Characterization of a measurement-based noiseless linear amplifier and its applications

A noiseless linear amplifier (NLA) adds no noise to the signals it processes, which works only in a probabilistic way. It can be realized approximately with either a physical implementation that truncates the working space of the NLA on a photon-number basis or a measurement-based implementation that realizes the truncation virtually by a bounded postselection filter. To examine the relationship between these two approximate NLAs, we characterize in detail the measurement-based NLA and compare it with its physical counterpart in terms of their abilities to preserve the state Gaussianity and their probability of success. The link between these amplifiers is further clarified by integrating them into a measure-and-prepare setup. We stress the equivalence between the physical and the measurement-based approaches holds only when the effective parameters, the amplification gain, the cutoff, and the amplitude of the input state, are taken into account. Finally, we construct a 1-to-infinity cloner using the two amplifiers and show that a fidelity surpassing the no-cloning limit is achievable with the measurement-based NLA.

Figure 1

Two equivalent noiseless linear amplifier schemes. (a) An ideal NLA immediately followed by a dual-homodyne measurement. (b) Measurement-based NLA that emulates the NLA by applying Gaussian postselection on the measurement outcomes.

Figure 2

Procedures of an MB-NLA. (a) Distribution of the input state, revealed by a direct dual-homodyne measurement. (b) Probability distribution after implementing the filter function with cutoff ${\alpha }_c$. (c) The final probability distribution after rescaling which leads to the target distribution with the input mean being amplified by a factor of $g$.

Figure 3

Output distribution of a measurement-based NLA. The dashed blue line depicts the $Q$ distribution of an input coherent state with a real amplitude of 0.5, while the red solid line refers to the distribution of the outcomes of an MB-NLA with different cutoffs and effective gains. As a reference, the output distribution of an ideal NLA is superimposed by the dotted black line. Note that the output distribution of an MB-NLA consists of two Gaussian distributions joined at the dot-dashed green circle with a radius of $\left|\alpha \right|={\alpha }_c/g$. The $x$ and $y$ axes are $\mathrm{Re}\left(\alpha \right)$ and $\mathrm{Im}\left(\alpha \right)$, respectively. The probability density contours are plotted with incremental steps of 0.1 (i.e., 0.1, 0.2, 0.3, and so forth). Some output contours have higher local values due to the more confined distribution in phase space.

Figure 4

Output distribution of a physical NLA. The dashed blue line gives the distribution of the input coherent state with an amplitude of ${\alpha }_0=0.5$. The dotted black line gives the output distribution of an ideal NLA with infinite photon number cutoff. The red solid line is the distribution of the outcome of a physical NLA with different photon number cutoffs and amplification gains. For a fixed amplification gain, a larger cutoff would lead to a better approximation of an ideal NLA. The contour levels are 0.1, 0.2, and 0.3. The $x$ and $y$ axes are $\mathrm{Re}\left(\alpha \right)$ and $\mathrm{Im}\left(\alpha \right)$, respectively.

Figure 5

Schematic of embedding (a) P-NLA and (b) MB-NLA into a measure-and-prepare setup. AM: amplitude modulator; PM: phase modulator.

Figure 6

Left: MB-NLA with varying gains and cutoffs ${\alpha }_c=4$, 6, and 8, represented, respectively, by dot-dashed blue, dotted black, and dashed red curves. Right: P-NLA with varying gains and photon number cutoffs $N_c$=1, 3, and 5 (dot-dashed blue, dotted black, and dashed red curves, respectively). The probability of success is plotted in the logarithmic scale. For all plots, the input state has an amplitude of ${\alpha }_0=0.5$. For comparison, the thick orange line is shown to illustrate the results based on an ideal NLA with an inifinite cutoff.

Figure 7

MB-NLA (left column) and P-NLA (right column) with varying cutoffs. We plot the probability of success here in the logarithmic scale. For all plots, we have an input amplitude of ${\alpha }_0=0.5$ and varying effective gains ($g$=1.5, 2.0, 2.5, and 3.0, shown in brighter to darker blue).

Figure 8

Performance of the MB-NLA (left) and P-NLA (right) with different input amplitudes as a function of cutoffs. For all plots, the effective gain is fixed at $g=1.5$, and the results of an ideal NLA with infinite cutoff are shown as orange dotted lines for comparison. Plotted curves (brighter to darker blue) are for ${\alpha }_0=0.5$, 1.0, 1.5, and 2.0.

Figure 9

MB-NLA filter as a function of $|{\alpha }_m|$ (${\alpha }_c=2.0$): $g=1.3,2.0,5.0,10.0,13.0$, shown in darker to lighter blue. The acceptance probability here is plotted in the logarithmic scale.