# 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

- Wildfeuer, C.F.; Pearlman, A.J.; Chen, J.; Fan, J.; Migdall, A.; Dowling, J.P. Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector. Quantum Limits in Optical Interferometry. Phys. Rev. A
**2009**, 80, 043822. [Google Scholar] [CrossRef] - Demkowicz-Dobrzański, R.; Jarzyna, M.; Kołodyński, J. Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector. Prog. Opt.
**2015**, 60, 345. [Google Scholar] - Von Helversen, M.; Böhm, J.; Schmidt, M.; Gschrey, M.; Schulze, J.-H.; Strittmatter, A.; Rodt, S.; Beyer, J.; Heindel, T.; Reitzenstein, S. Quantum metrology of solid-state single-photon sources using photon-number-resolving detectors. New J. Phys.
**2019**, 21, 035007. [Google Scholar] [CrossRef][Green Version] - Wu, J.-Y.; Toda, N.; Hofmann, H.F. Observation of squeezed states with strong photon-number oscillations. Phys. Rev. A
**2009**, 100, 013814. [Google Scholar] - Cattaneo, M.; Paris, M.G.A.; Olivares, S. Hybrid quantum key distribution using coherent states and photon-number-resolving detectors. Phys. Rev. A
**2018**, 98, 012333. [Google Scholar] [CrossRef][Green Version] - Stucki, D.; Ribordy, G.; Stefanov, A.; Zbinden, H.; Rarity, J.G.; Wall, T. Photon counting for quantum key distribution with peltier cooled ingaas/inp apds. J. Mod. Opt.
**2001**, 48, 1967. [Google Scholar] [CrossRef] - Kilmer, T.; Guha, S. Boosting linear-optical Bell measurement success probability with predetection squeezing and imperfect photon-number-resolving detectors. Phys. Rev. A
**2019**, 99, 032302. [Google Scholar] [CrossRef][Green Version] - Ren, M.; Wu, E.; Liang, Y.; Jian, Y.; Wu, G.; Zeng, H. Quantum random-number generator based on a photon-number-resolving detector. Phys. Rev. A
**2011**, 83, 023820. [Google Scholar] [CrossRef][Green Version] - Applegate, M.J. Efficient and robust quantum random number generation by photon number detection. Appl. Phys. Lett.
**2015**, 107, 071106. [Google Scholar] [CrossRef] - Eisaman, M.D.; Fan, J.; Migdall, A.; Polyakov, S.V. Single-photon sources and detectors. Rev. Sci. Instrum.
**2011**, 82, 071101. [Google Scholar] [CrossRef] - Jönsson, M.; Björk, G. Evaluating the performance of photon-number-resolving detectors. Phys. Rev. A
**2019**, 99, 043822. [Google Scholar] [CrossRef][Green Version] - Mirin, R.P.; Nam, S.W.; Itzler, M.A. Single-Photon and Photon-Number-Resolving Detectors. IEEE Photonics J.
**2011**, 4, 629. [Google Scholar] [CrossRef] - Marsili, F.; Bitauld, D.; Gaggero, A.; Jahanmirinejad, S.; Leoni, R.; Mattioli, F.; Fiore, A. Physics and application of photon number resolving detectors based on superconducting parallel nanowires. New J. Phys.
**2009**, 11, 045022. [Google Scholar] [CrossRef] - Marsili, F.; Najafi, F.; Dauler, E.; Bellei, F.; Hu, X.; Csete, M.; Molnar, R.J.; Berggren, K.K. Single-Photon Detectors Based on Ultranarrow Superconducting Nanowires. Nano Lett.
**2011**, 11, 2048. [Google Scholar] [CrossRef] [PubMed] - Jahanmirinejad, S.; Frucci, G.; Mattioli, F.; Sahin, D.; Gaggero, A.; Leoni, R.; Fiore, A. Photon-number resolving detector based on a series array of superconducting nanowires. Appl. Phys. Lett.
**2012**, 101, 072602. [Google Scholar] [CrossRef][Green Version] - Matekole, E.S.; Vaidyanathan, D.; Arai, K.W.; Glasser, R.T.; Lee, H.; Dowling, J.P. Room-temperature photon-number-resolved detection using a two-mode squeezer. Phys. Rev. A
**2017**, 96, 053815. [Google Scholar] [CrossRef][Green Version] - Ma, J.; Masoodian, S.; Starkey, D.A.; Fossum, E.R. Photon-number-resolving megapixel image sensor at room temperature without avalanche gain. Optica
**2017**, 4, 1474. [Google Scholar] [CrossRef] - Zolotov, P.; Divochiy, A.; Vakhtomin, Y.; Moshkova, M.; Morozov, P.; Seleznev, V.; Smirnov, K. Photon-number-resolving SSPDs with system detection efficiency over 50 at telecom range. AIP Conf. Proc.
**2018**, 1936, 020019. [Google Scholar] - Cai, Y.; Chen, Y.; Chen, X.; Ma, J.; Xu, G.; Wu, Y.; Xu, A.; Wu, E. Metamodelling for Design of Mechatronic and Cyber-Physical Systems. Appl. Sci.
**2019**, 9, 2638. [Google Scholar] [CrossRef] - Paul, H.; Törma, P.; Kiss, T.; Jex, I. Photon Chopping: New Way to Measure the Quantum State of Light. Phys. Rev. Lett.
**1996**, 76, 2464. [Google Scholar] [CrossRef] - Kok, P.; Braunstein, S.L. Detection devices in entanglement-based optical state preparation. Phys. Rev. A
**2001**, 63, 033812. [Google Scholar] [CrossRef][Green Version] - Rohde, P.P.; Webb, J.G.; Huntington, E.H.; Ralph, T.C. Photon number projection using non-number-resolving detectors. New J. Phys.
**2007**, 9, 233. [Google Scholar] [CrossRef] - Řeháček, J.; Hradil, Z.; Haderka, O.; Peřina, J., Jr.; Hamar, M. Multiple-photon resolving fiber-loop detector. Phys. Rev. A
**2003**, 67, 061801(R). [Google Scholar] [CrossRef] - Banaszek, K.; Walmsley, I.A. Fiber-assisted detection with photon number resolution. Opt. Lett.
**2003**, 28, 52. [Google Scholar] [CrossRef] [PubMed] - Fitch, M.J.; Jacobs, B.C.; Pittman, T.B.; Franson, J.D. Photon-number resolution using time-multiplexed single-photon detectors. Phys. Rev. A
**2003**, 68, 043814. [Google Scholar] [CrossRef][Green Version] - Achilles, D.; Silberhorn, C.; Sliwa, C.; Banaszek, K.; Walmsley, I.A.; Fitch, M.J.; Jacobs, B.C.; Pittman, T.B.; Franson, J.D. Photon-number-resolving detection using time-multiplexing. J. Mod. Opt.
**2004**, 51, 1499. [Google Scholar] [CrossRef] - Avenhaus, M.; Laiho, K.; Chekhova, M.V.; Silberhorn, C. Accessing Higher Order Correlations in Quantum Optical States by Time Multiplexing. Phys. Rev. Lett.
**2010**, 104, 063602. [Google Scholar] [CrossRef] - Kruse, R.; Tiedau, J.; Bartley, T.J.; Barkhofen, S.; Silberhorn, C. Limits of the time-multiplexed photon-counting method. Phys. Rev. A
**2017**, 95, 023815. [Google Scholar] [CrossRef][Green Version] - Teo, Y.S. Introduction to Quantum-State Estimation; World Scientific Publishing Co.: Singapore, 2015. [Google Scholar]
- Řeháček, J.; Teo, Y.S.; Hradil, Z. Determining which quantum measurement performs better for state estimation. Phys. Rev. A
**2015**, 92, 012108. [Google Scholar] [CrossRef]

**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