Practical tests for sub-Rayleigh source discriminations with imperfect demultiplexers

Konrad Schlichtholz; Tomasz Linowski; Mattia Walschaers; Nicolas Treps; Łukasz Rudnicki; Łukasz Rudnicki; Giacomo Sorelli; Giacomo Sorelli

doi:10.1364/OPTICAQ.502459

1. Introduction

Statistical hypothesis testing is an important tool in the analysis of scientific data. A typical hypothesis testing problem in optical imaging is source discrimination, i.e., establishing whether an image originates from one or two light sources [1]. This is relevant in astronomy, e.g., for efficient exoplanet and binary stars detection [2–6], and fluorescence microscopy, e.g., for counting the exact number of molecules in a sample [7–9]. For source separations smaller than the width of the point spread function of the optical apparatus, the efficiency of source discrimination protocols based on spatially resolved intensity measurements, i.e., direct imaging, drops significantly [10,11]. In this sub-Rayleigh regime, source discrimination could be performed with the help of quantum-inspired measurement techniques [12].

Recently, motivated by the super-resolving power of spatial demultiplexing (SPADE) in the closely related problem of source separation estimation [13,14], it was shown that SPADE is also quantum-optimal in source discrimination [15], even when the sources are not point-like [16]. Furthermore, it was demonstrated that, due to the symmetry of the problem, detecting even just one photon in a fixed antisymmetric mode allows to accept one of the hypotheses with zero probability of error, leading to a near-optimal, separation-independent decision strategy [15]. These findings, however, were obtained assuming ideal measurements. In practice, the obtained results are significantly affected by experimental imperfections [17–23]. In particular, in the case of SPADE, misalignment, defects in the fabrication of the demultiplexer, and other imperfections induce a finite probability of detecting photons in the incorrect output, i.e., cross talk [24–26].

In this article, we show that cross talk has a strong impact on SPADE for discriminating between one and two equally bright sources in the practically relevant regime of small separations. In particular, we prove that the simple separation-independent test discussed above changes from being quantum-optimal in the ideal case, to be as good as flipping a coin in the presence of arbitrarily small cross talk. Moreover, we find that even though it is possible to design a class of meaningful separation-independent tests even in the presence of cross talk, the associated error probabilities are hard to predict without previous knowledge of the source separations. As an alternative, we propose a semi-separation-independent test with easily tractable maximal probability of error.

2. Hypotheses and Measurement Setting

We are interested in distinguishing between two hypotheses, H0 and H1, as illustrated in Fig. 1. According to hypothesis H1, two weak, incoherent light sources of equal brightness (e.g., faraway thermal sources) are separated by a distance $d$ in the object plane. The coordinate system is chosen in such a way that the sources’ positions are given by $\pm \vec {r}_0=\pm (d/2, 0)$. We consider a diffraction-limited imaging system with a Gaussian point spread function

(1)$$u_{00}(\vec{r})=\sqrt{2/(\pi w^2)}\exp\{-r^2/w^2\},$$

so that the spatial distribution of the electromagnetic field in the image plane coming from a source at $\pm \vec {r}_0$ is given by $u_{00}(\vec {r}\mp \vec {r}_0)$ [27]. For weak sources, most of the photon detection events are single-photon events. Therefore, results in the image plane are effectively described as repeated measurements on $N$ copies of the single-photon state [13]

(2)$$\hat{\rho}_{\textrm{H1}}(d)\approx \frac{1}{2}\Big( |\phi({d})\rangle\langle\phi({d})| +|\phi(-{d})\rangle\langle\phi(-{d})|\Big),$$

where $|\phi (\pm {d})\rangle =\int d\vec {r}\;u_{00}(\vec {r}\mp \vec {r}_0)|\vec {r}\rangle$ and $|\vec {r}\rangle$ stands for the single-photon position eigenstate in the image plane. According to hypothesis H0, there is only one source in the object plane, centered at the origin of the coordinate system, and with the same total brightness as the two sources from hypothesis H1. In this case, the measurement results are effectively described by repeated measurements on the state

(3)$$\hat{\rho}_{\textrm{H0}} = \lim_{d\to 0}\hat{\rho}_{\textrm{H1}}(d) =|\phi(0)\rangle\langle\phi(0)|.$$

The efficiency of a given strategy for deciding which one of the two hypotheses is true is captured by the average probability of error

(4)$$P_{\textrm{e}}(N)=P_{\textrm{H0}}\alpha(N)+P_{\textrm{H1}}\beta(N),$$

where $P_{\textrm {H0(1)}}$ are the a priori probabilities for the respective hypothesis and $\alpha (N)$, $\beta (N)$ are the probabilities of error of the first and second kind, i.e., assuming H1 when H0 is correct and vice versa, for a sample of size $N$. If no a priori information is available, there is no reason to assign a higher probability to any of the hypotheses. For clarity, we concentrate on such a case, in which $P_{\textrm {H0}}=P_{\textrm {H1}}=1/2$. Note that abandoning this assumption has no qualitative impact on our results.

Fig. 1. Schematic representation of the measurement scenario. Depending on the hypothesis, there is one (H0) or two (H1) weak light sources in the object plane, resulting in diffraction-broadened spatial field distributions in the image plane. To decide whether H0 or H1 is true, the image-plane field distribution is analyzed via photon counting after spatial-mode demultiplexing affected by cross talk.

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3. Asymptotic Probability of Error

Effective source discrimination for sub-Rayleigh separations, i.e., for

(5) $$x:=d/2w<1,$$

requires a large number $N$ of samples. When $N\gg 1$, the probability of error minimized over all possible decision strategies for a specific measurement decays exponentially as $P_{\textrm {e}}^{\min }\sim \exp \{-N\xi \}$, where

(6)$$\xi:={-}\ln\Big(\min_{0\leq s\leq1}\sum_{k}p(k|\textrm{H0})^s p(k|\textrm{H1})^{1-s}\Big)$$

is the Chernoff exponent [28]. Here, $p(k|\textrm {H0})$ [$p(k|\textrm {H1})$] denote the probability of obtaining the measurement outcome $k$ conditioned on the hypothesis H0 [H1] being true.

An emblematic measurement for source discrimination is direct imaging, which can be treated as a benchmark for new measurement schemes. The effectiveness of these measurements for discriminating between one and two weak incoherent sources could be judged based on the Chernoff exponent in Eq. (6). For ideal, i.e., continuous and noiseless, direct imaging, the latter is given by

(7)$$\xi_{\textrm{DI}}(d) ={-}\ln\left[ \min_{0\leq s\leq 1} \int_{\mathbf{R}^2} d\vec{r}\, p(\vec{r}\,|0)^s p(\vec{r}\,|d)^{1-s} \right],$$

where

(8)$$p(\vec{r}\,|d)= \frac{1}{2}\left( |u_{00}(\vec{r}-\vec{r}_0)|^2 + |u_{00}(\vec{r}+\vec{r}_0)|^2\right)$$

stands for the probability of detecting the photon at the location $\vec {r}$. For small separations, i.e., $x\ll 1$, we can approximately calculate Eq. (7). Expanding the integrand into a series in $d$ and integrating the dominant terms in polar coordinates, we obtain

(9)$$\xi_{\textrm{DI}} \approx{-}\ln\left[ \min_{0\leq s\leq 1}1+ 4 (s-1) s x^4 \right].$$

Minimization yields $s_{\min }=1/2$, which after another expansion to leading order in $x$ results in

(10)$$\xi_{\textrm{DI}}\approx x^4.$$

However, it is important to note that in the most relevant regime of small separations, this measurement is not optimal for the task considered. This can be directly seen through the quantum Chernoff exponent which stands for the Chernoff exponent obtained by maximizing Eq. (6) over all possible measurements. Assuming that the quantum states associated with the two hypotheses are represented by the density operators $\hat {\rho }_{\textrm {H0}}$, $\hat {\rho }_{\textrm {H1}}$, the quantum Chernoff exponent takes the form [29]

(11)$$\xi_Q={-}\log\min_{0\leq s\leq 1} \textrm{Tr}{\hat\rho_{\textrm{H1}}^s\hat\rho_{\textrm{H0}}^{1-s}}.$$

It was shown that for the considered hypothesis testing problem (with weak incoherent sources), the quantum Chernoff exponent is given by $\xi _Q=x^2$ and that it can be saturated by spatial demultiplexing (SPADE) in modes such that whenever H0 is true, only one of the basis modes is occupied, i.e., $u_{00}$ is chosen to be one of the measurement modes [15]. In a general spatial demultiplexing scenario, one performs a measurement of the field in the image plane in some orthogonal spatial mode basis $v_k$. Then, the probabilities of a given outcome $k$ for the state in Eq. (2) which is associated with detecting a photon in mode $v_k$ can be calculated as

(12)$$p(k|d) = \frac{1}{2}\left(|f_{+ k}(d)|^2 + |f_{- k}(d)|^2\right),$$

where

(13)$$f_{{\pm} k}(d) = \int_{\mathbb{R}^2} d^2\vec{r} \, v_{k}^*(\vec{r}) \, u_{ 00}(\vec{r}\mp\vec{r}_0)$$

are the overlap integrals of the measurement basis function $v_{k}$ and the spatial field distribution $u_{ 00}(\vec {r}\mp \vec {r}_0)$ resulting from the source positioned at $\pm \vec {r}_0$. Let us now assume that one wants to use the Hermite–Gauss modes $u_{nm}$ centered at the origin of our coordinate system as a measurement basis. Note that since this basis contains $u_{00}$, it leads to optimal measurement, as mentioned above. In such a case, one should simply set $v_k=u_{nm}$, with $k$ standing now for the double index $(n,m)$.

However, in experimental settings, any measurement is unavoidably subject to apparatus misalignment, design and fabrication defects of the demultiplexer, and other imperfections. Accordingly, there is a small cross talk probability, i.e., it is possible that a measured photon is transmitted to an incorrect mode (see Fig. 1). Consequently, the real measurement basis deviates from the ideal one as

(14)$$v_{nm} = \sum_{k,l=0}^{D-1} C_{nm,kl}u_{kl},$$

where $C$ stands for the (unitary) cross talk matrix, which is close to unity and $D$ restricts the number of measured modes [24]. We remark that the restriction of measurement to a finite number of modes requires renormalization of the probabilities in Eq. (12) within the chosen subset of possible outcomes. To quantify the severity of imperfections, one can use the cross talk strength $\epsilon ^2$, defined as the mean absolute value square of the off-diagonal elements of the cross talk matrix [24]:

(15)$$\epsilon^2:= \frac{1}{D^2(D^2-1)}\sum_{\substack{n,m,k,l=0\\nm\neq kl}}^{D-1} |C_{nm,kl}|^2.$$

One can see that the cross talk strength $\epsilon ^2$ describes the average probability of cross talk from one measurement mode to another.

We now proceed to assess the impact of cross talk on SPADE. We focus on the practically relevant regime of small separations, $x\ll 1$, and weak cross talk $\epsilon \ll 1$. As it will become apparent, in this regime, the optimal probability of error for ideal implementation of SPADE in Hermite–Gauss modes is achieved already with measurement restricted to modes with $n,m\in \lbrace 0,1\rbrace$. Therefore, for simplicity, from now on, we assume $D=2$. Accordingly, we can approximate the Chernoff exponent in Eq. (6) by a series expansion in these parameters. Analogously to previous findings for separation estimation [26], we find that the expansion depends on the ratio $x/\epsilon$ (see Supplement 1):

(16)$$\xi\approx \begin{cases} \left\{1 - \left[\ln\ln q(x)-1\right]/\ln q(x) \right\} x^2, & x\gg \epsilon, \\ x^4/(8 p_0), & x\ll \epsilon, \end{cases}$$

where we restricted ourselves to leading terms in $x$ and $\epsilon$. Here, $q(x):= x^2/p_0$ and $p_0:= |C_{10,00}|^2\sim \epsilon ^2$ is the probability of cross talk from mode $u_{00}$ to $v_{10}$. Observe that for $x\approx \epsilon$, the obtained series converges too slowly to constitute a reliable approximation. In the range $x\ll \epsilon$, cross talk changes the scaling of $\xi$ from $x^2$ to merely $x^4$, i.e., the same as for ideal direct imaging. Nonetheless, despite the same scaling, SPADE is still superior to direct imaging due to a larger scaling coefficient [for weak cross talk, $1/(8p_0)\gg 1$]. More surprisingly, cross talk has a significant impact on the Chernoff exponent even in the range of relatively large separations $x\gg \epsilon$. Indeed, while the upper line of Eq. (16) approaches the ideal scaling $x^2$ with vanishing cross talk, it does so logarithmically slowly. This shows the importance of cross talk in hypothesis testing at any separation scale. A graphical comparison between the Chernoff exponents for cross talk-affected SPADE $\xi$, ideal direct imaging $\xi _{\rm DI}$, and the quantum bound $\xi _{\rm Q}$ in the sub-Rayleigh regime is provided in Fig. 2.

Fig. 2. Comparison between the median of the Chernoff exponents for SPADE $\xi$ from a sample of $500$ random unitary cross talk matrix (green, shaded area stands for the interquartile range), approximate Chernoff exponent in Eq. (16) for $x\ll \epsilon$ (green, dashed), asymptotic Chernoff exponent for perfect direct imaging $\xi _{\rm DI}$ (pink, dashed), and the quantum bound $\xi _{\rm Q}$ (blue, dot-dashed) versus $x := d/2w$. Calculations performed with $\epsilon ^2=0.0033$ and $D=2$. The black vertical line indicates $x=\epsilon$.

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4. Practicality of Statistical Tests

The optimal decision strategy for a specific measurement is given by the likelihood-ratio test [30], according to which H1 is accepted if and only if

(17)$$\prod_{k}\left(\frac{p(k|\text{H1})}{p(k|\text{H0})}\right)^{N_{k}} > 1,$$

where $N_{k}$ is the number of events in which the outcome $k$ was obtained. Unfortunately, the likelihood ratio test has some drawbacks. Due to its assumption of a fixed separation, the latter has to be first estimated from the data, using, e.g., the method of moments [25]. Errors in this estimation inevitably deviate this decision strategy from its optimality, and more importantly can lead to underestimation of the probability of error, which is in addition hard to calculate. Finally, the optimal test in Eq. (17) requires measuring in many modes, which is not always feasible in experiments.

A seemingly more practical test was introduced in Ref. [15]. Let us denote by $p_x$ the probability of measuring a photon in the mode $v_{10}$ conditioned on the separation $x$. The key observation behind this test is that, assuming ideal measurements, under H0 this probability takes the value $p_0=|C_{10,00}|^2=0$, while under H1 one has $p_x> 0$. In other words, photons can be measured in mode $v_{10}$ only if hypothesis H1 is true. This leads to the following test:

(18)$$N_{10} \underset{H0}{\overset{H1}{\gtrless}} 0,$$

where whenever equality holds, one assumes H0. One can easily calculate that for such a test, the probability of accepting hypothesis H1 whenever H0 is true takes the value $\alpha (N)=0$, since when hypothesis H0 is true, it is impossible to detect photons in the mode $v_{10}$. Therefore, whenever H0 is true, the test in Eq. (18) always results in accepting H0 [15]. One can also find the probability of accepting hypothesis H0 when H1 is true: $\beta (N) = \left (1 - p_x\right )^N$, which is simply the probability of not detecting any photons in the mode $v_{10}$ across all $N$ trials. Clearly, both $\alpha (N)$ and $\beta (N)$ vanish for large photon numbers $N$ regardless of $x$, and therefore, so does the total probability of error. We thus have a test that is simple, separation-independent, and requires measuring in only one mode. Moreover, for small separations $x \ll 1$, this simple test is also quantum optimal, as we can show that Eq. (18) is in fact equivalent to Eq. (17) in this regime (see Supplement 1). Unfortunately, as we will now prove, this test goes from optimal to completely useless in the presence of any amount of cross talk, no matter how small.

To see this, we observe that even for very weak cross talk $\epsilon \ll 1$, the probability $p_0$ of measuring a photon in mode $v_{10}$ under hypothesis H0 is no longer zero. As long as $p_0$ is non-zero, however small, the asymptotic probability of error is drastically changed. It is easily seen that $N_{10}$ for the test in Eq. (18) is a random variable with binomial distribution, with probabilities given by

(19)$$p(N_{10}=k|x)=\left(\begin{array}{@{}l@{}}{N}\\{k}\end{array}\right) p_x^k\left(1-p_x\right)^{N-k},$$

where the corresponding probability for hypothesis H0 is obtained by setting $x=0$. As an immediate consequence, the probability $\alpha (N)$ of assuming H1 when H0 is true is no longer zero. According to the test in Eq. (18), we should assume H1 whenever even a single photon is in mode $v_{10}$, meaning that to get $\alpha (N)$, we must sum Eq. (19) with $x=0$ over $0<k\leqslant N$, obtaining $\alpha (N)= 1 - (1-p_0)^N$. Similarly, $\beta (N)=\left (1 - p_x\right )^N$ is obtained by considering $k=0$ in Eq. (19), corresponding to no photons in mode $v_{10}$. The total probability of error of the test in Eq. (18) is now

(20)$$P_{\textrm{e}}(N) = \frac{1}{2}\left(1 - (1-p_0)^N \right) + \frac{1}{2}\left(1 - p_x\right)^N.$$

Clearly, whenever $p_0\neq 0$, indicating non-zero cross talk, this approaches $1/2$ as $N\to \infty$. The introduction of any experimental error changes the test from nearly optimal to as bad as flipping a coin.

The failure of the test in Eq. (18) does not necessarily disqualify all separation-independent tests. In fact, it is possible to design a class of separation-independent tests for source discrimination with imperfect demultiplexers that yields $\lim _{N\to \infty }P_{\textrm {e}}(N)=0$ (see Supplement 1). Such tests take the following form:

(21)$$N_{10} \underset{H0}{\overset{H1}{\gtrless}} N p_0+\zeta(N),$$

where $\zeta (N)>0$ are $x$-independent functions increasing faster than $\sqrt {N}$, but slower than $N$. The probabilities of error for such tests can be calculated analogously using a binomial distribution as in the case of Eq. (20) (see Supplement 1). Note that the natural generalization of the test in Eq. (18) given by Eq. (21) with $\zeta (N)=0$ results in the suboptimal $\lim _{N\to \infty }P_{\textrm {e}}(N)=1/4$ (see Supplement 1). Unfortunately, the family in Eq. (21) appears only marginally more practical than the original test in Eq. (18), as the corresponding rates of convergence of the probability of error to zero vary strongly with $x$, from nearly optimal to orders of magnitude worse [see Fig. 3(a)]. Accordingly, despite the tests being $x$-independent, a sensible estimation of their probability of error, for a given number $N$ of detected photons, would require some prior knowledge of $x$, severely undermining their practical utility.

Fig. 3. Probability of error versus $N$ assuming uniform cross talk (see Supplement 1) with $\epsilon ^2=0.01$ and using Gaussian approximation to the binomial distribution. In all figures, from right to left, the dashed lines correspond to the optimal test in Eq. (22) for $x=0.02,\,0.03, \,0.05, \,0.10$ (blue, magenta, green, orange). (a) Solid lines of corresponding colors stand for the distance-independent test in Eq. (21) for $\zeta =N^{4/5}/100$. We see that the effectiveness of the distance-independent test is severely affected by the actual value of $x$. (b) Solid lines stand for the modified test in Eq. (23) with $x_{\min }=0.02$. Accordingly, the curve for $x=0.02$ (blue) provides an upper bound for all other curves. The inset shows how, in the $P_{\textrm {e}}<0.05$ region, all solid curves in panel (b) feature approximately the same rate of convergence. (c) Solid lines stand for estimation of the probability of error for ideal direct imaging using the approximate Chernoff exponent $\xi _\textrm {DI}=x^4$. Note that the blue dashed line coincides with the upper bound on the probability of error for the modified test in Eq. (23) with $x_{\min }=0.02$, i.e., the blue solid line from Fig. 3(b). The fact that there is a solid curve corresponding to $x> x_{\min }$ to the right of this dashed line means that the semi-separation-independent test can outperform ideal direct imaging even for some separations $x\geq x_{\min }$.

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These findings motivate us to search for more practical tests for source discrimination. To achieve this goal, we go back to the optimal likelihood-ratio test in Eq. (17). Given that most information on the difference between the image of one source (H0) and that of two closely separated ones (H1) is contained in mode $v_{10}$, we rewrite Eq. (17) in terms of only two outcomes: photon in mode $v_{10}$ and photon in any other mode. This yields

(22)$$\left(\frac{p_x}{p_0}\right)^{N_{10}} \left(\frac{1-p_x}{1-p_0}\right)^{N-N_{10}} > 1.$$

Solving for $N_{10}$ and expanding to second order in $x$ and $\epsilon$, we obtain

(23)$$N_{10} \underset{H0}{\overset{H1}{\gtrless}} N\left(p_{0} + \gamma x^2/2\right),$$

where $\gamma = 1 - \mathcal {O}(\epsilon ^2)$. The modified test in Eq. (23) is particularly appealing since (for small separations and weak cross talk) it inherits the optimality of the likelihood ratio test while being simple and based on a single-mode measurement, like the tests in Eqs. (18) and (21).

On the downside, Eq. (23) is not separation-independent.To obviate to this problem, let us replace $x$ on the right-hand side of Eq. (23) by some $x_{\min }$ and then test for the modified hypotheses: (H0) there is only one source or (H1) there are two sources with separation $x\geqslant x_{\min }$. In this case, we will still obtain $\lim _{N\to \infty }P_{\textrm {e}}=0$ (see Supplement 1). Furthermore, for fixed $N$ and $x_{\min }$, $P_{\textrm {e}}$ is decreasing with growing $x$, meaning that the probability of error of the algorithm is upper bounded by $P_{\textrm {e}}$ calculated with $x=x_{\min }$. Therefore, the test in Eq. (23) is semi-separation-independent, in the sense that for a fixed $x_{\min }$, despite being not optimal for every value of $x$, it allows for an easy access to a maximal probability of error independent of any a priori knowledge of the separation. This, in particular, avoids any possible underestimation of the actual error probability $P_{\textrm {e}}$. All of this provides a reliable method of planning experiments: it is sufficient to set the minimal separation $x_{\min }$ into Eq. (23) to determine the number of photons $N$ to detect to be sure not to exceed a pre-established maximal tolerable probability of error. The minimal separation $x_{\min }$ can be dictated by multiple factors, such as physical constraints (e.g., atoms in molecules cannot be packed too tightly) or technical limitations of the experimental setups. Figure 3(b) shows a comparison between the real probability of error for this test and its upper bound. Additionally, we note that the semi-separation-independent test can outperform ideal direct imaging even for some separations $x\geq x_{\min }$. This can be seen in Fig. 3(c) where we present a comparison of the upper bound of the probability of error for semi-separation-independent test and probability of error for their ideal direct imaging estimated using the approximate Chernoff exponent in Eq. (10).

5. Conclusions

We have scrutinized the effectiveness of realistic SPADE-based discrimination between one and two closely separated light sources. We analytically showed that the presence of cross talk heavily affects the probability of successful discrimination, causing it to scale suboptimally with the source separation $d$ even for relatively large values of $d$. Similarly, any cross talk renders separation-independent hypothesis testing non-viable in practice. To remedy this, we proposed a simple semi-separation-independent algorithm based on the likelihood-ratio test, which, even for imperfect demultiplexers, gives access to the maximal probability of error without requiring separation estimation.

Our results suggest that it is mandatory to include the role of cross talk, and other experimental imperfection, e.g., electronic noise [21,25], in SPADE-based source discrimination. In particular, it would be interesting to see how significant experimental imperfections are in multiple-hypotheses testing [16], and when considering potentially unequal brightnesses of the sources [26]. Finally, our findings could be easily extended to the problem of pulse discrimination in the time-frequency domain [31,32]. In that context, demultiplexing can be performed using nonlinear optical devices [33], for which the finite width of the phase-matching function induces unavoidable cross talk.

Funding

Narodowe Centrum Nauki (UMO-2019/32/Z/ST2/00017); Agence Nationale de la Recherche; QuantERA (ApresSF); Horizon 2020 Framework Programme (731473); ERCIM (‘Alain Bensoussan’ Fellowship Programme).

Acknowledgments

Project ApresSF is supported by the National Science Centre (No. 2019/32/Z/ST2/00017), Poland; ANR under QuantERA, which has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 731473. This work was carried out during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

Supplemental document

See Supplement 1 for supporting content.

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Practical tests for sub-Rayleigh source discriminations with imperfect demultiplexers

Abstract

1. Introduction

2. Hypotheses and Measurement Setting

3. Asymptotic Probability of Error

4. Practicality of Statistical Tests

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (3)

Equations (23)

Optica Quantum