Figure 1

Ball-and-stick phase-space depictions of input and output noise for ideal (${\mu }^2=1$), perfect (${\mu }^2=\frac{1}{2}$), and immaculate (${\mu }^2=0$) amplifiers defined by the amplifier map (2.9) with initial ancilla “state” (2.13). Color and fill conventions: Solid (purple) fill is used for input noise; (red) fill with slanted lines for the output noise of an ideal amplifier; (blue) fill with dots for the output of a perfect amplifier; and solid (green) fill for the output of an immaculate amplifier. The primary-mode input is a coherent state $|\alpha \rangle$ with $\left|\alpha \right|=1$, and the gain is $g=4$, giving the output state a mean that lies on a circle of radius $g\left|\alpha \right|=4$. The input and output states are represented by noise circles centered at the mean complex amplitude (the stick) and having radius $\Sigma /2\sqrt{2}$ (the ball), where ${\Sigma }^2={\langle \left|\Delta \alpha \right|}^2\rangle$ is the variance of the complex amplitude calculated from the appropriate quasidistribution: for the normal ordering of the $P$ function, ${\Sigma }_P^2=\langle \Delta a^{†}\Delta a\rangle$; for the symmetric ordering of the Wigner $W$ function, ${\Sigma }_W^2=\frac{1}{2}(\langle \Delta a^{†}\Delta a\rangle +\langle \Delta a\Delta a^{†}\rangle )={\Sigma }_P^2+\frac{1}{2}$; for the antinormal ordering of the $Q$ distribution, ${\Sigma }_Q^2=\langle \Delta a\Delta a^{†}\rangle ={\Sigma }_W^2+\frac{1}{2}$. The $P$-function depiction is the one suggested by the amplifier map (2.9): The dot (${\Sigma }_P=0$) for the input coherent state $|\alpha \rangle$ is amplified by an immaculate amplifier to a dot for the output coherent state $|g\alpha \rangle$; the output for a perfect amplifier has additional noise ${\Sigma }_P^2=\frac{1}{2}(g^2-1)$, and the output for an ideal amplifier has additional noise ${\Sigma }_P^2=g^2-1$. The symmetrically ordered moments of the Wigner $W$ function give the traditional picture of amplifier noise: the input coherent state, represented by a circle corresponding to ${\Sigma }_W^2=\frac{1}{2}$, has its noise amplified by a perfect amplifier along the (gray) radial lines to the circle with ${\Sigma }_W^2=\frac{1}{2}{g}^2$; the output of an ideal amplifier has additional noise $\frac{1}{2}(g^2-1)$, giving total noise ${\Sigma }_W^2=g^2-\frac{1}{2}$, and the output of an immaculate amplifier has its noise reduced by $\frac{1}{2}(g^2-1)$ to the coherent-state value ${\Sigma }_W^2=\frac{1}{2}$. The antinormally ordered moments of the Husimi $Q$ distribution give a picture suited to discussion of simultaneous measurements of the quadrature components (see text): The input coherent state, represented by a circle corresponding to ${\Sigma }_Q^2=1$, has its noise amplified by an ideal amplifier along the (gray) radial lines to a circle with ${\Sigma }_Q^2=g^2$; the output of a perfect amplifier has less noise by $\frac{1}{2}(g^2-1)$, giving total noise ${\Sigma }_Q^2=\frac{1}{2}(g^2+1)$, and the output of an immaculate amplifier has its noise reduced by $g^2-1$ to the coherent-state value ${\Sigma }_Q^2=1$.Reuse & Permissions

Figure 2

Device that approximates an immaculate amplifier (figure based on Fig. 1 of Ref. [25]). An incident coherent state is split equally into $N$ modes at an $N$-port splitter. The state of each mode is a coherent state $|{\alpha }^{\prime }\rangle$, where ${\alpha }^{\prime }=\alpha /\sqrt{N}$; $N$ is chosen large enough that ${\alpha }^{\prime }=\alpha /\sqrt{N}\ll 1$, so that $\left|{\alpha }^{\prime }\right.〉=\left|0\right.〉+{\alpha }^{\prime }\left|1\right.〉+O\left(\right|{\alpha }^{\prime }{|}^2)$ is well approximated by its vacuum and one-photon pieces. Each of the $N$ modes enters a modified “quantum scissors” (MQS) [26], shown on the right, which is the heart of the amplifier. When the two detectors in the MQS get results 1,0 or 0,1, the MQS is said to work; a feedforward phase shift (FPS) by $\pi$, controlled on one of the two outcomes, is applied to the device's output mode. The result of these manipulations is that, conditioned on the MQS working, it implements the transformation $\left|{\alpha }^{\prime }\right.〉\to (1+g{\alpha }^{\prime }{a}^{†})\left|0\right.〉={\left|g{\alpha }^{\prime }\right.〉}_{\mathrm{trunc}}$, i.e., truncation of the state to the vacuum–one-photon sector and change of the relative weights of the vacuum and one-photon contributions so that the one-photon weight is increased; the gain is determined by the transmissivities and reflectivities of the beam splitters in the MQS. The amplified and truncated states, ${\left|g{\alpha }^{\prime }\right.〉}_{\mathrm{trunc}}$, are recombined at a second $N$-port splitter. Conditional on detecting vacuum in $N-1$ outputs of this splitter, the output mode is in the amplified state $\left|g\alpha \right.〉$ in the limit that $N\to \infty$. Successful immaculate amplification thus corresponds to heralding on the desired outcome of all of the MQSs, as well as vacuum detection in the $N-1$ ports of the final $N$-port splitter.Reuse & Permissions

Figure 3

Dependence of the ratio ${\overline{\alpha }}^2/M^2=a\left(\epsilon \right)$ on the deviation $\epsilon$ of the success probability $P\left(✓\right|\overline{\alpha },\mathrm{M})$ from unity: Numerical results are plotted as (red) circles; analytic approximation of Eq. (5.29) as (blue) squares. The analytic approximation works quite well for $\epsilon \in [0,0.5]$, but breaks down progressively beyond $\epsilon =0.5$.Reuse & Permissions

Figure 4

Success probability $P\left(✓\right|\overline{\alpha },\mathrm{M})={\mathrm{P}}^{\mathrm{B}}\left(✓\right)$. (Left column) As a function of $M$ with fixed ${\overline{\alpha }}^2$; (black) asterisks are the exact, numerically determined success probability (5.18); (red) circles give the approximate result (5.21) for many coherent states (small coherent-state amplitude); (blue) squares give the approximate result (5.28) for sparse coherent states (large coherent-state amplitude). (Right column) As a function of ${\overline{\alpha }}^2$ with fixed $M$; (black) solid line is the exact result; (red) dashed line, many coherent states; (blue) dotted line, sparse coherent states.Reuse & Permissions

Figure 5

Fidelity $F_k\left(\overline{\alpha }\right)$ of Eq. (6.30) (descending curves) and success probability $p_k\left(✓\right|\overline{\alpha })$ of Eq. (6.29) (ascending curves) plotted as functions of input amplitude $\overline{\alpha }$ for different extended Kraus operators ${{\rm Y}}_k$ with $k=0$ (solid lines), 1 (dashed lines), and 2 (dotted lines): (a) $g=\sqrt{2}$, $N=4$; (b) $g=3$, $N=9$. The inset in (b) illustrates the small differences in fidelity, undetectable in the main plot, among the three values of $k$.Reuse & Permissions

Figure 6

Fidelity $F_0\left(\overline{\alpha }\right)$ using the extended Kraus operator ${{\rm Y}}_0$ [Eq. (6.30)] (solid line with solid circles); corresponding success probability $p_0\left(✓\right|\overline{\alpha })$ [Eq. (6.29)] (dashed line with solid squares); fidelity $F\left(\overline{\alpha }\right)$ using the restricted Kraus operator $P_N{K}_0$ [Eq. (6.23)] (solid descending line); corresponding restricted success probability $p\left(✓\right|\overline{\alpha })$ [Eq. (6.22) with $k=0$] (dashed line); probability-fidelity product $p_0\left(✓\right|\overline{\alpha }){\mathrm{F}}_0\left(\overline{\alpha }\right)$ (solid humped line); and overlap ${\left|\langle \alpha |g\alpha \rangle \right|}^2$ (dotted line), all plotted as functions of input amplitude $\overline{\alpha }$: (a) $g=\sqrt{2}$, $N=2$; (b) $g=\sqrt{2}$, $N=4$; (c) $g=3$ $N=9$.Reuse & Permissions

Figure 7

$Q$ distribution of the output state of the immaculate linear amplifier given by the extended Kraus operator ${{\rm Y}}_0$, with $g=3$ and $N=9$, for four amplitudes of input coherent state: (a) $\overline{\alpha }=0.5$, (b) $\overline{\alpha }=1.5$, (c) $\overline{\alpha }=3$, (d) $\overline{\alpha }=5$. The (red) dot denotes the center of the input coherent state. The transition at input radius $\sqrt{N}/g=1$ is marked by a (red) arc, and its image at the output by the (black) arc at radius $\sqrt{N}=3$. Thus (a) lies within the high-fidelity region, and the output looks like an amplified coherent state; (b) lies beyond the transition, and its output is flattened along the arc of radius $\sqrt{N}$. A second transition occurs near $\overline{\alpha }\simeq \sqrt{N}$, as ${{\rm Y}}_0$ transitions to being the identity operator. Thus (c), lying right in the middle of this second transition, has output that is little amplified and is flattened along the radial direction, whereas (d), lying well beyond the second transition, has output that is nearly identical to the input coherent state.Reuse & Permissions

Figure 8

Antinormally ordered quadrature SNRs as a function of the input amplitude $\overline{\alpha }$. The SNR is defined as ${\text{SNR}}_1=〈x_1〉/\delta x_1$ for the amplitude (radial) quadrature $x_1$ or as ${\text{SNR}}_2=〈x_1〉/\delta x_2$ for the phase quadrature $x_2$. Four of the plots are for (i) the input state $\left|\alpha \right.〉$ (dotted line), for which ${\text{SNR}}_1={\text{SNR}}_2$ [this is also the bound given in Eq. (2.21)]; (ii) the output target state $\left|g\alpha \right.〉$ (solid line), for which ${\text{SNR}}_1={\text{SNR}}_2$; and (iii) and (iv) the output state of the ${{\rm Y}}_0$ immaculate amplifier (SNR${}_1$, solid line with crosses; SNR${}_2$, solid line with circles). The other two plots give the amplifier SNRs multiplied by the square root of the working probability, $\sqrt{p_0\left(✓\right|\overline{\alpha })}$, as described in the text: The dashed line with crosses plots $\sqrt{p_0\left(✓\right|\overline{\alpha })}{\text{SNR}}_1$, and the dashed line with circles plots $\sqrt{p_0\left(✓\right|\overline{\alpha })}{\text{SNR}}_2$. For the amplifier plots, (a) has $g=\sqrt{2}$, $N=2$, and (b) has $g=3$, $N=9$.Reuse & Permissions

Figure 9

Number-based SNR measure as a function of the input amplitude $\overline{\alpha }$. The four plots are ${\text{SNR}}_N$ for the input state $\left|\alpha \right.〉$ (dotted line); ${\text{SNR}}_N$ for the output target state $\left|g\alpha \right.〉$ (solid line); ${\text{SNR}}_N$ for the output state of the ${{\rm Y}}_0$ immaculate amplifier (solid line with crosses); and the root-probability–SNR measure $\sqrt{p_0\left(✓\right|\overline{\alpha })}{\text{SNR}}_N$ for the output state of the ${{\rm Y}}_0$ immaculate amplifier (dashed line with crosses). For the amplifier plots, (a) has $g=\sqrt{2}$, $N=2$, and (b) has $g=3$, $N=9$.Reuse & Permissions