Scaling of multicopy constructive interference of Gaussian states

TL;DR

This work introduces the gain-to-instability ratio (GIR) as a quantitative metric to assess the scalability of multicopy interference protocols using unstable, nonidentical Gaussian resources. By focusing on the signal-to-noise ratio (SNR) as the figure of merit for constructive interference, the authors analyze three linear-interferometer architectures (pyramidal, sequential, and dynamical loop) and show that, while mean SNR scales as $\sqrt{N}$ for both classical and squeezed inputs, the GIR reveals architecture-dependent stability: pyramidal/sequential/dynamical-loop schemes sustain $\mathrm{GIR}\sim \sqrt{N}$ under moderate squeezing fluctuations, whereas fixed-loop variants saturate with $\mathrm{GIR}\sim N^{1/4}$ and can even degrade under larger instability. The study also highlights that squeezing fluctuations can dramatically impact scaling, and that losses, while reducing SNR, can stabilize fluctuations to improve GIR. Supplementary results propose a harmonic-mean strategy and analyze losses, offering practical pathways to enhance scalability in realistic noisy settings and guiding future experimental validation of these scaling laws for Gaussian and non-Gaussian bosonic resources.

Abstract

Quantum technology advances crucially depend on the scaling up of essential quantum resources. Their ideal multiplexing offers more significant gains in applications; however, the scaling of the nonidentical, fragile and varying resources is neither theoretically nor experimentally known. For bosonic systems, multimode interference is an essential tool already widely exploited to develop quantum technology. Here, we analyze, predict and compare essential scaling laws for a constructive interference of multiplexed nonclassical Gaussian states carrying information by displacement with weakly fluctuating squeezing in different multimode interference architectures. The signal-to-noise ratio quantifies the increase in displacement relative to the noise. We introduce the gain-to-instability ratio to numerically estimate the effect of unexplored resource instabilities in a large scale interference scheme. The use of the gain-to-instability ratio to quantify the scaling laws opens steps for extensive theoretical investigation of other bosonic resources and follow-up feasible experimental verification necessary for further development of these platforms.

Figures (7)

• Figure 1: Gain-to-instability quantification of quantum protocols: (a) multicopy protocols and their applications use quantum resources parametrised by $\boldsymbol{r}= (r_1,...,r_N)$ in a number $N$ of finite, different and imperfect replicas of inputs quantified by the classical figure-of-merit $F_{in,i}(r_i)$, $i=1,\ldots,N$. The quantum states carrying resources are interacting in a quantum protocol that produces an output figure-of-merit $F_{out}^{(N)}(\boldsymbol{r})$ dependent on all the resources $r_1,\ldots,r_N$. Each input is characterized by the same gain as average figure of merit $\langle F_{in}^{(1)}\rangle = \langle F_{in,i}(r_i) \rangle$ where $i=1,...,N$. If the gains $\langle F^{(N)}_{out} \rangle > \langle F_{in}^{(1)} \rangle$ than any of the input, therefore it exhibits a multicopy gain. The gain may be considered nonclassical when it surpasses the classical benchmark $\langle F_{out, cl}^{(N)} \rangle$ and scalable if it is larger than $\langle F_{out, cl}^{(N-1)} \rangle$, for any $N$. (b) To thoroughly evaluate the protocol, we propose an additional quantifier, the gain-to-instability ratio (GIR) for both inputs and outputs. We also extend the multicopy, nonclassical, and scalable criteria for GIR. As depicted in (c), the protocols with comparably scalable gains exhibit qualitatively different scaling of stability in the GIR. Protocol 1 (in blue) exhibits a highly stable scalability and, on the other hand, the protocol 2 (in orange) behaviour a lowly stable scalability.
• Figure 2: Different architectures of interferometric schemes: (a) Pyramidal architecture is an interferometer composed solely of balanced beam splitters. Each pair of input modes interacts, and the constructive output modes of two adjacent beam splitters are utilised as the input modes of the next beam splitter in the interferometer. At the output, this pyramid ensures full constructive interference of mean inputs in the $X$ variable. The output is verified by detecting this variable. (b) Sequential architecture must employ transmittances $t_i = i/(i+1)$ to achieve the same constructive interference in the output as the pyramidal strategy, with identical mean and standard deviation. (c) Dynamical and fixed loop architectures, where the input modes are temporally multiplexed in time-bins, interfere on a single beam splitter with transmittance $t_i$ in such a way that the constructive interference output is re-injected into the first input and interacts with a new squeezed state. The transmittance $t_i$ can vary dynamically as $t_i = i/(i+1)$ to create a fully equivalent output to the pyramidal and sequential architectures. In simpler scenarios, the transmittance is optimised to a single fixed value $t^{*}$ according to the number $N$ of the interfering copies to achieve maximal output mean displacement, or alternatively is simply fixed to $t_i=0.5$. These simpler strategies do achieve maximal mean displacement at the output, however, they can still partially increase the SNR.
• Figure 3: Constructive interference of $N$ equally displaced squeezed states with varying standard deviation of noise for the different architectures. The parameter values are the same for all architectures and are set to $x_0 = 0.5$, $r_0 = 2$, $\sigma_r = 0.5$. (a) The mean signal-to-noise ratio $\langle \hbox{SNR}\rangle$: continuous lines represent the ideally stable squeezing for each architecture. The dotted lines account for the squeezing instability. The blue line corresponds to the pyramidal, sequential, and dynamical loop architectures as defined by Eq.(\ref{['eq:SNRsqueezedstable']}). For the fixed loop architecture outlined in Eq.(\ref{['eq:SNRSqueezingLoop']}), the orange line denotes a transmittance $t^*$, the green line represents a fixed transmittance $t_{i}=(N-1)/N$, while the purple line indicates the fixed $t_i=0.5$, independent of $N$. The blue dotted line relates to Eq.(\ref{['eq:SNRsqueezingfluctuation']}), and for comparison, the blue dashed line is provided by the approximate Eq.(\ref{['eq:SNRsqueezingapprox']}) (except for very small $N$, which is not shown on the graph). The dotted orange, green, and purple lines adhere to Eq.(\ref{['eq:SNRSqueezingLoop']}) for $t=t^*$, $t_i=1-1/N$, and $t_{i}=0.5$, respectively. For comparison, the yellow points indicate the average SNR for measurement-induced displacements from varying two-mode squeezed states fiurasek_conditional_2001schnabel_squeezed_2017. The black curve represents the classical limit as given by Eq. (\ref{['eq:SNRcoherentstable']}), where the input states are coherent states. (b) The gain-to-instability ratio $\hbox{GIR}_{\hbox{SNR}}=\langle \hbox{SNR} \rangle/\Delta \hbox{SNR}$: The blue points correspond to the GIR from numerical simulations, and the blue dashed line represents their exponential fit by $1.5 \times N^{0.51}$. For comparison, the blue continuous line is the approximate Eq.(\ref{['eq:SNRSNRsqueezingPyr']}); its asymptotic behaviour, as described in Eq.(\ref{['eq:SNRSNRsqueezingPyrAsympt']}), is indistinguishable from the blue continuous line. The yellow points correspond to the GIR for measurement-induced displacements from varying two-mode squeezed states. The orange points represent the GIR numerically simulated for the loop architecture with optimised transmittances $t = t^*(N)$. The dashed orange line is their exponential fit $3.5 \times N^{0.27}$. Finally, the green points curve represents the data for the loop architecture with transmittance fixed at $(N-1)/N$, which saturates quickly. It is similar to the purple curve for the loop architecture with $t=0.5$, which performs poorly. Note that all the parameters and exponential fitting values are in Tables \ref{['tab:SNRfit']} and \ref{['tab:GIRfit']}; see Appendix \ref{['appendixD:fitparametervalues']}.
• Figure 4: Constructive interference in the different architectures at Fig.2 as functions of (a,c) squeezing instability $\sigma_r$ and (b,d) average squeezing $r_0$. Parameters are $x_0 = 0.5$, $r_0 = 2$, $\sigma_r = 0.5$ as in Fig.\ref{['fig:3']}, unless free parameter. Note that the colors and thickness codes used are the same in all the panels of Fig.\ref{['fig:4']}. In Fig.\ref{['fig:SNRvssigma']}, the horizontal continuous blue line is for ideal behaviour in the pyramidal, sequential and dynamical loop architectures, orange lines corresponds to the loop architecture the optimized transmittance $t^*$ and green lines are for fixed transmittance equal $(N-1)/N$. Trios of lines are for $N= 3$ (thinner lines), $N= 30$ and $N= 300$ (boldest lines) input copies (from bottom to top lines). The points of corresponding colours are the results of numerical simulations for different strategies. The blue decreasing lines are for approximated (Eq. (\ref{['eq:SNRsqueezingapprox']}); dotdashed line) and asymptotic (Eq. (\ref{['eq:SNRsqueezingAsymptotic']}); dotted line) formulas. Note that when $N$ increase the approximated and asymptotic formula's for SNR become arbitrarily close and appears like a continuous line. This means that the asymptotic behaviour of the SNR is reached already for a number of modes $N$ of order $10$. In Fig.\ref{['fig:SNRvssqueezing']}, the colors and thickness codes are used for different architectures (colors) and number of input copies (thickness) for $N=3$ (bottom thin lines), $N=30$ (middle medium lines) and $N=300$ (top bold lines). The dots shows the numerical simulations. For the blue colors, approximated (dotdashed) and asymptotic (dotted) formulas are displayed. Note that when $N$ increase the approximated and asymptotic formula's for SNR become arbitrarily close and appears like a continuous line for $N=300$. For Fig.\ref{['fig:SNRSNRvssigma']}, the GIR is showed for different architectures (colors) and different number of input modes (thickness) as a function of the standard deviation $\sigma_r$. Here, only the numerical data is displayed. The dots are joined by a continuous line in order to emphasize the decreasing of the GIR with the standard deviation $\sigma_r$. Finally, in Fig.\ref{['fig:SNRSNRvssqueezing']}, the GIR is showed for different architectures (colors) and different number of input modes (thickness) as a function of the mean squeezing $r_0$. Here again, only the numerical is displayed and the dots are joined by straight lines in order to emphasize their behaviour. Here, one can see that, for all architecture (colors) and number of input modes $N$ (thickness), the GIR is not very sensitive to the mean squeezing $r_0$.
• Figure 5: By changing the protocol allowing for conditional measurement on all output modes but the constructive modes, on can change further enhance the scaling capability. Indeed, not only does the SNR slightly improves but, more importantly, the GIR improve by order of magnitudes. This confirms the predictions that this harmonic averaging of the second order moment enhance the stability of the constructive output squeezed state. In green is the harmonic mean and in blue the deterministic protocol.