Weak Coherent and Heralded Single Photon Sources for Quantum Secured Imaging and Sensing

1. Introduction

Quantum imaging and sensing protocols offer enhanced measurement schemes compared to traditional schemes, finding applications in measuring light-sensitive samples under low illumination [1,2,3,4,5,6,7,8]. Quantum communication, particularly through quantum key distribution (QKD), offers a fundamentally secure approach to information transfer against attacks such as intercept resend, photon number splitting(PNS), and unambiguous state discrimination(USD) [9,10,11,12]. Quantum secure imaging and sensing are emerging disciplines that enable secure information transfer and measurements [13,14,15,16,17,18,19,20,21]. However, for practical deployment, it is essential that these protocols remain robust during continuous operation, even in the presence of potential adversarial attacks, and are an active area of research [22,23,24,25,26].

Weak coherent sources (WCS) and spontaneous parametric down-conversion (SPDC) based sources have been widely employed in QKD, quantum imaging, and sensing applications [27,28,29,30,31,32,33,34]. High-dimensional(HD) quantum states have also been explored to improve the quantum communication, imaging, and sensing protocols [35,36,37,38,39,40,41]. At high operating speeds, generating all four quantum states (as in BB84 QKD protocol) becomes technically challenging due to the increased voltage demands of modulation devices. In contrast, two-state QKD protocols are more practical under such conditions, requiring less modulation effort. The two-state protocol, however, requires an additional monitoring detector to guard against advanced unambiguous state discrimination (USD) attacks [12].

This work expands the quantum secure information transfer landscape by examining SPDC-based heralded single-photon sources (HSPS) and WCS under decoy and non-decoy QKD protocol configurations. Additionally, we investigate both BB84 and B92 QKD protocols, analyzing their performance in quantum secure information applications. To further enhance the resilience of these systems, we investigate high-dimensional QKD protocols (HD-B92) to improve resistance against PNS and USD attacks. We adapt these countermeasures to our mathematical modelling by drawing on solutions developed in the HD-QKD literature.

2. Mathematical Modelling and Methods

2.1. Photon Number Distribution for Quantum States of a Weak Coherent Source and a Heralded Single Photon Source

The equation describing the photon number distribution for WCS is given by equation 1, and for SPDC-based HSPS(thermal) is given by equation 2 [42,43].

$$P_{k,x}^{weak}={\displaystyle \frac{e^{-x}\ast x^k}{k!}}$$

(1)

$$p_{k,x}^{HSP{S}_{per}}={\displaystyle \frac{x^k}{{\left(1+x\right)}^{k+1}}}\ast {\displaystyle \frac{\left(1-{\left(1-{\eta }_A\right)}^k+d_A\right)}{P_x^{post}}};$$

(2)

where $x$ is the mean photon number, knumber of photons, ${\eta }_A$ represents the efficiency the source end,$d_A$ is the dark count rate for the detectors, $P_x^{post}$​ is the post-selected probability given by $P_x^{post}={\displaystyle \frac{x\ast {\eta }_A}{1+x\ast {\eta }_A}}+d_A$; $p_{0,x}=1-P_x^{cor}+p_{0,x}^{per}{P}_x^{cor},$ and $p_{k,x}=p_{k,x}^{per}{P}_x^{cor}$. ${ P}_x^{cor}=0.3$;

Figure 1 illustrates the photon number distributions for thermal, WCS (Poissonian), and HSPS(thermal) at various mean photon numbers: 0.0001, 0.001, 0.01, and 0.1. As observed, the vacuum component (zero-photon probability) is significantly suppressed in the heralded single-photon source. However, as the mean photon number increases, multiphoton components begin to appear across all sources, highlighting the growing probability of more than one photon per pulse, which is critical when assessing security and performance in quantum secure communication, imaging, and sensing protocols.

2.2. Security Analysis for Non-Ideal Conditions to Obtain Secure Bit Rate vs Loss in dB

The secure bit rate without decoy state for BB84 and B92 protocol is given by equations 3 and 4, respectively [12,44,45,46]

$$R_{BB84}=q{Q}_{\mu }\{\left(1-\mathsf{\Delta }\right)\left\{\left({\log }_{2}d)-\left[H_d\left(e_1\right)\right]\right)\right\}-f\left(E_{\mu }\right)\ast H_d\left(E_{\mu }\right);$$

(3)

where q is parameter depending on QKD protocol, $q={\displaystyle \frac{1}{2}}$ for BB84 and $q={\displaystyle \frac{1}{4}}$for B92; $Q_{\mu }$ is the overall gain, d is the dimension, H is the binary Shannon entropy, $e_1$ is the error rate of single photons given by $e_1={\displaystyle \frac{E_{\mu }}{\mathsf{\Delta }}};$ $\mathsf{\Delta }={\displaystyle \frac{1-P_0-P_1}{Q_{\mu }}}$; here $E_{\mu }$ is the overall error rate,$\mathsf{\Delta }$ is term considering multiphoton probablity against PNS attack. $H_d\left(p\right)=-p\ast lo{g}_2\left[{\displaystyle \frac{p}{d-1}}\right]-\left(1-p\right)\ast {\log }_2\left(1-p\right)]$; and $f\left(E_{\mu }\right)=1.22$;

$$R_{B92}=q{Q}_{\mu }\{\left(1-{\mathsf{\Delta }}^{\prime }\right)\left\{\left({\log }_{2}d)-\left[H_d\left(e_1\right)\right]\right)\right\}-f\left(E_{\mu }\right)\ast H_d\left(E_{\mu }\right)-I_{AE}^{USD}\};$$

(4)

Here $I_{AE}^{USD}$ is the information leakage due to USD attack given by $I_{AE}^{USD}={\displaystyle \frac{\left(1-\eta \right){\left(1-cos\alpha \right)}^N}{\eta (1-{\left(1-cos\alpha \right)}^N)}}$;

N is the number of multiple qubit encoding which in high dimensional(hybrid encoding) terms is ${\mathrm{N}=\log }_{2}d$, ${\mathsf{\Delta }}^{\text{'}}={\displaystyle \frac{1-P_0-P_1-P_2}{Q_{\mu }}}$ is used by considering the contribution of 2 photon pulses in B92 protocol due to its robutness against PNS attack.

$$Q_x^{WCS}=Y_o+1-e^{-\mu \ast \eta }; E_x={\displaystyle \frac{\left(e_0-e_d\right)d_B}{Q_x^{WCS}}}+e_d;$$

$$Q_x^{HSPS}={\displaystyle \frac{1}{P_x^{post}}}\left[{\displaystyle \frac{x\eta \left(1+d_A\right)}{1+x\eta }}+{\displaystyle \frac{x{\eta }_A\left(1+d_B\right)}{1+x{\eta }_A}}+{\displaystyle \frac{1}{1+x\left(\eta +{\eta }_A-\eta {\eta }_A\right)}}-\left(1-d_A{d}_B\right)\right]; E_x={\displaystyle \frac{\left(e_0-e_d\right)d_B}{Q_x^{HSPS}}}+e_d.$$

The secure bit rate with decoy state for BB84 and B92 protocol is given by equation 5 [10,38,43]

$$R={q\{Q}_0\ast ({\log }_{2}d)+Q_1\ast \left[{\log }_{2}d-H_d\left(e_1\right)\right]-Q_{\mu }\ast f\left(E_{\mu }\right)\ast H_d(E_{\mu }\left)\right\};$$

(5)

where $Q_0=Y_0\ast p_0\left(\mu \right)$; $e_0={\displaystyle \frac{d-1}{d}}$;$Y_0=1-{\left(1-d_B\right)}^d$; $Q_1=Y_1\ast p_1\left(\mu \right)$;

$$Y_1={\displaystyle \frac{p_2\left(\mu \right)\ast Q_{\nu }-p_2\left(\nu \right)\ast Q_{\mu }-Y_0\left[p_0\left(\nu \right)\ast p_2\left(\mu \right)-p_0(\mu \right)\ast p_2\left(\nu \right)]}{\left(p_1\left(\nu \right)\ast p_2\left( \mu \right)-p_1\left(\mu \right)\ast p_2\left(\nu \right)\right\}}}; { e}_1={\{Q}_{\mu }\ast E_{\mu }-e_0{\ast Y}_0{\ast p}_0\left(\mu \right)\}/\{Y_1{\ast p}_1\left(\mu \right)\}.$$

(6)

The shared parameter values used in simulations are as follows: channel attenuation($\alpha )$= 0.21 (dB/km), detection efficiencyat receiver (${\eta }_b)$= 0.045, heralding arm efficiency(${\eta }_A)$ = 0.8, dark count probablity in heralding detector$\left(d_A\right)$=${10}^{-5}$, misalignment error $\left(e_d\right)$= 0.033.

3. Results and Discussion

Figure 2 presents the secure bit rate as a function of channel loss (in dB) for the BB84 QKD protocol using a WCS without decoy state analysis. Two scenarios are shown, corresponding to different dark count probabilities of Bob’s detector: (a) $d_b={10}^{-6}$ and (b) $d_b={10}^{-7}$. A lower dark count probability enhances the signal-to-noise ratio, thereby improving the maximum quantum secure information transfer distance. This performance improvement demonstrates the importance of low-noise detectors in practical quantum communication. When such low-noise conditions are combined with high-dimensional encoding schemes, secure distance and bit rate gains can be expected.

Figure 3 shows the secure bit rate versus channel loss (in dB) for the BB84 protocol using a HSPS without decoy state analysis. The results are plotted for two different dark count probabilities of Bob’s detector: (a) $d_b={10}^{-5}$and (b) $d_b={10}^{-6}$. It can be observed that the HSPS remains effective even in the presence of relatively high detector noise ($d_b={10}^{-5}$), achieving a reasonable secure distance. This robustness to noise highlights one of the key advantages of HSPS in practical quantum secure information transfer scenarios, particularly when high-performance detectors are not available. It is observed that the BB84 protocol using HSPS demonstrates an advantage in secure distance of around 10 dB over WCS without decoy states.

Figure 4 presents the secure bit rate versus channel loss for the B92 protocol using a WCS. Subplot (a) shows the case where two-photon contributions are excluded, while (b)–(d) incorporate the two-photon components. WCS-based B92 protocol benefits from its intrinsic resistance to PNS attacks, allowing the secure bits to be extracted even in the presence of multiphoton pulses using high dimensional (hybrid encoding). This demonstrates that B92, when used with a WCS, can maintain security without decoy state analysis and avoid the protocol to stop under PNS attack.

Figure 5 presents the secure bit rate versus channel loss for the B92 protocol without decoy state analysis for HSPS. Plot (a) shows the case where two-photon contributions are excluded from the key rate calculation, while plots (b)–(d) incorporate two-photon contributions. In the HD B92 with WCS, d=8 over d=2 has an advantage of around 9dB, while the HSPS-based HD B92 protocol achieves around 15 dB advantage for the same high dimensional (hybrid encoding) upgrade. Moreover, in secure information transfer distance, HSPS outperforms WCS by around 6-7 dB in the HDB92 configuration in terms of quantum secure information transfer distance.

Figure 6 compares the performance of (a) WCS and (b) HSPS under decoy state analysis using the BB84 protocol. Including decoy states significantly enhances the secure distance and enables detection of PNS attacks. While HSPS achieves a longer secure transmission distance, WCS benefits from a much higher photon emission rate. As a result, although HSPS provides superior distance performance, the overall bit rate is often higher for WCS when the overall figure of merit is considered concerning the photon counts obtained from the source, making it a practical choice in many real-world systems. Finally, implementing decoy states provides a further gain of 15–20 dB compared to schemes without decoy states.

HD quantum states enhance both security and bits per pulse in quantum communication. In quantum imaging and sensing, samples may exhibit sensitivity to high-dimensional photon degrees of freedom, or multiple degrees of freedom, such as polarization and orbital angular momentum, can be exploited for probing. WCS can generate higher photon rates, which are advantageous for key generation rates in communication. Considering the WCS advantage of higher photon counts of about three orders based on current technology, it gives a higher secure bit rate of 1-2 orders when considering the overall figure of merit for secure bits with repetition rate of the source. In contrast, HSPS, which have reduced vacuum contributions, are more suitable to securely probing samples over longer distances at low photon illumination.

4. Conclusion

Our analysis focuses on the photon number statistics of WCS and HSPS, examining their roles in quantum secure imaging, sensing, and communication. Nonorthogonal two state protocols offer resilience to PNS attack to a certain threshold of multi-photon components without halting the quantum secure information transfer in non-ideal source conditions. HSPS is particularly beneficial in high-loss settings due to its reduced vacuum component, allowing for extended secure distances. However, their practical advantage relies on highly efficient and low-loss components in encoding, detection, and coupling. In contrast, WCSs produce higher photon rates, making them advantageous for high-throughput applications, including precision quantum sensing. Despite a higher vacuum component, their compatibility with GHz clock rates and decoy-state methods makes them suitable for secure communication, imaging and sensing scenarios. Combining HD state encoding introduces a trade-off: while it improves security and bits per pulse in ideal conditions, performance declines with increasing channel loss. Also, the modulation speed of devices to encode high dimensional quantum states need to improve to outperform low dimensional high-speed counterparts. Still, such configurations support a broader range of secure imaging and sensing applications. Resistance to both quantum and classical jamming attacks is possible, where the framework can adapt by subtracting mutual information of an adversary from the secure bit rate equation, to estimate secure distances accordingly. Moreover, these findings have direct implications for the design of quantum networks, where flexible combinations of source types dimensionality and protocol choices can be optimized based on channel conditions and application goals—whether it be quantum secure imaging, sensing, or communication. Thus, the proposed analysis supports resilient architectures for quantum networks operating in real-world noisy and lossy conditions.