Estimating the Photon-Number Distribution of Photonic Channels for Realistic Devices and Applications in Photonic Quantum Information Processing

Characterizing the input-output photon-number distribution of an unknown optical quantum channel is a worthwhile task for many applications in quantum information processing. Ideally, this would require deterministic photon-number sources and photon-number-resolving detectors, but these technologies are still work in progress. In this work, we propose a general method to rigorously bound the input-output photon-number distribution of an unknown optical channel using standard optical devices such as coherent light sources and non-photon-number-resolving detectors and homodyne detectors. To demonstrate the broad utility of our method, we consider the security analysis of practical quantum key distribution systems based on calibrated single-photon detectors and an experimental proposal to implement time-correlated single-photon counting technology using homodyne detectors instead of single-photon detectors.

Figure 1

Typical scheme for estimating the input-output photon-number distribution $q(\overrightarrow{m}|\overrightarrow{n})$ of a photonic channel. Each of the $N$ input modes of the photonic channel is connected to a light source that supplies the input photons and each of the $M$ output modes is connected to a detector that gives an outcome related to the number of photons leaving the photonic channel. Note that the sources and detectors considered here may include some form of active modulation devices, such as intensity modulators.

Figure 2

We simulate the achievable secure key rate with two avalanche photodiode detectors, each one featuring a dark-count rate of ${10}^{-6}$ and a fixed channel error rate of 5% in all bases (depolarizing channel). The secure key rate in the source and detector modulation case is $K\ge p_0(\mu )q_{00|0}[1-r_0^{(0)}({\eta }_0)r_0^{(0)}({\eta }_1)]+p_1(\mu )p_{\det }^{x=y=0}H(A|E)-Q(\mu ,\eta )h_2[E(\mu ,\eta )]$ where $H(A|E)$ is the conditional entropy on Alice’s key bit given Eve’s side information for a qubit protocol. For BB84, we have $H(A|E)\ge 1-h_2(e_X)$, for six states we have $H(A|E)\ge 1-[H(\lambda )-h_2(e_Z)]$. To recompute the single-photon statistics, we use three intensity levels: ${10}^{-3},{10}^{-2},0.5$ and four efficiency levels per detector: $0.94,0.96,0.98,1$. See more details in Appendix pp2.

Figure 3

We compare the exact channel error rate value used for the simulation (5% here) to the upper bound provided by our proposed analytical method and the one proposed in Ref. [36]. The definition in Ref. [36] is including the noise from the detector dark counts while our is only considering the noise on the channel. As a result, the definition of Ref. [36] is increasing while ours is staying close to the exact value in the high-loss regime, hence a slight improvement in distance for the key rate. Note that this improvement is only affecting privacy amplification, since the noise due to dark counts still has to be corrected in the error-correction step.

Figure 4

The original TCSPC concept is based on the recording of single-photon detection events and construction of the corresponding histogram as shown above. As a result, there is an implicit postselection of the conclusive outcomes. That is, the experiment is repeated until the bin with the maximum number of events reaches a certain level. The histogram is then normalized to derive the probability of having a detection event in a particular bin $t\in \mathcal{T}$ given that there was a conclusive outcome $C$, i.e., $Pr(t|C)$. Nevertheless, due to the dead time effect, the recorded probability is not exactly the one aforementioned, but rather the probability of observing a detection in a bin and not recording any detection before. When the counting rate is low, the two probabilities are close though [24].

Figure 5

We consider here an ideal version of TCSPC where perfect PNRDs are used to record the exact number of incoming photons in each time bin. We label $Y^{\{t\}}\in \mathbb{N}$ the corresponding random variable for each time bin. After enough events have been recorded, it is possible to estimate the photon-number probability distribution at each time bin. By keeping only the single-photon events, it is easy to recover the same TCSPC information as in Fig. 4.

Figure 6

We consider here a practical implementation of TCSPC using a binned homodyne detector. We label $X^{\{t\}}\in \mathcal{X}$ the corresponding random variable for each time bin. After enough events have been recorded, it is possible to estimate the measurement PDF at each time bin. Then from this information, we show that it is possible to recompute the same photon-number distribution as in Fig. 5.

Figure 7

Simulation of an exponential decay intensity profile that can be reconstructed using homodyne detection only. The upper (dashed line) and lower (dotted line) bound on the probability of one photon emission are very close to the exact value (solid line) using our method while the bounds on the probability of two photon emission are looser due to the inability of the method to recover very low values. This simulation is using 16 bins evenly spaced in the range $[0,5]$ for the real outcome given by the homodyne detector.