1. Introduction

The laws of quantum physics impose fundamental limits on the performance of optical amplifiers. In particular, phase insensitive amplification of light unavoidably adds noise to the amplified signal. This can be understood by observing that the hypothetical noiseless amplification of coherent states $|\alpha \rangle \rightarrow |g\alpha \rangle$ with gain $g>1$ decreases the overlap between the output states and therefore it cannot be a deterministic physical operation, i.e. a trace-preserving completely positive map. These limits of deterministic amplifiers can be overcome by probabilistic amplifiers [1], where successful amplification is heralded by certain outcome of measurement on an auxiliary quantum system. Probabilistic noiseless amplifiers approximate the (still unphysical) conditional operation $g^{\hat {n}}$, where $\hat {n}$ is the photon number operator, that maps input coherent state $|\alpha \rangle$ onto an amplified coherent state, $\hat {g}^{n}|\alpha \rangle =e^{(g^2-1)|\alpha |^2/2}|g\alpha \rangle$. The heralded noiseless quantum amplifiers were theoretically proposed by Ralph and Lund [2] and soon afterwards experimentally demonstrated by several groups [3–8]. The probabilistic noiseless quantum amplifiers have attracted significant attention [9–16] because they can be used for suppression of losses in quantum optical communication [17–19], entanglement distillation [20,21], improvement of continuous-variable quantum teleportation [22], or breeding of larger highly non-classical Schrödinger cat-like states formed by superpositions of coherent states. Since the noiseless quantum amplifiers are non-Gaussian conditional quantum operations diagonal in Fock state basis, in a specific regime they can also be used to generate highly non-classical non-Gaussian quantum states [23].

The original noiseless quantum amplifier for weak coherent states $|\alpha \rangle$ was based on modified quantum scissors scheme [24], which truncates the Fock space to the subspace of vacuum and single-photon states. This, however, limits the performance of the amplifier and an amplification gain larger than $1$ can be achieved only for coherent states with amplitude $|\alpha |<0.5$. For larger coherent states one can either split the signal into several modes and utilize several elementary noiseless amplifiers in parallel [2], or consider a generalized quantum scissors scheme with higher Fock-state ancilla $|n\rangle$ [25,26], which however rapidly increases the experimental complexity of the amplifier. To avoid these drawbacks, an alternative approach to noiseless amplification of light was developed based on combination of single-photon addition and subtraction [6,9,10]. The resulting conditional noiseless amplifier modulates the amplitudes of Fock states and the full Fock space is preserved. With this approach, high-fidelity noiseless amplification of coherent states is achievable. The single-photon subtraction can be relatively easily realized by tapping off a small portion of the incoming signal with a highly unbalanced beam splitter with small reflectance $R\ll 1$ and conditioning on detection of a photon in the deflected beam [27–30], see Fig. 1(a). The single-photon addition can be experimentally implemented by injecting the light into the input signal port of a nonlinear crystal that serves as a two-mode squeezer operating in the weak squeezing limit [30,31], see Fig. 1(b). Conditioning on detection of a photon in the output idler port, we can conclude that a single photon has been coherently added to the signal beam. This scheme has the advantage that inefficient single-photon detection only reduces the success rate but does not reduce the quality of photon addition operation, provided that the nonlinear interaction in the crystal is sufficiently weak.

 

Fig. 1. (a) Conditional single-photon subtraction. BS - highly unbalanced beam splitter with small reflectance $R \ll 1$. APD - single photon detector. (b) Conditional single-photon subtraction. NLC- nonlinear crystal where the input signal and idler modes are coupled by weak two-mode squeezing interaction. L - pump laser beam. Successful photon addition or subtraction is heralded by click of the detector APD.

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In the present paper, we propose and analyze a new scheme for noiseless quantum amplification of light. Specifically, we exploit the concept of teleportation of quantum gates [32] to design a noiseless quantum amplifier based on a suitably shaped two-mode entangled state that serves as a quantum channel in continuous-variable quantum teleportation [33,34]. In contrast to ordinary deterministic quantum teleportation we must condition on the outcomes of homodyne measurements in the teleportation protocol [35] to achieve noiseless amplification. Interestingly, we find that the required resource entangled state can be prepared from a two-mode squeezed vacuum state solely by conditional photon subtractions. Explicit photon addition operation is avoided because, due to the perfect photon number correlations between the signal and idler mode of two-mode squeezed vacuum, addition of a photon to the signal mode becomes equivalent to photon subtraction from the idler mode. The motivation for investigation of teleportation-based noiseless quantum amplifier is at least two-fold. First, the required two-mode entangled state can be in principle prepared beforehand [26] and stored in a quantum memory until needed, which could significantly increase the success probability of the scheme as compared to the noisless amplification based on direct application of conditional photon addition and subtraction. Second, the noiseless teleamplification can be suitable for experiments with continuous-wave signals, where strongly squeezed states with very high purity can be generated [36–40] and photon subtraction is well mastered [28,29], while the conditional photon addition to a specific temporal mode may be more difficult to accomplish.

Previous works have investigated entanglement distillation by noiseless amplification of the two-mode squeezed vacuum state and the improvement of continuous-variable teleportation by utilizing the noiselessly amplified two-mode squeezed vacuum state as a quantum channel in teleportation [21–23]. Here, we make important step further beyond these previous studies and use the teleportation scheme for conditional noiseless amplification of the teleported state. We note that also the noiseless quantum amplifier based on the quantum scissors scheme can be interpreted as a quantum tele-amplifier which requires ancilla Fock states and is driven by projection on a suitable multimode $n$-photon state [26].

The rest of the paper is organized as follows. In Section 2 we present a pure-state description of the protocol which allows us to easily explain its main working principles. Subsequently, in Section 3 we develop a more realistic model of the teleportation-based noiseless amplifier that takes into account the various experimental imperfections. This model is based on the powerful and well established phase-space techniques, where the states are represented by their Wigner functions or Husimi $Q$-functions. In Section 4 we discuss noiseless teleamplification of highly nonclassical superpositions of coherent states. Finally, Section 5 contains a brief summary and conclusions.

2. Teleportation-based noiseless quantum amplifier

We start with a brief overview of the probabilistic noiseless amplification of coherent states via combination of conditional single-photon addition and subtraction. The simplest instance of such noiseless quantum amplifier consists of single-photon addition followed by single-photon subtraction [6,9,10], described by the operator

This amplifier achieves gain $g$ for weak coherent states, $\hat {G}(|0\rangle +\alpha |1\rangle )=|0\rangle +g\alpha |1\rangle$. The conditional operation (3) can be realized with an interferometric scheme, where photon subtraction is attempted both before and after the photon addition and the two subtracted beams are overlapped at a beam splitter before detecting the subtracted photon [41,42]. This erases the which way information about the origin of the subtracted photon and produces the coherent superposition (3), where the relative weight of the terms $\hat {a}\hat {a}^\dagger$ and $\hat {a}^\dagger \hat {a}$ can be controlled by the splitting ratio of the beam splitter where the two subtracted beams interfere.

In this paper we show that the noiseless amplification operation $\hat {G}$ can implemented via continuous-variable teleportation-based scheme. Our approach requires auxiliary Gaussian squeezed states and conditional photon subtraction, but it does not require auxiliary single-photon states and we avoid injection and mode-matching of the amplified signal beam in a nonlinear crystal. Specifically, we can encode the conditional operation $\hat {G}$ into a two-mode squeezed vacuum state

 

Fig. 2. Teleportation-based noiseless quantum amplifier. Modes $A$ and $B$ are prepared in a suitable entangled state $|\Psi _G\rangle$ that encodes the amplification operation $\hat {G}$. The input coherent state is injected in mode $C$. Modes $A$ and $C$ interfere on balanced beam splitter and are measured with two balanced homodyne detectors BHD. Mode $B$ can be coherently displaced and the output state is accepted or rejected depending on the measurement outcome $\beta$.

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We observe that for $\beta =0$ the noiseless amplification is implemented perfectly up to an additional attenuation due to finite squeezing, i.e. the output state is $\hat {G}|\lambda \alpha \rangle$ for any input state $|\alpha \rangle$. To see this, define a non-normalized infinitely squeezed state

For $\beta =0$, the modes $A$ and $C$ in Fig. 2 are projected on state $|\Psi _{\mathrm {EPR}}\rangle$, and the teleported state can be expressed as

Taking into account the perfect photon number correlations in states $|\Psi (\lambda )\rangle$ and $|\Psi _{\mathrm {EPR}}\rangle$ and expanding the coherent state $|\alpha \rangle$ in Fock basis,

We can see that the effective amplification gain is reduced by the finite squeezing strength $\lambda <1$, $g_{\mathrm {eff}}=\lambda g$. The performance of the conditional noiseless amplifier can be more comprehensively characterized by the actual amplitude dependent amplification gain

Let us now discuss the preparation of the resource entangled state (5). Consider first the noiseless amplification via the conditional operation (1). The target entanged state reads

Remarkably, this state can be obtained by subtracting a single photon from each mode of the two-mode squeezed vacuum state [43],

Due to the perfect photon number correlations between the two modes the addition of a photon to mode $B$ becomes equivalent to subtraction of a photon from mode $A$. The joint subtraction of two photons from a two-mode squeezed vacuum state has already been successfully demonstrated experimentally [44,45]. For weakly squeezed states, the local photon subtractions can conditionally increase the entanglement of the state and can thus serve for continuous-variable entanglement distillation.

Consider now preparation of the resource state $|\Psi _G(\lambda )\rangle$ for the more general noiseless amplification operation $\hat {G}$ given by Eq. (3). We have

This state can be prepared from a two-mode squeezed vacuum state by a modified joint photon subtraction from each mode, where an auxiliary weak two-mode squeezed vacuum state is injected into the auxiliary input ports of the beam splitters that serve for photon subtraction [46]. The scheme is illustrated in Fig. 3 and the successful state preparation is heralded by simultaneous click of both single-photon detectors APD. Let us assume that perfect photon number resolving detectors are employed that project the output modes $C$ and $D$ on single-photon Fock states $|1,1\rangle _{CD}$. Let $\mu$ denote the squeezing strength of the auxiliary two-mode squeezed vacuum state $|\Psi (\mu )\rangle _{CD}$ and let $T=1-R$ denote the intensity transmittance of the two tapping beam splitters BS in Fig. 3. The conditionally prepared state of modes $A$ and $B$ can be expressed as

 

Fig. 3. Preparation of entangled state for teleportation-based noiseless amplification via generalized photon subtraction [46]. The optical parametric amplifiers OPA generate two-mode squeezed vacuum states with different squeezing strengths $\lambda$ and $\mu$. The squeezed states interfere on beam splitters BS and the output modes $C$ and $D$ are measured with single photon detectors APD. Successful state preparation is heralded by simultaneous click of both detectors APD.

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The experimentally available single-photon detectors usually only distinguish the presence or absence of photons but do not resolve their number. Moreover, the detectors have only limited detection efficiency $\eta$. The photon subtraction scheme shown in Fig. 3 can faithfully generate the desired target state (15) even under such imperfect experimental conditions provided that the beam splitters BS are highly unbalanced and the auxiliary input state $|\Psi (\mu )\rangle _{CD}$ is only very weakly squeezed,

Under this condition, the auxiliary input state of modes $C$ and $D$ can be approximated as $|0,0\rangle +\mu |1,1\rangle$. The two dominant events leading to the simultaneous clicks of the two detectors APD in Fig. 3 are that either the auxiliary modes $C$ and $D$ are in vacuum state and a single photon is subtracted from both input modes $A$ and $B$ (probability scaling $R^2$), or that a single photon is present in each of the auxiliary input modes $C$ and $D$ and these photons are transmitted through the beam splitters BS and detected by APDs (probability scaling $|\mu |^2$). The dominant unwanted event where two photons impinge on one of the detectors APD while a single photon arrives to the other detector APD has probability that is much smaller than the probability of the desired events provided that the inequality (20) holds. A detailed analysis of the whole teleportation-based noiseless amplifier that takes into account these experimental limitations is provided in the next section. Here we just note that in the limit $|\mu |\ll 1$ and $R \ll 1$ the expression for the effective gain simplifies and we can write

Note that $\mu$ and $\lambda$ can have opposite signs and therefore the effective gain $g_{\mathrm {eff}}$ can be arbitrarily high. We observe that the effective amplification gain is rather sensitive to small changes of $\mu$. To see this explicitly, we evaluate $dg_{\mathrm {eff}}/d\mu$ and obtain

For typical configurations with $R\ll 1$ and $|\mu |\ll |\lambda |$ we find that $dg_{\mathrm {eff}}/d\mu \approx -(g_{\mathrm {eff}}-\lambda _{\mathrm {eff}})^2 /(\lambda ^2 R)$. Especially achieving large effective gains $g_{\mathrm {eff}}$ requires destructive quantum interference, i.e. $\lambda >0$ and $\mu <0$, and the resulting small term $R\lambda +T\mu$ in the denominator of Eqs. (19) and (21) implies high sensitivity of $g_{\mathrm {eff}}$ to changes of $\mu$.

Figure 4 illustrates the performance of teleportation-based noiseless quantum amplifier for two different gains. We consider noiseless amplification with pure entangled state (15) and conditioning on $\beta =0$, which leads to the best possible performance of the amplifier. The results depicted in Fig. 4 thus provide a benchmark for comparison with predictions of more realistic models discussed in the next section. For small $\alpha$, the noiseless amplification is practically perfect. For larger $|\alpha |$, the resulting amplification gain and fidelity begins to decrease and the uncertainty product $V_x V_p$ grows, indicating that the amplified state is no longer a minimum uncertainty state. Note that a deterministic amplifier would result in an uncertainty product $V_xV_p=(g^2(\alpha )+|g^2(\alpha )-1|)^2/4$, indicated by dashed lines in Fig. 4(d).

 

Fig. 4. Performance of teleportation-based noiseless quantum amplification of coherent states $|\alpha \rangle$ with pure resource entangled state (15) with $\lambda =0.5$. The results are plotted for two different effective gains $g_{\mathrm {eff}}=2$ (red lines) and $g_{\mathrm {eff}}=1.5$ (blue lines). Projection on $\beta =0$ is assumed in the teleportation protocol. (a) The amplitude-dependent amplification gain $g(\alpha )$. (b) Fidelity of the amplified state with coherent state $|g(\alpha )\alpha \rangle$. (c) Variances $V_x$ and $V_p$ of the amplitude and phase quadratures of the amplified state. (d) The uncertainty product $V_xV_p$. The dashed lines show the uncertainty product for the optimal linear deterministic amplifier: $(g^2(\alpha )+|g^2(\alpha )-1|)^2/4$.

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The conditional operation (3) approximates the noiseless amplifier well only for relatively weak coherent states with $|\alpha |<1$. Better approximation can be achieved with more sophisticated filters that approximate the unphysical operation $g^{\hat {n}}$ on a larger subspace of the Fock states. In particular, one can consider the following operation where the amplification is truncated at Fock state $|N\rangle$ [1,15,23],

The corresponding normalized resource entangled state for teleportation-based noiseless amplification reads

As a preparatory step for further discussion let us recall some facts about the continuous-variable teleportation of coherent states in the Vaidman-Braunstein-Kimble scheme [33,34]. The measurement outcome $\beta$ indicates that the modes $A$ and $C$ were projected onto coherently displaced EPR state $\hat {I}_A\otimes \hat {D}_C(\beta )|\Psi _{\mathrm {EPR}}\rangle _{AC}$, where $\hat {D}(\beta )=\exp (\hat {a}^\dagger \beta -\hat {a}\beta ^{\ast })$. This can be equivalently interpreted as a coherent displacement of the input state in mode $C$, $\hat {D}(-\beta )|\alpha \rangle _C=e^{(\alpha \beta ^\ast -\alpha ^\ast \beta )/2}|\alpha -\beta \rangle _C$, followed by projection of modes $A$ and $C$ onto the EPR state $|\Psi _{\mathrm {EPR}}\rangle$. This in turn is equivalent to the projection of mode $A$ onto the re-normalized coherent state $\frac {1}{\sqrt {\pi }} |\alpha ^\ast -\beta ^\ast \rangle$. The measurement outcomes $\beta$ span the whole complex plane and the overcompleteness of projectors onto coherent states is accounted for by the prefactor $1/\sqrt {\pi }$ in our definition of the EPR state $|\Psi _{\mathrm {EPR}}\rangle$, which ensures that we obtain correct probability density of measurement outcomes.

Let us first consider teleportation with the Gaussian two-mode squeezed vacuum state (4). Conditional on measurement outcome $\beta$, which occurs with probability density

Let us now turn to teleportation with the entangled state (24). If the coherent amplitude $\alpha -\beta$ satisfies the condition

This formula is meaningful only if $\sigma$ and $|\alpha |$ are sufficiently small such that the following conditions are satisfied

The first inequality ensures that the Gaussian integral in (30) converges and the second inequality guarantees that the range of complex amplitudes $\beta$ that non-negligibly contribute to the integral (30) includes only $\beta$ for which the condition (28) is satisfied. The factor $s$ appearing in Eq. (32) means that we take $s$ standard deviations as the effective width of the involved Gaussian distribution of $\beta$. A conservative choice is to set $s=3$.

Performance of the teleportation-based noiseless quantum amplification with the resource entangled state $|\Psi _N(\lambda )\rangle$ is illustrated in Fig. 5 for $N=10$, $\lambda =0.5$ and effective gain $g_{\mathrm {eff}}=\lambda g=1.5$. We numerically simulate the teleamplification where the teleportation outcomes are always accepted if $|\beta |<\sigma$ and rejected otherwise. The corrective coherent displacement is set to $\hat {D}(\lambda g\beta )$. In Fig. 5(a) we plot the fidelity of the teleamplified state with the ideal amplified coherent state $|\lambda g \alpha \rangle$, and he corresponding success probabilities are plotted in Fig. 5(b). The results are shown for three different values of $\sigma$. As expected, larger $\sigma$ leads to higher success probability but also to lower output state fidelity.

 

Fig. 5. Teleamplification of coherent states $|\alpha \rangle$ with resource entangled state (24) with $\lambda =0.5$ and $g\lambda =1.5$. The teleportation fidelity (a) and success probability (b) are plotted for $N=10$ and three different acceptance thresholds $\sigma =0.5$ (solid blue line), $\sigma =0.75$ (red dashed line), and $\sigma =1$ (green dot-dashed line). Panel (c) displays the maximum probability $P_0$ achievable by a noiseless teleamplifier that amplifies all coherent states with amplitude $|\alpha |\leq |\alpha _{\mathrm {th}}|$ with fidelity $F \geq F_{\mathrm {th}}=0.999$ (blue line). For comparison, the red line is the maximum $P_0$ achievable by the noisleess amplification operation $\hat {G}_N$, Eq. (23) , and the green lines represent predictions of the analytical model (31) constrained by inequalities (32) with $s=3$ (solid green line) and $s=1$ (dashed green line).

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The efficiency of the noiseless amplification can be conveniently quantified by its probability of success for the vacuum state $P_0$. If a coherent state $|\alpha \rangle$ is essentially perfectly noiselessly amplified with gain $g$ then the success probability of its noiseless amplification can be expressed as $P(\alpha )=e^{(g^2-1)|\alpha |^2}P_0$. For the noiseless amplifier described by operator (23) we have $P_0=g^{-2N}$. In order to compare the efficiency of the teleamplification based on state $|\Psi _N(\lambda )\rangle$ with the reference operation $\hat {G}_N$ for the same effective amplification gain we proceed as follows. We introduce a fidelity threshold $F_{\mathrm {th}}$ and for any chosen threshold amplitude $\alpha _{\mathrm {th}}$ we maximize the probability $P_0$ of the teleamplifier over $N$ and $\sigma$ under the constraint $F(\alpha _{\mathrm {th}})=F_{\mathrm {th}}$, while keeping $g$ and $\lambda$ fixed. The same procedure is applied also to the noiseless amplifier (23), where we optimize only over the threshold $N$ and set $g=g_{\mathrm {eff}}$. The results are reported in Fig. 5(c). The two curves exhibit similar slopes but the teleapmplifier is less efficient by almost two orders of magnitude which is the price we pay for the teleportation-based protocol. Note that the probability curve for the noiseless amplifier (23) has the shape of discrete steps, because we optimize only over a discrete variable $N$.

For completeness, we also display in Fig. 5(c) the predictions of the analytical model (31). We maximize the probability (31) for the vacuum input, $\alpha =0$, over $N$ and $\sigma$ under the constraints (32), where we set $\alpha =\alpha _{\mathrm {th}}$. The result is plotted as green solid line in Fig. 5(c). The analytical model with $s=3$ reproduces the slope of the probability curve but underestimates the actual success probability achievable by the teleamplifier. If we impose a less stringent constraint and set $s=1$ in the second inequality (32) then we obtain a very good quantitative agreement with the protocol where a hard threshold $\sigma$ for acceptance or rejection is introduced.

A two-mode entangled state of the form (24) can be generated from a two-mode squeezed vacuum state by repeated application of the generalized joint photon subtraction scheme depicted in Fig. 3. This method of two-mode quantum state engineering is inspired by the previously proposed schemes for preparation of arbitrary single-mode states of light by repeated conditional photon addition or subtraction combined with coherent displacements [47,48]. We assume very weakly squeezed state in input auxiliary modes $C$ and $D$, $|\mu |\ll 1$, hence it can be approximated as $|\psi \rangle _{CD}\approx |0,0\rangle +\mu |1,1\rangle$. Conditioning on detection of a single photon in each auxiliary output mode $C$ and $D$, the following operation is applied to the input state of modes $A$ and $B$,

In order to simplify the subsequent discussion and focus on core aspects of the quantum state generation procedure, we will assume that $R \ll 1$ and neglect all terms of the order of $\mu R$ in Eq. (33). We also neglect the exponential attenuation factor, $T^{(\hat {n}_A+\hat {n}_B)/2} \approx 1$. The conditional operation (33) then simplifies to

If we apply the operation $\hat {Z}_N$ to a two-mode squeezed vacuum state with squeezing parameter $\lambda$, we get a state with preserved perfect photon number correlations of modes $A$ and $B$ and the amplitude of each pair Fock state $|n,n\rangle$ is modulated by a factor that is a polynomial of degree $N$ in $n$,

This procedure allows us to engineer any desired polynomial $H_{N}=\sum _{k=0}^N h_k n^k$ normalized such that $h_N=\lambda ^N$. For any given $h_k$ and $\lambda$ the formula (37) yields a system of linear equations for $z_k$ that can easily be solved. Subsequently, the parameters $\kappa _j$ can be determined as roots of polynomial $\sum _{k=0}^N z_k(-\kappa )^k$. Specifically, if we choose $H_{N}(n)=(g\lambda )^n/g^N$ for $n=0,1,\ldots,N$, then we will generate a state whose structure will coincide with the state (24) for the low Fock states up to $|N\rangle$, while the amplitudes of the higher Fock states will also be modulated by the polynomial function $H_N(n)$. The probability of successful state preparation will typically decrease exponentially with $N$ and with current technology it could be difficult to go beyond $N=2$. If highly efficient photon number resolving detectors would be available, then one can go beyond the limit of low $R$ and $\mu$ and optimize $R$ to maximize the success probability of the state preparation.

3. Phase-space description of teleportation-based noiseless quantum amplifier

In this section we present results of a more realistic model of the teleportation-based quantum noiseless amplifier that takes into account the main experimental imperfections. The model is based on the well-established phase-space approach, where the Wigner function of a non-Gaussian state generated by photon subtractions from some Gaussian state can be expressed as a linear combination of several Gaussian Wigner function [48,49]. Here we choose to work with the Husimi $Q$-function instead of the Wigner function, because with the $Q$-function it is particularly straightforward to calculate projections onto coherent states. Let us begin with some useful definitions. we consider system of $N$ modes, we denote by $\hat {x}_j$ and $\hat {p}_j$ the amplitude and phase quadrature operators of the $j$th mode, respectively, and and we collect the quadrature operators into a vector $\hat {\boldsymbol{q}}=(\hat {x}_1,\hat {p}_1,\ldots, \hat {x}_N,\hat {p_N})^T$. The covariance matrix $\gamma$ of the state is a $2N\times 2N$ square matrix with elements

Note that the covariance matrix of vacuum state is equal to the identity matrix, $\gamma _{\mathrm {vac}}=I$.

Let us first describe the preparation of the two-mode resurce state (15) according to the scheme shown in Fig. 3. Our model assumes that the input states of modes $A,B$ and $C,D$ are Gaussian noisy two-mode squeezed thermal states that can be equivalently represented as initial pure two-mode squeezed states transmitted through a lossy channel with some effective transmittance $\eta$. The covariance matrices of the resulting mixed Gaussian states read

The single-photon detectors APD in Fig. 3 are modeled as on-off detectors that can only distinguish the presence or absence of photons and have overall detection efficiency $\eta _{\mathrm {APD}}$. It is convenient to account for the inefficient detection by letting the detected modes propagate through a lossy channel with transmittance $\eta _{\mathrm {APD}}$ followed by perfect on-off detectors whose click is described by POVM element $\hat {I}-|0\rangle \langle 0|$. In the state preparation scheme shown in Fig. 3 we condition on simultaneous clicks of both APDs and the corresponding POVM element therefore reads

The effective covariance matrix of the overall Gaussian state of the four modes $A,B,C,D$ just before the measurement on modes $C$ and $D$ can be expressed as

The Husimi $Q$-function of a two-mode state $\hat {\rho }_{AB}$ is defined as

Here the two-mode and three-mode covariance matrices $\gamma _{\mathrm {eff}, AB}$, $\gamma _{\mathrm {eff}, ABC}$ and $\gamma _{\mathrm {eff}, ABD}$ are submatrices of the full covariance matrix (42). The matrices $\Gamma _j$ appearing in the exponent in Eq. (45) are then extracted from $\tilde {\Gamma }_{j}$ as two-mode submatrices corresponding to modes $A$ and $B$. Note that, in particular, $\Gamma _1=\tilde {\Gamma }_1$. Finally, the probability $P_{AB}$ of conditional state preparation is given by

With the phase-space representation of the resource entangled state we can proceed to analysis of the teleportation-based noiseless amplification of coherent states $|\alpha \rangle$. As discussed in the previous section, the teleported state corresponding to measurement outcome $\beta$ can be obtained by projecting the mode $A$ of the two-mode entangled state (45) onto the re-normalized coherent state $\frac {1}{\sqrt {\pi }}|\alpha ^\ast -\beta ^\ast \rangle$. We introduce two real displacement vectors

To illustrate the calculations, consider teleportation with a generic two-mode Gaussian state with $Q$-function

We introduce a bipartite $A|B$ splitting of the matrix $\Gamma$,

The $Q$-function of a non-normalized state of mode $B$ corresponding to projection of mode $A$ onto $\frac {1}{\sqrt {\pi }}|\alpha ^\ast -\beta ^\ast \rangle$ reads

The output Gaussian state (53) has covariance matrix $\gamma _B=2\Gamma _B^{-1}-I$ and coherent displacement $D \boldsymbol{d}_\alpha$.

Let us now turn to teleportation with a non-Gaussian entangled resource state (45). We can write the normalized output teleported state conditioned on $\beta =0$ as a linear combination of four Gaussian states,

The coherent displacement of the output state (56) can be calculated as

The symmetry of the state-preparation scheme in Fig. 3 and the teleportation protocol in Fig. 2 ensures that for the considered class of the resource entanged states (45) the matrices $D_j$ are all proportional to the identity matrix, $D_j= d_j I$, and the effective state-dependent amplification gain can be expressed as

Following a similar procedure, one can also calculate the covariance matrix of the output amplified state (45),

Finally, a fidelity of the output state (53) with an amplified coherent state $|g\alpha \rangle$ can be easily evaluated from the Husimi $Q$-function (53),

We have verified that if we consider pure input squeezed vacuum states ($\eta _{AB}=\eta _{CD}=1$), perfect detectors with $\eta _{\mathrm {APD}}=1$ and tapping beam splitters with very small reflectance $R<0.01$ , then for suitable choices of $\lambda$ and $\mu$ we recover the results predicted by pure state model and presented in the previous Section in Fig. 4. Let us now consider a more realistic scenario with noisy squeezed states, $\eta _{AB}=\eta _{CD}=0.9$ and superconducting single-photon detectors with total detection efficiency $\eta =0.85$. We assume $\lambda =0.5$, which corresponds to $4$ dB of squeezing and $4.5$ dB of anti-squeezing in the resulting noisy two-mode squeezed thermal state with covariance matrix (40). The results are plotted in Fig. 6 which displays the same quantities as Fig. 4. The results are again shown for two different values of the effective gain $g=1.5$ and $g=2$. The values of squeezing strength $\mu$ that yield these gains were determined numerically and read $\mu =-0.0150$ and $\mu =-0.0197$, respectively. As expected, noisy squeezed states and imperfect single-photon detection with on-off detectors lead to reduced performance of the noiseless amplifier in comparison to the ideal pure-state scenario considered in the previous section. Most notably, the state fidelity is not a monotonously decreasing function of $|\alpha |$ but instead grows with $|\alpha |$. Note that this behavior is observed even if we consider fidelity with state $|g_{\mathrm {eff}}\alpha \rangle$ instead of $|g(\alpha )\alpha \rangle$. Similarly, the product of quadrature variances decreases with increasing $\alpha$.

 

Fig. 6. Performance of realistic noiseless amplifier with parameters $\lambda =0.5$, $T=0.95$, $\eta _{\mathrm {APD}}=0.85$, $\eta _{AB}=\eta _{CD}=0.9$, and two different values of $\mu$ yielding effective gains $g_{\mathrm {eff}}=1.5$ ($\mu =-0.0150$, blue curves) and $g_{\mathrm {eff}}=2$ ($\mu =-0.0197$, red curves). Conditioning on $\beta =0$ is assumed. The quantities plotted are the same as in Fig. 4: (a) amplitude-dependent amplification gain; (b) fidelity of amplified state with coherent state $|g(\alpha )\alpha \rangle$, (c) variances of amplitude and phase quadratures of the amplified state; (d) product of quadrature variances of the amplified state.

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To understand this behavior, we can consider setup with perfect photon number resolving detectors, vacuum in auxiliary modes $C$ and $D$ ($\mu =0$), and low thermal noise with mean number of thermal photons $\bar {n}_{\mathrm {th}}\ll 1$ in modes $A$ and $B$. In this limit, the amplified coherent state can be approximately expressed as a mixture of three states $(\hat {n}+1)|\lambda \alpha \rangle$, $\alpha (\hat {n}+2)|\lambda \alpha \rangle$, and $(\hat {n}+1)\hat {a}^\dagger |\lambda \alpha \rangle$ where the probability of the last two states in the mixture is proportional to $\bar {n}_{\mathrm {th}}$. For $\alpha =0$ the state $\hat {a}^\dagger |\lambda \alpha \rangle$ becomes equal to the Fock state $|1\rangle$ that is orthogonal to the vacuum state. On the other hand, for $\alpha \neq 0$ the state $\hat {a}^\dagger |\lambda \alpha \rangle$ has a nonzero overlap with $|\lambda \alpha \rangle$ and this overlap increases with $|\alpha |$. This explains the behavior of fidelity observed in Fig. 6(b).

In practice, a finite acceptance window for the outcomes $\beta$ will be required to achieve a non-zero overall success probability of the noiseless amplification. In order to obtain analytical results, we shall again assume that the outcomes $\beta$ are accepted with Gaussian probability $\exp (-|\beta |^2/\sigma ^2)$. Simultaneously, we include a corrective coherent displacement proportional to $\beta$, that is applied to the output state. For teleportation with entangled Gaussian state (49), the Husimi $Q$-function of the resulting output single-mode state can be expressed as

In order to write down the formulas for the various matrices appearing in Eqs. (63) and (64) we first specify the matrix $\Gamma _\beta$,

With the help of these definitions we can write down explicit formulas for the remaining matrices,

If we apply the above Gaussian integration to each of the four terms in the formula (45) for the $Q$-function of the photon subtracted two-mode squeezed state, we obtain the final expression for the $Q$-function of the teleported state,

Here

Figure 7 illustrates the performance of the teleportation-based noiseless amplifier for a finite acceptance window of measurement outcomes $\beta$ ($\sigma ^2=0.08$). We compare the protocols with ($k=1$) and without ($k=0$) corrective coherent displacement. To make a fair comparison, we choose the auxiliary squeezing parameter $\mu$ and the transmittance $T$ of the tapping beam splitters separately for each case to achieve the same effective gain $g_{\mathrm {eff}}=1.5$ and the same total success probability $P_{\mathrm {tot}}$ for small $\alpha$. We can see that the gain curves practically overlap, but the protocol with corrective displacement achieves higher fidelity and lower quadrature fluctuations of the amplified state. These results therefore clearly demonstrate the usefulness and advantage of the corrective displacement.

 

Fig. 7. Performance of realistic noiseless amplifier with parameters $\lambda =0.5$, $\eta _{\mathrm {APD}}=0.85$, $\eta _{AB}=\eta _{CD}=0.9$, $g_{\mathrm {eff}}=1.5$, and finite acceptance window for measurement outcomes $\beta$, $\sigma ^2=0.08$. The blue curves indicate results for a protocol with corrective displacement: $k=1$, $T=0.95$, $\mu =-0.0179$. For reference, the red curves display results for a protocol without any corrective displacement: $k=0$, $T=0.955$, and $\mu =-0.01475$. The panels (a)-(d) display the same quantities as in Figs. 4 and 6. In the two additional panels (e,f) we plot the success probabilities $P_{\mathrm {tele}}$ (e), and $P_{\mathrm {tot}}$ (f).

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As shown in Fig. 7(e), the success probability of the teleportation protocol itself $P_{\mathrm {tele}}$ is of the order of $1\%$ for the parameters considered. As discussed above, the preparation of resource entangled state can be accomplished before the teleamplification is attempted and in this sense the probability $P_{\mathrm {tele}}$ characterizes the probabilistic noiseless teleamplifier. For comparison, in the experiment by Bellini group [6] the overall success probability of conditional noiseless amplification via sequence of single photon addition and subtraction was about $P_{\mathrm {NA}}\approx 10^{-6}$, which is several orders of magnitude lower than $P_{\mathrm {tele}}$. In the experiment [6], the success probability was mainly reduced by the very low probability of photon-pair generation in a nonlinear crystal that served for conditional photon addition. The teleportation-based noiseless amplification can thus be potentially more efficient than direct application of conditional photon addition and subtraction. However, with current technology it would be very difficult to store the prepared entangled state in a quantum memory while maintaining the required high quality of the state. Practical implementation of the protocol would therefore require simultaneous success of the conditional state preparation and teleportation, which is characterized by the total probability of success $P_{\mathrm {tot}}$, that is mainly reduced by the small probability of preparation of the photon subtracted state, here $P_{AB} \approx 3\times 10^{-4}$.

For a fixed input state $|\alpha \rangle$, the success probability $P_{\mathrm {tele}}$ scales as $\sigma ^2$ for small $\sigma$ and then it saturates and approaches $1$ when $\sigma \rightarrow \infty$, which corresponds to the limit of deterministic teleportation. The amplification gain $g(\alpha )$ quickly decreases with increasing $\sigma$ and for the data used to produce Fig. 7 we observe that even for weak coherent states $g(\alpha )$ drops below $1$ at $\sigma \approx 0.5$. Asymptotically, at large $\sigma$, the teleamplification gain becomes independent on $\alpha$ and is equal to the strength of the corrective displacement $k$ in the teleportation protocol. The teleamplification fidelity $F$ and the quadrature uncertainty product $V_xV_p$ can exhibit non-monotonic dependence on $\sigma$, especially for small $\alpha$. In particular, at small $\sigma$ the fidelity can be an increasing function of $\sigma$ for some range of $\alpha$, and the quadrature uncertainty product can be a decreasing function of $\sigma$. This is consistent with our earlier observation that the vacuum state can be more affected by the teleamplification imperfections than coherent states with non-zero $\alpha$.

In Fig. 8 we compare the performance of the teleportation-based noiseless quantum amplifier with noisless quantum amplifier based on combination of single-photon addition and subtraction [6,9,10] and noiseless quantum amplifier based on generalized quantum scissors scheme [4]. The quantum scissors require ancilla single-photon state that can be prepared by conditioning on detection of a single photon from a correlated photon pair generated in the process of spontaneous parametric down conversion. All three approaches then involve conditioning on detection of two photons in total. All three schemes also require interferometric stability and squeezing. However, weak squeezing is needed for noiseless amplification based on photon addition and for the quantum scissors scheme, while the teleportation-based noiseleless quantum amplifier requires rather strong squeezing to achive good performance and avoid fast attenuation of the higher Fock-state amplitudes by the factor $\lambda _{\mathrm {eff}}^n$.

 

Fig. 8. Comparison of the amplification gain (a), fidelity (b), and quadrature uncertainty product (c) of amplified coherent states for three different noiseless quantum amplifiers with effective nominal amplification gain $g_{\mathrm {eff}}=1.5$. The results are plotted for a noiseless amplifier based on combination of photon addition and subtraction (red line, $T=0.99$ and $\nu =0.1$), noiseless amplifier based on the generalized quantum scissors scheme (green line), ideal noiseless teleamplifier with pure resource entangled state (15) with $\lambda =0.75$ and conditioning on $\beta =0$ (blue solid line), and a realistic noiseless teleamplifier with $\lambda =0.787$, $\mu =0$, $T=0.99$, $\eta _{AB}=0.95$, $\eta _{\mathrm {APD}}=0.85$, $\sigma =0.2$, and $k=1$ (dashed blue line).

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The quantum scissors amplifier truncates the state at Fock state $|1\rangle$ and for input coherent state $|\alpha \rangle$ it outputs the state

The effective amplification gain reads

These features are illustrated in Fig. 8. The ideal noiseless amplifier based on combination of photon addition and subtraction achieves the best performance and the ideal teleportation based amplifier performs only marginally worse. Both schemes produce output states that are very close to amplified coherent states, as indicated by high fidelities $F$ and low uncertainty product $V_xV_p$ that confirms that the output state is very close to a minimum uncertainty Gaussian state. The amplifier based on the quantum scissors scheme exhibits significantly worse performance, especially for larger $|\alpha |$. To make a full comparison, we also provide results for a noiseless teleamplifier affected by realistic imperfections. We consider here the state preparation scheme in Fig. 3 without the second weakly squeezed state, $\mu =0$. This requires higher squeezing, but we avoid the need to control interference of two squeezed beams and precisely control the squeezing strength $\mu$. The results reported in Fig. 8 are obtained for $\lambda =0.787$ and $\eta _{AB}=0.95$, which corresponds to $7.9$ dB of squeezing and $9$ dB of anti-squeezing. This level of squeezing is experimentally demanding, but in principle accessible [38]. Even when we take into account the experimental imperfections the teleamplifier can outperform the quantum scissors scheme for lager $|\alpha |$, where the latter scheme does not amplify the coherent states at all. The amplifiers based on combination of photon addition and subtraction or generalized quantum scissors are constructed such that for input vacuum stat they again output vacuum state even in presence of experimental imperfections such as reduced interference visibility or other effects. The only condition is that the squeezing in a nonlinear crystal that is used for photon addition or for generation of ancilla photon pair is sufficiently weak so that the probability of photon-pair generation is very small. By contrast, in the teleamplification scheme the experimental imperfections affect the vacuum state and disturb it, and the fidelity can become an increasing function of $|\alpha |$ for small $|\alpha |$.

4. Noiseless teleamplification of superpositions of coherent states

In this section we show that the techniques presented in the previous section can be extended beyond the input coherent states and can be utilized to investigate noiseless teleamplification of highly nonclassical states. Specifically, we will consider the even and odd Schrödinger cat-like states formed by superpositions of two coherent states,

In particular, if we set $\beta =-\alpha$ we get

With the help of formulas (76) and (68) the $Q$ function of the teleamplified cat-like states (72) can be expressed as a linear combination of $4\times 4=16$ Gaussian functions.

Since the superpositions of coherent states (72) are eigenstates of $\hat {a}^2$ with eigenvalue $\alpha ^2$, the effective gain of the noiseless teleamplification of these states can be evaluated as

We have numerically investigated noiseless teleamplification of even and odd cat-like state for realistic resources including noisy two-mode squeezed states and imperfect single photon detectors and the results are reported in Fig. 9. Similarly as in Fig. 8 we again consider the simplest state preparation scheme without auxiliary squeezed state, $\mu =0$, and sufficiently strong squeezing, $\lambda =0.787$ and $\eta _{AB}=0.95$. This yields $g_{\mathrm {eff}}=1.5$, see Fig. 8. Figure 9(a) displays the effective amplification gain evaluated with the use of Eq. (77). The observed ampification gains are for low $\alpha$ in very good agreement with the theoretical formulas (78). The noiselessly teleamplified cat-like states have well preserved shape and coherence, as witnessed by the high fidelity of the output states with even or odd cat-like state with amplitude $g_{+}(\alpha )\alpha$ or $g_{-}(\alpha )\alpha$, respectively. The Wigner function of the odd cat-like state is negative at the origin of phase space, $W_{-}(0,0)=-\frac {1}{\pi }$. This negativity is preserved for the teleamplifierd state, as illustrated in Fig. 9(c). On the other hand, the even cat-like state is a superposition of even Fock states and its Wigner function at the origin of phase space is positive, $W_{+}(0,0)=\frac {1}{\pi }$. For comparison, we plot in Fig. 9 also results for unity-gain teleportation where the initial noisy two-mode squeezed Gaussian state of modes $A$ and $B$ serves as the quantum channel. The fidelity of this latter deterministic teleportation protocol is smaller than the fidelity of the conditional teleamplification scheme, the unity gain teleportation does not amplify the teleported states, and the negativity of Wigner function of the teleported odd cat-like state is reduced due to the noise added in the teleportation.

 

Fig. 9. Teleamplification of superpositions of coherent states. We plot the effective amplification gain (a), the fidelity of the teleamplified states (b), and the value of the Wigner function of the teleamplified state at the origin of phase space (c). The results are shown for even cat-like states (solid blue lines) and odd cat-like state (solid red lines). The parameters of the teleamplifier are $\lambda =0.8$, $\mu =0$, $\eta _{AB}=0.95$, $T=0.99$, $\eta _{\mathrm {APD}}=0.85$, and $\sigma =0.2$. For comparison, the dashed lines display results for state transfer via deterministic teleportation with the initial Gaussian two-mode squeezed state.

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5. Conclusions

In summary, we have proposed and theoretically analyzed a scheme for noiseless quantum amplification of coherent states of light via probabilistic quantum teleportation with a suitably designed two-mode entangled state. With this approach we can implement noiseless amplifiers similar to those based on combination of conditional single photon addition and subtraction while avoiding the potentially experimentally costly photon addition operation. Instead, the protocol requires auxiliary two-mode squeezed vacuum states and balanced homodyne detection in addition to the conditional single photon subtraction. In its simplest version, the noiseless amplification is achieved with a two-mode squeezed vacuum with single photon subtracted from each of its modes. However, the achievable gain $2\lambda$ is limited by the available squeezing and cannot exceed $2$. More control over the amplification gain can be obtained by utilizing an additional second two-mode squeezed vacuum state with low squeezing $\mu$. With this extended scheme, arbitrary high gain (for low $|\alpha |$) is in principle achievable, but the setup may become sensitive to fluctuations of $\mu$. Therefore, the simplest variant with $\mu =0$ is experimentally most feasible. For $\mu =0$ it could be releatively easy to experimentally target $g_{\mathrm {eff}}=1$, which would enable high-fidelity unity-gain probabilistic teleportation of weak coherent states with teleportation fidelity outperforming deterministic teleportation that would use comparable resources, i.e. the same squeezed or photon subtracted state as used in the noiseless probabilistic teleportation protocol.

The teleportation-based noiseless amplifier requires high purity squeezing to operate properly. During recent years we have witnessed tremendous progress in generation of highly pure strongly squeezed states of light, see. e.g. [36–40]. For instance, in Ref. [38], generation of squeezed state with $10$ dB of squeezing and only $11$ dB of anti-squeezing is reported. In our numerical simulations, we have considered lower squeezing for which even higher purity is achievable. Imperfections of single photon detectors mainly limit the success probability of the protocol, while the quality of the noiseless amplifier is preserved if beam splitters with sufficiently low reflectance $R$ are employed. With recent significant development of superconducting single photon detectors, quantum detection efficiencies exceeding $90\%$ become available, with recently reported system detection efficiency of $98\%$ [51]. Furthermore, the spatial or temporal multiplexing can be used to turn on-off detectors into approximate photon number resolving detectors [52], which in combination with higher reflectances $R$ of the tapping beam splitters could significantly increase the success probability of photon subtraction while preserving high quality of the conditionally generated state.

Finally, we would like to point out that the teleportation of quantum gates can be applied also to other important elementary conditional continuous-variable quantum operations such as single photon addition. This requires as a resource state a two-mode squeezed vacuum state with a single photon subtracted from mode A,

For arbitrary input state $|\psi \rangle _C$, conditioning on $\beta =0$ in the teleportation scheme in Fig. 3, the conditionally prepared output state in mode $B$ becomes $\hat {b}^\dagger \lambda ^{\hat {n}}|\psi \rangle _B$. Similarly to the teleportation-based noiseless amplification, conditioning on finite range of measurement outcomes $\beta$ will unavoidably introduce some noise and imperfections into the implemented conditional operation. Nevertheless, the effective replacement of photon addition by photon subtraction can make this approach appealing for certain applications.