# On the Prospects of Multiport Devices for Photon-Number-Resolving Detection

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## Abstract

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## 1. Introduction

## 2. General Physics of Multiport Devices

## 3. Informational Completeness of Photon-Number Distribution Measurements

## 4. General Framework for the Reconstruction Accuracy of Multiport Devices

#### 4.1. Mean Squared-Error and Its Cramér–Rao Bound

#### 4.2. A Measure of Tomographic Performance

## 5. Multiport Device of Equal Port Efficiencies and $\mathit{s}\to \infty $ Output Ports

#### 5.1. Perfect Multiport Devices Without Losses

#### 5.2. Imperfect Multiport Devices with Losses

#### 5.3. Noisy Photon-Number Resolution of Multiport Devices with $s\to \infty $ and $\u03f5>0$

## 6. Multiport Device of Equal Port Efficiencies and $\mathit{s}$ Output Ports

#### 6.1. Perfect Multiport Devices without Losses

#### 6.2. Imperfect Multiport Devices with Losses

#### 6.3. Noisy Photon-Number Resolution of Multiport Devices with $s<\infty $ and $\u03f5>0$

## 7. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Optimality of the Fock-State Measurement for Noiseless Multiport Devices

## Appendix B. Averages over the Probability Simplex

## References

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**Figure 1.**Numerical (colored markers) and theoretical (colored dashed curves) values of TTF (logarithmically scaled) for infinitely large multiport devices of various $\u03f5$ and Hilbert-space dimensions d. A total of 1000 random pure states were used to evaluate each numerical plot point. The convergence to the optimal TTF in Equation (40) at $\u03f5=1$ is as expected. It therefore comes as no surprise that losses monotonically lowers reconstruction accuracy.

**Figure 2.**Informational completeness phase diagrams for various d in the ${d}_{\mathrm{res}}$-$\u03f5$ plane with ${\mu}_{\mathrm{thres}}={10}^{-3}$. Subspaces of dimensions below the boundary are resolvable, and hence render the multiport device of $s\to \infty $ and ${\eta}_{j}=(1-\u03f5)/s$ IC. Those of dimensions above the boundary are unresolvable with such a multiport device. The thick dashed curves represent the analytically calculated boundaries using the approximation in (52), which provide conservative underestimates for the maximum ${d}_{\mathrm{res}}$ compared to the numerically computed boundaries. Clearly, the range of $\u03f5$ for which the entire d-dimensional Hilbert space is completely resolvable reduces as d increases.

**Figure 3.**A plot of the critical $\u03f5$ value (${\u03f5}_{\mathrm{crit}}$) against d that shows the maximum amount of photon losses a multiport device can tolerate before losing its informational completeness property. Here, ${\mu}_{\mathrm{thres}}={10}^{-3}$. The simple ${\u03f5}_{\mathrm{crit}}\sim 1/d$ behavior serves as a back-of-the-envelope solution for designing such devices.

**Figure 4.**Numerical (colored markers) and theoretical (colored dashed curves) values of the TTF (logarithmically scaled) for finite-size multiport devices of various s values and Hilbert-space dimensions d. A total of 2000 random pure states were used to evaluate each numerical plot point. The $s\ge d$ regime illustrates the TTF for IC multiport POVMs only, which is the regime an observer would be interested in for the purpose of photon-number-distribution tomography.

**Figure 5.**Numerical (colored markers) and theoretical (colored dashed curves) values of the TTF (logarithmically scaled) for finite-size multiport devices of various s values, a fixed $\u03f5=0.3$, and Hilbert-space dimensions d. A total of 2000 random pure states were used to evaluate each numerical plot point. As in the case of $\u03f5=0$, the TTF for $d=2$ takes a constant value of 0.3809 for this particular $\u03f5$ value. The worsening of the tomographic performance with a finite loss probability is clearly manifested as an overall increase in the TTF values.

**Figure 6.**A plot of the critical $\u03f5$ value (${\u03f5}_{\mathrm{crit}}$) against d for various number of outputs s of the multiport device. The threshold ${\mu}_{\mathrm{thres}}={10}^{-3}$ is chosen. For a given d-dimensional Hilbert subspace, increasing s also raises ${\u03f5}_{\mathrm{crit}}$, although for reasonable values of s such an increase is not dramatic even when $d\ll s$.

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## Share and Cite

**MDPI and ACS Style**

Teo, Y.S.; Jeong, H.; Řeháček, J.; Hradil, Z.; Sánchez-Soto, L.L.; Silberhorn, C.
On the Prospects of Multiport Devices for Photon-Number-Resolving Detection. *Quantum Rep.* **2019**, *1*, 162-180.
https://doi.org/10.3390/quantum1020015

**AMA Style**

Teo YS, Jeong H, Řeháček J, Hradil Z, Sánchez-Soto LL, Silberhorn C.
On the Prospects of Multiport Devices for Photon-Number-Resolving Detection. *Quantum Reports*. 2019; 1(2):162-180.
https://doi.org/10.3390/quantum1020015

**Chicago/Turabian Style**

Teo, Yong Siah, Hyunseok Jeong, Jaroslav Řeháček, Zdeněk Hradil, Luis L. Sánchez-Soto, and Christine Silberhorn.
2019. "On the Prospects of Multiport Devices for Photon-Number-Resolving Detection" *Quantum Reports* 1, no. 2: 162-180.
https://doi.org/10.3390/quantum1020015