}}\rangle =\sqrt {1-\lambda ^2}\sum _{n=0}^\infty \lambda ^n |n,n\rangle _{AB}$, where $\lambda =\tanh r$ and $|n\rangle$ denote the Fock states. The mode C that contains the input coherent state $|\alpha \rangle$ is combined with mode A on a balanced beam splitter BS and the output modes are measured with homodyne detectors BHD$_1$ and BHD$_2$. After suitable scaling, the measured quadratures read

The mode quadrature operators satisfy the canonical commutation relations $[\hat {x}_j,\hat {p}_k]=i\delta _{jk}$, where $j,k\in \{A,B,C\}$. For each particular measurement outcome, Bob’s mode is prepared in a pure coherent state $|\lambda \alpha -\lambda \beta \rangle$ whose complex amplitude depends on the Bell measurement outcome,

Here, the symbols without hats denote the measured values of the quadrature operators. In standard unity-gain teleportation, the amplitude of Bob’s mode is coherently displaced by $\beta$, which at the level of quadrature operators corresponds to the transformation

The complex amplitude of the output state is equal to $\alpha$, but the state becomes mixed. The unity-gain teleportation adds thermal noise to the teleported state and the mean number of added thermal photons is given by $\bar {n}=e^{-2r}$. Note that the state of Bob remains correlated with the measurement outcome on Alice’s side. Specifically, if Alice records measurement outcome $\beta$ and Bob applies the unity-gain corrective displacement $\hat {D}(\beta )$, then Bob’s mode is prepared in coherent state $|\lambda \alpha +(1-\lambda )\beta \rangle$. Upon averaging over all measurement outcomes $\beta$, an effective displaced thermal state is obtained in mode B. The probability distribution $P(\beta )$ of Alice’s measurement outcomes $\beta$ can be calculated as the convolution of the Wigner function of the input state with certain Gaussian function [13]. For the input coherent state $|\alpha \rangle$ the probability distribution $P(\beta |\alpha )$ is Gaussian and its mean and covariance matrix can be directly determined from Eq. (2). The distribution is centered on $\alpha$, $\langle \beta \rangle =\alpha$. The real and imaginary parts of $\beta$ exhibit the same variance, $\frac {1}{2}\langle (\Delta \hat {x}_{-})^2\rangle =\frac {1}{2}\langle (\Delta \hat {p}_{+})^2\rangle =\frac {1}{2}\cosh ^2 r$, and they are not correlated, $\langle \Delta \hat {x}_{-}\Delta \hat {p}_{+}\rangle =0$. The distribution $P(\beta |\alpha )$ thus reads

 

Fig. 1. Continuous-variable quantum teleportation of optical mode C [13,14]. Alice and Bob share entangled two-mode state $|\Psi _{\mathrm {EPR}}\rangle$ in modes A and B. Mode A interferes with mode C on a balanced beam splitter BS. Two homodyne detectors BHD measure the amplitude quadrature $\hat {x}_{-}$ of the first output mode and the phase quadrature $\hat {p}_{+}$ of the second output mode. Measurement results are communicated to Bob, who applies a corrective coherent displacement operation $\hat {D}(g\beta )$ with gain $g$. Performance of the teleportation protocol can be conditionally improved by applying a specific filter $Z(\beta )$ to the measurement outcomes [18], together with a suitable choice of the displacement gain.

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Instead of the unity gain, one can choose gain $\lambda$, apply corrective displacement $\hat {D}(\lambda \beta )$ and recover a pure coherent state $|\lambda \alpha \rangle$ that is not correlated with the measurement outcomes of Alice. The teleportation then becomes equivalent to a purely lossy channel [15,16]. At the level of quadrature operators, this corresponds to transformation

It is straightforward to verify that for coherent state in the input mode C and two-mode squeezed vacuum in modes A and B the variances of the output quadratures (5) remain equal to coherent state variances. For finite squeezing, the homodyne measurements on Alice’s side reveal some information about the teleported coherent state. In particular, the measured quadratures allow for estimation of the mean values of the quadratures of teleported mode C with variance $\cosh ^2 r$, c.f. the probability distribution (4). Interestingly, the same signal-to-noise ratio can be obtained by performing eight-port homodyne detection on an attenuated coherent state with amplitude $\sqrt {1-\lambda ^2}\alpha$, since $1-\lambda ^2=1/\cosh ^2 r$. Therefore, as shown in Ref. [15], the coherent state teleportation via two-mode squeezed vacuum with finite squeezing is equivalent to splitting the teleported coherent state at a fictitious beam splitter with amplitude transmittance $\lambda$. The transmitted mode represents the output of teleportation, and the reflected mode is measured with eight-port homodyne detector. The measured data can then be used for feedforward displacement, which reduces the attenuation in teleportation and adds thermal noise, going from a purely lossy channel with intensity transmittance $\lambda ^2$ to a channel with unit transmittance and mean number of added thermal photons $\bar {n}=e^{-2r}$.

After these preliminary recapitulations we are now in a position to analyze the improvement of teleportation of coherent states by measurement-based, or emulated, noiseless amplification. The protocol proposed and experimentally tested in Ref. [18] is based on probabilistic filtering in dependence on the measurement outcomes $\beta$. An inverse Gaussian filter with cut-off $\beta _C$ is applied, which results in acceptance probability

Here $\kappa ^2$ determines the strength of the noiseless amplification. The authors of Ref. [18] found that the filtering (6) can enhance the teleportation fidelity at the cost of making the protocol probabilistic. In Ref. [18], the coefficient $\kappa ^2$ in the filter function $Z(\beta )$ was parameterized as

Taking into account the picture of teleportation reviewed above, we can see that the actual effect of the filter (6) is to probabilistically improve the estimation of the coherent state amplitude $\alpha$ from the measured data. For the sake of simplicity let us neglect the cut-off in the filter (6). If we apply the filtering, the probability (4) changes to $P_F(\beta |\alpha ) \propto Z(\beta ) P(\beta |\alpha )$. After normalization, we get

The filtering increases both the mean value of $\beta$ and its variance by factor of $(1-\kappa ^2\cosh ^2 r )^{-1}$. This means that the signal-to-noise ratio is increased and the complex amplitude $\alpha$ can be estimated from the filtered data with better precision. To quantitatively analyze this effect, we take as the starting point the coherent displacement $\lambda \beta$ which leads to a complete decorrelation between Bob’s state $|\lambda \alpha \rangle$ and Alice’s measurement outcomes $\beta$. This means that for this particular displacement gain any filtering is useless and cannot improve the output state. However, if we aim at unity gain teleportation, the filtering can reduce the added thermal noise. To achieve unity gain teleportation, the attenuated coherent state $|\lambda \alpha \rangle$ needs to be additionally displaced by $(1-\lambda )[1-\kappa ^2\cosh ^2 r]\beta$. The total required displacement gain thus reads

Mean photon number of the resulting added thermal noise can be calculated as follows,

In the limit $\kappa ^2 =0$ we recover the ordinary unity gain teleportation, while in the limit $\kappa ^2 \rightarrow 1/\cosh ^2 r$ the teleportation becomes perfect and the thermal noise vanishes. Note, however, that in this limit both the mean value and the variance of filtered $\beta$ become infinite. In practice, a finite cut-off $\beta _C$ needs to be imposed to preserve finite probability of filtering, c.f. Eq. (6), which limits the applicability of the procedure to a subset of coherent states with sufficiently small amplitude. For coherent states with amplitude well above the cut-off the filtering will have no effect.

The fidelity of unity-gain teleportation of coherent states can be expressed as a function of the added thermal noise $\bar {n}$,

On inserting the formula (10) into the expression for fidelity, we obtain

This is similar to the formula for fidelity provided as Eq. (27) of the Supplementary Information of Ref. [18], $F=1/(1+e^{-2r}/G^2)$. Note, however, that according to our analysis we get $F\rightarrow 1$ already for $\kappa ^2=1/\cosh ^2 r$, i.e. for $G^2=1/(1-\kappa ^2)=1/\tanh ^2 r$. This is consistent with the fact that noiseless amplification with gain $1/\tanh r$ applied to one mode of two-mode squeezed vacuum state converts it to infinitely squeezed EPR state. The fidelity $F$ can approach unity even in the absence of entanglement [18]. In the limit $r=0$ the scheme acts as a probabilistic measure-and-prepare amplifier, which was discussed in Ref. [33]. A possible source of the discrepancy between the fidelity expressions obtained here and in Ref. [18] is that in the analysis presented in the Supplementary Information of Ref. [18] it is assumed that the measurement outcomes $\beta$ follow a distribution $\frac {1}{\pi }\exp (-|\beta -\alpha |^2)$. However, the correct distribution of $\beta$ is given by Eq. (4) of the present paper and its variance depends on the two-mode squeezing constant $r$.

The success probability of teleportation with applied measurement-based amplification can be expressed as

For states with sufficiently small amplitude, $|\alpha | \ll (1-\kappa ^2\cosh ^2r)|\beta _c|^2$, the cut-off can be neglected and the integral can be well approximated by a Gaussian integral. This yields

The success probablity is lowest for vacuum and increases exponentially with $|\alpha |^2$, which is a characteristic feature of noiseless amplification.

3. General Gaussian states

The results obtained in the previous section for coherent states are largely valid for arbitrary Gaussian states. Consider teleportation of a class of Gaussian states with a fixed covariance matrix $\boldsymbol{\gamma }_C$ and variable coherent displacement $\vec {d}_C=(\langle \hat {x}_C\rangle,\langle \hat {p}_C\rangle )^T$ . Recall that for an arbitrary multimode state its covariance matrix is defined as $\gamma _{jk}=\langle \hat {r}_j\hat {r}_k+\hat {r}_k\hat {r}_j\rangle -2 d_j d_k$, where $\hat {r}_j$ denote the quadrature operators and $d_j=\langle \hat {r}_j\rangle$. Since the variance of vacuum quadrature operators is equal to $\frac {1}{2}$, the covariance matrix of vacuum is equal to the identity matrix, $\gamma _{\mathrm {vac}}=I$. Let us assume that the quantum channel is formed by an entangled Gaussian state with covariance matrix $\boldsymbol{\gamma }_{AB}$ and vanishing displacement. The interference on a balanced beam splitter BS and the homodyne detections are Gaussian unitary operations and measurements, respectively. It then follows from the general formalism of Gaussian quantum operations [36,37] that the covariance matrix of the teleported state does not depend on the outcomes of homodyne detections, while the coherent displacement of the state is linearly proportional to the measured quadrature values and also linearly proportional to the coherent displacement of the input teleported state. Let

Here

The results presented in this section can be straightforwardly extended also to teleportation of one part of two-mode entangled Gaussian state, i.e. to Gaussian entanglement swapping. The entanglement of Gaussian states is fully determined by their covariance matrix and does not depend on the coherent displacement. Therefore, the amount of Gaussian entanglement that can be teleported is fundamentally limited by the shared Gaussian entangled state that serves as the quantum channel in quantum teleportation, and cannot be improved by filter $Z(\beta )$ applied to Alice’s measurement. Indeed, for each measurement outcome of Alice $\vec {d}_M$, the covariance matrix of the state after entanglement swapping will be determined by a formula similar to Eq. (16) and will not depend on the measurement outcome $\vec {d}_M$, only the coherent displacement will depend on $\vec {d}_M$. This displacement can be compensated to deterministically obtain a state with the optimal covariance matrix.

In particular, pure entangled Gaussian state can be again teleported to a pure entangled state if we use pure two-mode squeezed vacuum for teleportation, as is well known from the analysis of CV entanglement swapping [16]. The resulting teleportation will not be a unity gain teleportation, as far as the coherent displacement is concerned. Application of the filter $Z(\beta )$ to Alice’s measurements can only help to achieve unity gain teleportation with respect to the coherent displacement with reduced added thermal noise. However, any such attempt to achieve unity gain teleportation will add extra noise to the state, similarly to the teleportation of the single-mode coherent states.

Going beyond Gaussian states, in Ref. [38] an effective quantum channel was derived for the teleportation combined with conditioning on Alice’s measurement outcomes with the filter function $Z(\beta )$. The resulting probabilistic operation is a combination of the noiseless amplification $G_{\mathrm {eff}}^{\hat {n}}$, a lossy channel or a linear amplification channel and a channel that adds thermal noise. The first operation is probabilistic while the other two channels are deterministic. Expressions for parameters of this effective channel were derived in Ref. [38]. The strength of this representation lies in its ability to handle arbitrary input state. On the other hand, the physical picure that we have established here for Gaussian states and that clearly reveals what the filtering with function $Z(\beta )$ can achieve, is not so easily extractable from the effective channel representation.

For unity gain teleportation of coherent states as discussed in Section 2, the parameters of the effective channel can be relatively easily obtained. Since

Since we consider unity gain teleportation, the noiseless amplification with gain (20) is followed by a lossy channel with amplitude transmittance $1/G_{\mathrm {eff}}$ and this is further followed by addition of a thermal noise with mean number of thermal photons $\bar {n}_F$ given by Eq. (10).

4. Displaced thermal states and displaced squeezed states

In this section we analyze the effect of measurement-based noiseless amplification on the teleportation of displaced thermal states with mean number of thermal photons $\bar {n}$ and variable coherent amplitude $\alpha$ [18]. Similarly as for the coherent states, we assume that pure two-mode squeezed vacuum serves as the quantum channel in teleportation. We could use the formulas provided in the previous section to identify the displacement gain $g$ for which the teleported state becomes decorrelated from the outcomes of homodyne measurements. Instead, we choose an alternative approach that provides more physical insight into the value of the optimal gain $g$. Namely, this gain can be found as the gain that minimizes the variance of the quadrature $gx_C-gx_A+x_B$ (we could equivalently take also the phase quadratures). A straightforward calculation yields

The teleported state is a displaced thermal state with reduced mean number of thermal photons

Teleportation of thermal states results in broader distribution of $\beta$ as compared with the teleportation of coherent states,

If we emulate noiseless amplification and apply the inverse Gaussian filter (6) to the measured complex amplitudes $\beta$, we obtain Gaussain distribution with rescaled mean value and variance

To achieve unity gain teleportation, additional displacement by $(1-g_{\mathrm {th}})[1-\kappa ^2(\cosh ^2r+\bar {n})]\beta$ is required, which adds some thermal noise. The total mean number of thermal photons in the teleported state can be expressed as

After some algebra, we get

The teleported state becomes purified if $\bar {m}<\bar {n}$ which is equivalent to

Note that $\kappa ^2 <1/(\cosh ^2 r+\bar {n})$ must hold to ensure that after the emulation of noiseless amplification the distribution of $\beta$ remains Gaussian with positive variance. For the sake of completeness we also provide an explicit formula for the total gain required to achieve unity gain teleportation of displaced thermal states with measurement-based noiseless amplification,

In the analysis of purification of displaced thermal states in Ref. [18] a specific situation is considered in which, according to Ref. [18], the protocol can be treated as a concatenation of a lossy channel and a noiseless linear amplifier. The expression for the output complex ampitude provided in Ref. [18] predicts that this amplitude vanishes when the noiseless gain $G$ approaches zero, i.e. when $\kappa ^2\rightarrow -\infty$. However, in this limit the filter function $Z(\beta )\propto [|\beta |^2(1-G^{-2})]$ approaches the Dirac delta distribution which means that one postselects the single outcome $\beta =0$ [17]. This means that Alice projects her two modes A and C on the infinitely squeezed EPR state $\sum _{n=0}^\infty |nn\rangle$. The teleported state becomes a noiselessly attenuated state, $\hat {\rho }_{\mathrm {tele}}\propto \lambda ^{\hat {n}}\hat {\rho }_{\mathrm {in}}\lambda ^{\hat {n}}$, where $\lambda =\tanh r$. For input displaced thermal state with amplitude $\alpha$, the amplitude of the teleported state in this limit is given by $g_\mathrm {th}\alpha$, and does not vanish. Note that the minimum-noise state that we deterministically obtain by displacement with gain $g_{\mathrm {th}}$ is equal to the state obtained for $\beta =0$, which is precisely the noiselessly attenuated state. The thermal state purification arises from the combination of the effect of noiseless attenuation, which reduces the mean number of thermal photons to $\bar {n}_T$, and the improved precision of estimation of the coherent displacement thanks to filtering, which reduces the thermal noise added by the additional displacement required for unity gain operation.

We can use the above approach to investigate also the teleportation of pure squeezed coherent states [18] with squeezing constant $s$. Let

Observe that the optimal displacement gain satisfies $g_{\pm }< 1$ irrespective of $V_{\pm }$ for any finite two-mode squeezing $r$. Quadrature variances of the teleported state are given by

If we perform the measurement-based noiseless amplification and apply inverse Gaussian filter (6) to the measured homodyne data, the following additional displacement gains are needed to achieve unity gain teleportation

The total displacement gains read $g_{\mathrm {tot},\pm }=g_{\pm }+g_{\mathrm {ad},\pm }$ and after some algebra we get

The resulting variances of the amplitude and phase quadratures of the conditionally teleported state read

The above formulas remain valid also for mixed Gaussian states with $V_+V_{-}>\frac {1}{4}$. In particular, if we set $V_{+}=V_{-}=\frac {1}{2}(1+2\bar {n})$ in Eqs. (33) and (34), we recover formulas (28) and (26), respectively.

In case of teleportation of squeezed states the squeezing of the teleported state is usually the most important property. The analysis here and in Section 3 illustrates that the measurement-based noiseless amplification applied to Alice’s measurement results cannot improve the squeezing beyond the deterministically achievable limits set by Eq. (31). If one wants to teleport a single Gaussian squeezed state or perform entanglement swapping of fixed Gaussian states, then the filtering of Alice’s measurement results is not useful. The measurement-based noiseless amplification becomes relevant when one deals with an ensemble of states with unknown coherent amplitudes and wants to conditionally improve the teleportation fidelity of the whole ensemble.

5. Discussion and conclusions

We have analyzed the role of measurement-based noiseless amplification in continuous-variable teleportation of Gaussian states [18]. We have seen that application of the inverse Gaussian filter $Z(\beta )$ to results of Alice’s homodyne measurements can conditionally improve the precision of estimation of coherent displacement of the teleported state. Therefore, unity-gain teleportation can be achieved with a lower added noise in comparison to ordinary unity gain deterministic teleportation. For a fully Gaussian scheme, the noise in the teleported state is fundamentally limited by the covariance matrices of the input state and the shared entangled state, c.f. Eq. (16), but the measurement-based noiseless amplification allows to conditionally achieve teleportation arbitrary close to unity displacement gain even for this minimal added noise. In quantum teleportation, the measurement results have to be sent from Alice to Bob via a classical communication channel. Since the measurement-based noiseless amplification operates on data that are subsequently transmitted over this classical channel, it is questionable whether it could be useful in protocols where eavesdropping is an issue. We also recall that due to the cut-off in the filter $Z(\beta )$ the measurement-based noiseless amplification will have effect only on states with small enough coherent amplitude.

Conditional improvement of continuous variable quantum teleportation via measurement-based noiseless amplification [18] should be distinguished from an earlier proposal [39] to improve the fidelity of quantum teleportation by noiseless amplification of the shared two-mode squeezed vacuum state $|\Psi _{\mathrm {EPR}}\rangle _{\mathrm {AB}}$ [40,41], which can increase the squeezing of this state. Nevertheless, there is an interesting connection between these two approaches. Consider teleportation of the vacuum state. Application of the filter $Z(\beta )$ to Alice’s measurement outcomes can then be interpreted as a measurement-based noiseless amplification of mode A that is measured with eight-port homodyne detector. One can thus say that the measurement-based noiseless amplification affects the shared entangled state. The upper bound on admissible $\kappa ^2$ obtained for teleportation of coherent states in Section 2 can be interpreted in this context such that in the limit $\kappa ^2\rightarrow 1/\cosh ^2 r$ the noiseless amplification of mode A would convert the two-mode squeezed vacuum state $|\Psi _{\mathrm {EPR}}\rangle _{\mathrm {AB}}$ with squeezing constant $r$ into an infinitely squeezed state.

To conclude our discussion, we remark that the quantum teleportation can serve for implementation of noiseless amplification [42,43] by encoding the regularized noiseless amplification operation into the entangled state that forms the quantum channel in the teleportation [44]. In the present case, the coherent states could also be teleported with gain larger than unity while keeping the thermal noise in the teleported state arbitrarily low. However, in the limit $\kappa ^2\rightarrow 1/\cosh ^2 r$ the complex amplitude of the input coherent states (up to some cut-off) can be determined from the filtered homodyne data with arbitrary precision. The protocol then becomes reminiscent of perfect unambiguous discrimination of quantum states, and the state transfer becomes essentially classical, although the coherent states form an overcomplete basis and are not all linearly independent.