Quantum enhancement of signal-to-noise ratio with a heralded linear amplifier

Jie Zhao; Josephine Dias; Jing Yan Haw; Thomas Symul; Mark Bradshaw; Rémi Blandino; Timothy Ralph; Syed M. Assad; Ping Koy Lam

doi:10.1364/OPTICA.4.001421

1. INTRODUCTION

The question of quantum noise in linear amplifiers has stirred considerable interest not only because of its technical significance, but also owing to its intimate connection with the most fundamental features of quantum theory. A perfect linear amplifier (PLA) increases the power of an incoming signal without introducing a degradation to its signal-to-noise ratio (SNR). This is achievable easily for classical signals. However, in the quantum world, a PLA cannot function deterministically. Due to the bosonic nature of photons, an optical amplifier unavoidably introduces noise to any signal it processes. The noise penalty arises from the interaction between the initially independent input mode and the internal modes of an amplifier. This quantum property of amplifiers was theoretically elucidated by Haus and Mullen [1] and was quantitatively expressed as the amplifier uncertainty principle [2]. In particular, for a phase-insensitive amplifier, the minimum amount of additional noise is equivalent to $| G - 1 |$ units of vacuum noise, where $G$ denotes the power gain for the input signal. This noise penalty prevents the increase of distinguishability of quantum states under amplification. It therefore ensures that by means of the amplify-and-split approach [3], two orthogonal quadrature amplitudes of a bosonic mode cannot be measured simultaneously with arbitrary precision, in compliance with the Heisenberg uncertainty principle.

One way to circumvent the excess noise is to instead apply phase-sensitive amplification. One such example is to utilize an optical parametric amplifier to squeeze either the input mode or the internal mode such that the amplified output has reduced noise in one quadrature at the expense of degrading the conjugate quadrature [4–7]. Besides, phase-sensitive amplification can also be realized using a series of light emitter detectors in conjunction with high-quantum-efficiency photodetectors [8]. This device can achieve, in principle, a signal transfer limited only by the photodetector efficiency ( ${SNR}_{out} / {SNR}_{in} \approx η_{d}$ , where $η_{d}$ is the quantum efficiency of the photodetector) for a sufficiently large number of emitters. However, while the intensity of light is amplified, all phase information is destroyed. Another method of low-noise amplification is to use an electro-optic feed-forward loop [9]. The setup avoids the requirement of a nonlinear optical process, and due to the fact that not all of the input light is destroyed, some of the phase information can be retained.

If one demands an amplification of both quadratures equally, an alternative way to evade the noise penalty is to allow a probabilistic operation. Fiurášek proposed a probabilistic amplification method that could be applied to coherent states of fixed amplitude but unknown phase [10]. Ralph and Lund extended this idea and proposed independently a noiseless linear amplifier (NLA) [11] that could in principle be applied to arbitrary ensembles. This amplifier outperforms the perfect linear amplifier by preserving the noise characteristic of the input state, and is hence, from a classical point of view, a noise-reduced amplifier, as illustrated in Fig. 1. The price to pay is that the process has to be probabilistic and approximate in terms of the output states produced. A better approximation is attainable at the expense of a lower success probability [11,12]. This compromise guarantees that, on average, the Heisenberg’s uncertainty relation remains satisfied. Nevertheless, the successfully amplified quantum states can be heralded and thus are valuable in extending the range of loss-sensitive protocols. We note that given the access to correlated inputs, it is possible to achieve phase-insensitive noise reduction deterministically via the cancellation of the internal noise, albeit the total SNR is, at best, conserved [13]. This entanglement-based quantum amplifier was demonstrated in [14] and applied in [15] as an enhanced interferometer for phase estimation.

Fig. 1. Wigner function contours of input and output coherent states for a continuum of linear amplifiers. The green dashed circle here refers to the best possible deterministic linear amplifier, which adds the minimum amount of noise imposed by quantum mechanics; any amplifier that introduces less noise is necessarily probabilistic. One example of the probabilistic amplifiers is the perfect linear amplifier (PLA) that preserves the SNR of an incoming signal while amplifying its power. Amplifiers capable of enhancing SNR are called noise-reduced amplifiers (shaded area in orange, including the NLA) and the extreme case of the noise-reduced amplifier is an NLA that not only amplifies the amplitude of an input state, but also preserves its noise characteristics.

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Various physical implementations of NLA have been proposed and experimentally demonstrated, including the quantum scissor setup [16–19] as well as the photon-addition and photon-subtraction [20,21] and noise addition [22] schemes. In all these approaches, a large truncation is often imposed on the unbounded amplification operator in the photon-number basis. The high-fidelity operating region of the amplification is consequently restricted to small input amplitude and small gains [23,24]. The current realizations require nonclassical light sources and non-Gaussian operations like photon counting, thereby rendering their application to many systems and protocols very challenging. Intriguingly, as recently proposed [25,26] and experimentally demonstrated [27], the benefits of noise-reduced amplification can be retained via classical post-processing, provided that the NLA precedes a dual homodyne measurement directly. Although the simplicity of this measurement-based noiseless linear amplifier (MB-NLA) is appealing, its post-selective nature confines it to point-to-point applications such as quantum key distribution. To overcome this drawback, the concatenation of an MB-NLA and a deterministic linear amplifier (DLA) and yet outputs a quantum state was proposed recently and studied in the context of quantum cloning [28], where the production of clones with fidelity surpassing the deterministic no-cloning bound was demonstrated.

In the current paper, we realize for the first time a quantum enhancement of SNR for arbitrary coherent states and amplification gains using a heralded noise-reduced linear amplifier. This amplifier combines the advantages of a DLA and an MB-NLA. Owing to the fully tunable cutoffs and independent control of the NLA and DLA gains, great versatility in the effective gain and in the input amplitude is attained, mitigating therefore the undesirable constraints in previous physical implementations. We show a signal transfer of 110% from input to output with an amplification gain of 6.18 when Gaussian statistics are maintained. Furthermore, by marginally compromising the Gaussianity of the output state, we demonstrate an SNR enhancement of more than 4 dB for a coherent state amplitude of $| α | = 0.5$ with an amplification gain of 10.54. Unlike the previous measurement-based NLA scheme [27], a heralded and free-propagating amplified state is produced with our amplifier. It is worth stressing that the setup uses Gaussian elements and a post-selection algorithm only, and hence has a better compatibility with other continuous-variable protocols.

2. THEORY

A. Conceptual Scheme

Here we study the behavior of our amplifier on both an ensemble of coherent states and a single coherent input. Its behavior is dominated by the interfacing between the two intrinsically different amplifiers. A larger DLA gain would contribute to a higher success probability, but also introduce a larger noise penalty, while a larger NLA gain is the requisite to attain an increase of SNR, however, at the expense of reducing the success probability.

The effect of our linear amplifier on an unknown input state ${\hat{ρ}}_{in}$ is to transform the state as follows:

{\hat{ρ}}_{out} = N {Tr}_{v} {{\hat{U}}_{g_{DLA}} g_{NLA}^{\hat{n}} {\hat{ρ}}_{in} \otimes | 0 ⟩ {⟨ 0 |}_{v} g_{NLA}^{\hat{n}} {\hat{U}}_{g_{DLA}}^{†}},

where the constant

N

is a normalization factor. The operator

g_{NLA}^{\hat{n}}

here models the action of the NLA on the input density operator, while

{\hat{U}}_{g_{DLA}} = e^{- θ (\hat{a} {\hat{a}}_{v} - {\hat{a}}^{†} {\hat{a}}_{v}^{†})}

is a unitary transformation acting on the input mode and an ancillary vacuum mode that models the action of the DLA. The parameter

θ

relates to the gain of the DLA via

g_{DLA} = \cosh (θ)

. The ancillary mode is traced out to give the final output. We can characterize the outcome of this interaction by considering the expectation value of an observable

\hat{M} (\hat{a}, {\hat{a}}^{†})

,

⟨ \hat{M} ⟩ = Tr {\hat{M} {\hat{ρ}}_{out}} = Tr {\hat{M} {\hat{U}}_{g_{DLA}} g_{NLA}^{\hat{n}} {\hat{ρ}}_{in} g_{NLA}^{\hat{n}} {\hat{U}}_{g_{DLA}}^{†}} = Tr {{\hat{M}}_{DLA} {\hat{ρ}}_{NLA}},

where we use the cyclic permutation of the trace and

{\hat{M}}_{DLA} = {\hat{U}}_{g_{DLA}}^{†} \hat{M} {\hat{U}}_{g_{DLA}}

,

{\hat{ρ}}_{NLA} = g_{NLA}^{\hat{n}} {\hat{ρ}}_{in} g_{NLA}^{\hat{n}}

.

We first consider the input ${\hat{ρ}}_{in}$ to be an ensemble comprised of a Gaussian distribution of coherent states,

{\hat{ρ}}_{in} (λ) = \frac{1}{π} \frac{1 - λ^{2}}{λ^{2}} \int d^{2} α e^{- \frac{1 - λ^{2}}{λ^{2}} {| α |}^{2}} | α ⟩ ⟨ α |,

where

λ

(

0 \leq λ < 1

) relates to the variance of the distribution by

V = \frac{1 + λ^{2}}{1 - λ^{2}}

. Due to the linearity of the NLA operator, the distribution

{\hat{ρ}}_{in} (λ)

changes as

g_{NLA}^{\hat{n}} {\hat{ρ}}_{in} (λ) g_{NLA}^{\hat{n}} \propto \hat{ρ} (g_{NLA} λ)

under noiseless linear amplification [29]. That is, if Alice sends a distribution of coherent states of width

λ

, the conditional state after the successful operation of NLA is proportional to a distribution of width

g_{NLA} λ

. Correspondingly, the variance of the ensemble of coherent states becomes

V = \frac{1 + g_{NLA}^{2} λ^{2}}{1 - g_{NLA}^{2} λ^{2}}

. We note that for the amplified distribution to be physical,

g_{NLA}^{2} λ^{2}

must be less than one. The state

{\hat{ρ}}_{NLA} (g_{NLA} λ)

is then amplified by the DLA to give the final output state. The expectation value of an arbitrary observable

M ({\hat{a}}_{out}, {\hat{a}}_{out}^{†})

can then be constructed using [2,30]

{\hat{a}}_{out} = {\hat{U}}_{g_{DLA}}^{†} {\hat{a}}_{in} {\hat{U}}_{g_{DLA}} = {\hat{a}}_{in} g_{DLA} + {\hat{a}}_{v}^{†} \sqrt{g_{DLA}^{2} - 1} .

So far, we have described how an ensemble of coherent states evolves by our amplifier. We now examine the action of our amplifier on each individual coherent state $| α ⟩$ .

The NLA probabilistically amplifies the complex amplitude of an input coherent state $| α ⟩$ to $| g_{NLA} α ⟩$ with a gain $g_{NLA} > 1$ . The DLA then performs the deterministic transformation as shown in Eq. (4). The mean of the amplitude ${\hat{X}}_{+} = \hat{a} + {\hat{a}}^{†}$ and phase ${\hat{X}}_{-} = - i (\hat{a} - {\hat{a}}^{†})$ quadratures of the electric field are therefore amplified by

{⟨ {\hat{X}}_{\pm} ⟩}_{out} = g_{NLA} g_{DLA} {⟨ {\hat{X}}_{\pm} ⟩}_{in} .

To quantify the amplification of the signal, we define $g_{eff} = g_{NLA} g_{DLA}$ as the effective gain. Since the NLA incurs no additional noise, the overall output noise is only a function of the DLA gain (where the quantum noise level is 1),

{⟨ {(δ {\hat{X}}_{\pm})}^{2} ⟩}_{out} = 2 g_{DLA}^{2} - 1 .

B. Equivalent Experimental Scheme

Figure 2 shows the experimental scheme of the amplifier, where an input coherent state is first fed through a beam splitter with a transmissivity of (see more details in Supplement 1)

T^{'} = g_{NLA}^{2} / g_{DLA}^{2} .

Fig. 2. Experimental schematic of our heralded noise-reduced linear amplifier achieved with a feed-forward loop. HD, homodyne measurement; EOM, electro-optic modulator.

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The reflected mode is then subjected to a dual-homodyne setup locked to simultaneously measure two conjugate quadratures. An MB-NLA, consisting of a filter function and a rescaling factor, amplifies the mean of the measured statistics by $g_{NLA}^{'}$ without changing its noise feature. More specifically, a probabilistic Gaussian filter given by

P (α_{m}) = {\begin{matrix} \exp ({| α_{m} |}^{2} - {| α_{c} |}^{2}) (1 - \frac{1}{{(g_{NLA}^{'})}^{2}}) & if | α_{m} | \leq α_{c} \\ 1 & otherwise \end{matrix}

is applied to the measurement outcomes

α_{m} = (x_{m} + i p_{m}) / \sqrt{2}

of the dual-homodyne station. The cutoff parameter

α_{c} > 0

acts as the truncation on the working phase space of the unbounded amplification operator. More specifically, all measurement outcomes

α_{m}

with magnitude less than

α_{c}

are selected or rejected with probability specified in Eq. (8), while those

α_{m}

falling beyond the cutoff amplitude are kept with unit probability [31]. The cutoff,

α_{c}

, therefore determines how closely the filter approximates an ideal NLA and also the success probability of the protocol. This filter function heralds the successful amplification and over-amplifies both the mean and the variance of the measured statistics by

g_{NLA}^{' 2}

. Thus, to retrieve the target mean and eliminate the additional noise, a rescaling factor of

1 / g_{NLA}^{'}

is applied to the filtered statistics. The entire functionality of the measurement-based NLA is symbolized as the tunable gain

g_{NLA}^{'}

in Fig. 2. The output signal of the measurement-based NLA is further amplified electronically by

g_{DLA}^{'} = \sqrt{2 (g_{DLA}^{2} - 1)}

and coupled to the transmitted input beam to fulfill the displacement operation (the relationships between

g_{NLA}^{'}

,

g_{DLA}^{'}

and

g_{NLA}

,

g_{DLA}

are addressed in more detail in Supplement 1).

The output mean and variance of the quadrature amplitudes can be derived as

{⟨ {\hat{X}}_{\pm} ⟩}_{out} = (\sqrt{\frac{1 - T^{'}}{2}} g_{DLA}^{'} g_{NLA}^{'} + \sqrt{T^{'}}) {⟨ {\hat{X}}_{\pm} ⟩}_{in},

{⟨ {(δ {\hat{X}}_{\pm})}^{2} ⟩}_{out} = 1 + {(g_{DLA}^{'})}^{2} .

We quantify the performance of our amplifier by introducing the signal transfer coefficient,

T_{s} = {SNR}_{out} / {SNR}_{in},

which is equal to

g_{eff}^{2} / (2 g_{DLA}^{2} - 1)

for a quantum-limited amplification [30] and is larger than 1 for a noise-reduced operation. From Eqs. (9) and (10), we obtain the theoretical

T_{s}

for both quadratures for our setup,

T_{s} = \frac{{(\sqrt{\frac{1 - T^{'}}{2}} g_{DLA}^{'} g_{NLA}^{'} + \sqrt{T^{'}})}^{2}}{1 + {(g_{DLA}^{'})}^{2}} .

Another performance metric—the success probability $P_{S}$ —is also calculated (see Supplement 1) so as to define the operating region where the amplification is experimentally feasible.

The key features of our hybrid linear amplifier are threefold: first, the output is a free-propagating amplified physical state; second, the setup only depends on linear optics; third, the two cascading gains can be tuned independently, and our cutoff is fully adjustable. This introduces more flexibility in optimizing the success rate while preserving high fidelity with an ideal implementation of NLA. It also largely extends the operating region of the amplifier by alleviating the constraints of previous physical implementations where amplification is confined to small input amplitudes and low amplification gains.

Figure 3 illustrates the operational degrees of freedom of our noise-reduced linear amplifier. The amount of noise reduction depends on both the product and the ratio of $g_{NLA}$ and $g_{DLA}$ , which correspond to, respectively, the values of the effective gain $g_{eff}$ and the transmittivity $T^{'}$ in Fig. 2. Intuitively, for a fixed effective gain $g_{eff}$ , a higher signal transfer coefficient $T_{s}$ is obtainable with a larger $g_{NLA}$ , since the associated noise determined by $g_{DLA}$ decreases while the input amplitude undergoes the same amount of amplification. Hence, under the same effective gain, a higher $T^{'}$ would always lead to a larger signal transfer coefficient.

Fig. 3. Tunability of the amplifier. Signal transfer coefficient (blue contours), various effective gains (red contours), and $T^{'}$ (green lines) as the function of $g_{NLA}$ and $g_{DLA}$ . The blue-dotted line denotes the amplification process where the input SNR is preserved, while the enclosed shaded area refers to the region where additional noise is introduced. We note that, without a sufficiently high NLA gain, increasing $g_{DLA}$ alone would not suffice to approach the noise-reduced amplification.

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We note that there is an ultimate limit of our current setup embodied in Eq. (7). Since $T^{'} < 1$ , $g_{NLA}$ must be smaller than $g_{DLA}$ . This, in terms of the effective gain, poses a limit on the signal transfer as

T_{s} < \frac{g_{eff}^{2}}{2 g_{eff} - 1}

(see Supplement 1). Nonetheless, as shown in Fig. 3, an arbitrarily high

T_{s} > 1

is attainable using the current setup by applying well-tailored

T^{'}

and

g_{eff}

.

3. EXPERIMENTAL SETUP

The light source for this experiment is an Nd:YAG laser producing continuous-wave single-mode light at 1064 nm. The coherent state at a sideband frequency is generated by sending modulation signals at 4 MHz to a pair of electro-optical modulators (EOMs) on the signal beam. The laser was found to be shot-noise-limited at this frequency, and the amplitudes of the modulation signals determine the complex amplitude of the coherent state. To amplify the coherent state, we first inject the input state into a beam splitter with transmittivity of $T^{'}$ where it is split to the transmitted and reflected modes. A dual-homodyne measurement is then performed on the reflected mode, and the measurement outcomes serve two purposes. First, they are used to extract the 4-MHz modulation and to reveal the term $| α_{m} |$ in Eq. (8), which is used to provide the heralding signal. To this end, the outcome is demodulated by mixing it with an electronic local oscillator, before being low-pass filtered at 100 kHz and oversampled on a 12-bit analog-to-digital converter at 625 kSa per second. Second, the outcomes of the dual-homodyne measurement are also employed to accomplish the feed-forwarding. They are amplified electronically with a gain $g_{ele} = g_{DLA}^{'} / g_{NLA}^{'}$ and fed into a pair of EOMs modulating a bright auxiliary beam. This intense beam is then coupled in phase with the transmitted signal beam by an asymmetric beam splitter of transmissivity 98% to realize the displacement operation.

The combined beam is then characterized by a homodyne measurement, locked alternatively to amplitude and phase quadratures. The homodyne measurement goes through the same signal processing, and at least $5 \times 10^{7}$ data points are acquired.

4. RESULTS

A. Linearity of the Amplifier

Figure 4 shows the performance of our linear amplifier for input coherent states with different complex amplitudes $| (x + i p) / 2 ⟩$ . As illustrated in Fig. 4(a), we demonstrate the phase-preserving property of the amplifier and observe the symmetric noise spectrum of the amplitude and phase quadratures. These results emerge from the linearity and phase invariance of the present setup, as is also clearly demonstrated in Fig. 4(b). In particular, under the same $T^{'}$ (0.6), by selecting input states with different complex amplitudes $(x, p) = (- 0.71, 0.72), (- 0.01, - 1.51), (2.23, 2.19), (5.26, - 0.02)$ , we plot the output magnitudes against the input magnitudes as we vary the cutoff values, or alternatively as we vary the effective gains. The amplifier behaves linearly in either circumstance, thus verifying the independence of the amplification on the input states.

Fig. 4. Linearity of the amplifier. (a) Amplification for coherent states with different amplitudes. Left panels: noise contours (one standard deviation width) of the amplified states, depicted in red. Right panels: normalized probability distribution for amplitude and phase quadratures of the output states. (b) Output magnitudes versus input magnitudes as we reduce the cutoff while maintaining the values of $g_{NLA}^{'}$ and $g_{D LA}^{'}$ . Inset: output magnitudes versus input magnitudes with cutoff $α_{c} = 4.42$ at different effective gains.

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Apart from the relationship between $| α_{out} |$ and $| α_{in} |$ shown in Fig. 4, we also notice that as we reduce the cutoff, the output states start to exhibit nonuniform noise between the in-phase and out-of-phase fluctuations. More specifically, as the cutoff is decreased from 4.42 to 0.50, we observe the output noise $[{⟨ {(δ {\hat{X}}_{+})}^{2} ⟩}_{out}, {⟨ {(δ {\hat{X}}_{-})}^{2} ⟩}_{out}]$ reduces from [1.83, 1.83] to [1.59, 1.70] for input $(x, p) = (- 0.01, - 1.51)$ and from [1.87, 1.86] to [1.70, 1.58] for input $(x, p) = (5.26, - 0.02)$ . In these cases, the cutoff with respect to the effective gain no longer suffices to preserve the Gaussianity, i.e., a Gaussian probability distribution, of the output state, and the amplified states start to squash along the radial direction [23]. This produces mixed, non-Gaussian states. Nevertheless, the amplification remains phase-insensitive due to the fact that it is always the variance of the quadrature along the radial direction that becomes classically “squeezed,” while that of the orthogonal quadrature inclines to be anti-squeezed. Interestingly, it is worth emphasizing that, even in this operating region, the amplifier still works linearly regardless of the insufficient cutoff [refer to the light blue line in Fig. 4(b)]. This special property would be of great benefit for coherent states discrimination. For example, consider multiple weak coherent states in a quadrature phase-shift-keyed format [32] as inputs of our linear amplifier. Regardless of the phases of the input states, the amplifier increases their complex amplitudes consistently and, meanwhile, suppresses the added noise along the radial direction. The amplification works conditionally, whereas as long as a heralding signal reveals that the amplification is successful, the distinguishability of these states would be enhanced.

B. Quadrature Independence and the Success Probability

Figure 5 demonstrates the tunability and versatility of our amplifier. The signal transfer coefficients of the amplitude and phase quadratures, superimposed by the success probability, are plotted as a function of increasing effective gains. We examine an input coherent state with complex amplitude of $(x, p) = (1.51, 1.54)$ for all plots. Two different transmissions, $T^{'} = 0.60$ and 0.45, are picked to test the amplifier in different settings.

Fig. 5. Amplifier performance: noise properties and contour plot of the success probability $P_{S}$ in logarithmic scale as a function of $T_{s}$ and $g_{eff}$ . The success probability decreases as we increase the effective gain. The signal transfer coefficient for amplitude quadrature (blue symbols) is also superimposed as a function of $g_{eff}^{2}$ for varying $T^{'}$ : 0.6, 0.45. The theoretical prediction, assuming infinite cutoff, is depicted in crosses. It is clearly shown that the experimental $T_{s}$ increases in compliance with the prediction, demonstrating that the cutoff ( $α_{c} = 4.3$ ) selected is sufficient and no over- or underestimation of $T_{s}$ appears. For the sake of comparison, the best achievable $T_{s}$ of an optimal deterministic linear amplifier (also termed as the quantum noise limit) is shown in the orange solid line. This illustrates that our amplifier surpasses the quantum limit for a phase-insensitive amplifier, and this superiority becomes more distinguished as we increase the $g_{eff}$ . Inset: the experimental data superimposed with its theoretical prediction for phase quadrature.

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In accordance with Fig. 3, data points with the same $T^{'}$ illustrate evidently the improvement of $T_{s}$ as $g_{eff}$ increases, corresponding to moving along the green lines in Fig. 3. Alternatively, when keeping $T_{s}$ constant, lowering $T^{'}$ results in a smaller success probability, which also coincides with Fig. 3, because this decrease of the success probability results from the increase of $g_{NLA}$ .

We note that all $T_{s}$ , for both amplitude and phase quadratures, exceed the quantum limit regardless of the values of $T^{'}$ , among which the maximum achieved $T_{s}$ are $0.830 \pm 0.025$ and $0.860 \pm 0.024$ for $T^{'} = 0.6$ and $T^{'} = 0.45$ , respectively. These results significantly surpass the maximum allowable signal transfer in the deterministic regime [c.f. Eq. (12)] by around 10 and 12 standard deviations, respectively. All the observed values of $T_{s}$ show good agreement with the theoretical model, assuming infinite cutoff and taking into account the experimental imperfections (see Supplement 1). The corresponding success probability ranges are between $10^{- 3}$ and 0.3, rendering the amplifier still relatively practicable. Slight discrepancies are observed between $X_{+}$ and $X_{-}$ owing to the different losses experienced by the two quadratures during feed-forwarding (see Fig. S2, Supplement 1). Small deviations of the experimental data from the prediction are attributed to other in-line electronic noise.

C. High Signal Transfer Coefficients

In Fig. 6, we summarize our experimental results when our amplifier is operating in the large gain domain for an input state $(x, p) = (0, 1.01)$ ( $| α | = 0.5$ ) and $T^{'} = 0.155$ . As is shown in Fig. 6(a), a higher $T_{s}$ is obtained at the expense of a lower success probability. For $g_{eff} < 8.5$ , we see that the increasing of $T_{s}$ as a function of $g_{eff}$ coincides with the theoretical model based on an infinite cutoff (see Supplement 1), indicating that the cutoff employed is sufficient to encompass the amplified distribution and thus exclude any distortion of the output. In this high-fidelity operating region, a $T_{s}$ larger than 1 (specifically, $1.10 \pm 0.04$ ) is observed, thus verifying a clear fulfillment of the noise-reduced amplification. As $g_{eff}$ keeps increasing, a wider discrepancy appears between the experimental value of $T_{s}$ and its theoretical prediction, as the result of an insufficient cutoff. In [31], it was shown that the Gaussian profile of the output state of a measurement-based NLA depends on the amplitude of the input state, the NLA gain, and the cutoff in the filter function Eq. (8). If Gaussian-output statistics are desired, the cutoff value can be adjusted according to $g_{NLA}^{'}$ set by the target $T_{s}$ and the maximum input size of the ensemble. To be more explicit, it was proposed in [31] that

α_{c} = {(g_{NLA}^{'})}^{2} | α_{m} | + β \sqrt{0.5} g_{NLA}^{'} .

Fig. 6. Signal transfer coefficients in the large gain domain. (a) $T_{s}$ exceeding 1 with increasing $g_{eff}$ for $α_{c} = 4.5$ and a coherent state amplitude of $| α | = 0.5$ . The experimental $T_{s}$ shows good agreement with the theory plot (in crosses) until around $g_{eff} = 6.5$ , where the data points start to depart, thereby indicating that the cutoff no long suffices to maintain the output Gaussianity. Inset: probability distribution of the amplified state labeled in red. (b) Probability distributions of the phase quadrature of the amplified state with cutoffs given by Eq. (15). Data points are the post-selected ensemble out from $2.7 \times 10^{9}$ homodyne measurements, while the red curves indicate the corresponding best-fitted Gaussian distributions.

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The parameter $β$ quantifies how well is the cutoff circle able to embrace the distribution of the amplified state. We note that a sufficient amount of data points should be retained to characterize the output properly.

To complete the investigation of our setup, we also explore the relationship between $T_{s}$ and the output Gaussianity while keeping the success probability unchanged (around $10^{- 6}$ ), as shown in Fig. 6(b). In this case, as we relax the requirement for the output Gaussianity, it is possible to enjoy a higher effective gain and therefore achieve a considerably larger $T_{s}$ without decreasing the success probability. We experimentally obtained a signal transfer of $T_{s} = 2.55 \pm 0.08$ from input to output with an amplification gain of 10.54.

5. DISCUSSION AND CONCLUSIONS

In this paper, we demonstrate an enhancement of SNR for arbitrary coherent states with a noise-reduced linear amplifier that profitably combines a measurement-based noiseless linear amplifier and a deterministic linear amplifier. We also investigate the possibility of applying our amplifier to an ensemble of coherent states. The hybrid nature of the amplifier retains the flexible and operational characteristics of the measurement-based NLA, which, as opposed to the physical implementations, evades the demand of nonclassical light sources and the restriction to small input states and low amplification gains. It also preserves the free-propagating amplified states and thus circumvents the drawback of a pure measurement-based setup whose output can only be classical statistics. Even though the amplifier works conditionally, a heralding signal is generated for successful events. We circumvent the additional noise associated with a phase-insensitive deterministic optical amplifier with a success probability ranging from 0.1% to 30%. We further demonstrate the superiority of our amplifier over an ideal phase-sensitive amplifier in the amplified quadrature by observing a signal transfer coefficient $T_{s}$ larger than 1, clearly showing that the amplification is noise-reduced. We show that higher $T_{s}$ —more specifically, $T_{s} = 2.56$ with $g_{eff} = 10.54$ —is attainable if one is willing to accept a lower success probability (around $10^{- 6}$ ) or instead to compromise slightly the output Gaussianity. Interestingly, we also notice that there exists an operating region where the amplifier works linearly, regardless of the relatively small distortion of the output. This would provide a useful coherent state discrimination machine.

Owing to the composability, tunability, and ease of implementation of our amplifier, it provides several interesting avenues for future research in loss-sensitive quantum information protocols. First, the access to the two variable knobs—deterministic and probabilistic—provides a spectrum of effective gain and success probability. For protocols with high SNR demands, such as long-distance quantum communication [33], the signal transfer coefficient can be enhanced by intensifying the probabilistic gain. When signal transfer speed is the critical requirement, the deterministic amplification can play the leading role while maintaining the same effective gain. An interesting extension of this work would be to study the optimality of these gains for a given channel loss and excess noise in various quantum communication protocols, including quantum key distribution [29], entanglement distillation [16,21,27], and quantum repeaters [34,35]. Second, by having a priori information about the input alphabet, the amplification cutoff value can be easily tailored to achieve maximum SNR gain for a given success probability. Such is the case for a phase covariant input with fixed amplitude [22] or an input distribution with finite energy [36]. Furthermore, the preservation of the quantum state over the heralded channel opens up the possibility of overcoming deterministic bounds of various feed-forward-based protocols, such as quantum teleportation [37,38], upon suitable modification of our scheme. Lastly, since our scheme only relies upon linear optics and feed-forwarding, it is wavelength agnostic. Hence, it has the potential to improve the transmission distance of optical communication at telecom wavelengths, and in particular to enhance the signal transfer coefficient per distance when applied in conjunction with an ultra-low-loss fiber [39].

Funding

Australian Research Council (ARC) (CE110001027).

Acknowledgment

We thank G. Guccione and Nelly Huei Ying Ng for their helpful discussions.

See Supplement 1 for supporting content.

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Quantum enhancement of signal-to-noise ratio with a heralded linear amplifier

Abstract

1. INTRODUCTION

2. THEORY

A. Conceptual Scheme

B. Equivalent Experimental Scheme

3. EXPERIMENTAL SETUP

4. RESULTS

A. Linearity of the Amplifier

B. Quadrature Independence and the Success Probability

C. High Signal Transfer Coefficients

5. DISCUSSION AND CONCLUSIONS

Funding

Acknowledgment

REFERENCES

Supplementary Material (1)

Cited By

Figures (6)

Equations (15)

Optica