Quantum Correlation Enhanced Optical Imaging

1. Introduction

2. Building Blocks of Quantum Optical Imaging

2.2. Quantum Photonic Sources

Light sources in quantum imaging exhibit different photon number statistics, characterized by two key parameters: the average photon count and the variance. Classical light sources follow a super-Poissonian distribution, where the variance exceeds the mean photon count. In these cases, the photon count is highly variable. In contrast, non-classical light sources, a concept rooted in quantum optics, exhibit sub-Poissonian distribution, meaning that the variance is less than the mean photon count. Coherent states, commonly used to approximate laser radiation, occupy a unique position. They have Poissonian photon number statistics, where variance equals mean photon count [49,50]. Single-photon sources may be categorized into two types: deterministic and probabilistic. Deterministic sources, such as hexagonal boron nitride (hBN), provide single photons as wanted, ensuring reliability [51]. Probabilistic sources include both HSPS and WCS. Deterministic sources provide easy integration with current technology. However, their complex manufacturing process presents significant technical hurdles. Probabilistic sources, while they provide higher count rates, have efficiency challenges. This requires improvements in source purity and operational control.

Table 1. Quantum states of light and respective a few notable experiments.

Table 1. Quantum states of light and respective a few notable experiments.

Source of light	Areas of application	Comments
Squeezed states of light [52]	Gravitational wave detection	Squeezed light has been employed to augment the sensitivity of gravitational wave detectors, such as the Laser Interferometer Gravitational-Wave Observatory (LIGO). Squeezed states enable the detection of gravitational waves of lower intensity compared to classical light by mitigating quantum noise.
Squeezed local oscillator [53]	Particle tracking	Using a squeezed local oscillator in particle tracking investigations can enhance the accuracy of measurements, enabling more precise monitoring of particle dynamics at the microscopic level.
Entangled states of light [54]	QKD	The use of entangled photons is employed for the safe transmission of encryption keys. The phenomenon of entanglement guarantees that any illicit efforts to intercept information may be identified by the perturbations occurring in the entangled states.
Entangled N00N state (N=2) [55]	Concentration measurement of protein	This application employs entangled N00N states, where N is the number of entangled particles, to conduct high-precision measurements, such as determining protein concentration in a solution. Entangled states enable measurements with resolutions that are beyond the diffraction limit.

2.2.1. Quantum Entangled Photon Sources

Entangled states are a fundamental concept in quantum mechanics. They are multi-particle states that cannot be described as a simple product of individual states $|{\mathsf{\varphi }}_{\mathrm{N}}⟩=|{\mathsf{\psi }}_1⟩|{\mathsf{\psi }}_2⟩...|{\mathsf{\psi }}_{\mathrm{N}}⟩$. [45]. Entanglement can be created in various degrees of freedom, including polarization, time-energy, and position-momentum. In the context of quantum imaging, position-momentum entangled photons are particularly important. These photons can be described in momentum representation as $\left\{|\mathsf{\varphi }⟩\right\}=\int \int {\mathrm{d}\mathrm{q}}_1{\mathrm{d}\mathrm{q}}_2\mathsf{\psi }\left({\mathrm{q}}_1,{\mathrm{q}}_2\right)\left|{\mathrm{q}}_1\right.⟩{\mathrm{q}}_2⟩$, where $q_i$ represents the transverse component of the wave vector$k_i$. The function $\mathsf{\psi }({\mathrm{q}}_1,{\mathrm{q}}_2)$ is defined based on the paraxial approximation, and it is characterized by strong anti-correlation in momentum. Entangled photons have found extensive applications in various domains, including quantum computing, quantum communication, quantum metrology, sensors, and quantum imaging [39,56,57,58,59,60].

Generation of quantum entanglement between two or more photons has long posed challenges. Historically, the entanglement of photons was achieved via sequential emissions from distinct atoms [61,62]. However, these methods were intricate and soon overshadowed by advancements in nonlinear optics. Through nonlinear optics, photon pairs can be produced in materials with nonlinear attributes, notably in nonlinear crystals. The feasibility of this approach was supported both theoretically and experimentally, where a singular pump photon in conjunction with a nonlinear medium was shown to generate photon pairs [63,64,65]. Nonlinear optical processes have grown in importance because they enable the creation of non-classical light states, now commonly used in quantum sensing. The optical medium's polarization response can be described by the equation. $P_i={ϵ}_0{\chi }_{ij}^{\left(1\right)}{E}_i+{ϵ}_0{\chi }_{ijk}^{\left(2\right)}{E}_{ijk}{+}_{\dots \dots \dots }$ indicates the nth-order susceptibility of that medium[66]. Hence, materials with susceptibilities beyond the first order can facilitate interactions between optical fields of varying frequencies. The second and third-order non-linear interactions are typical, often called three- and four-wave mixing. Examples include second harmonic generation, sum frequency generation, and difference frequency generation[67,68,69]. Three-wave mixing, or SPDC is facilitated in non-centrosymmetric crystals with a non-zero ${\chi }^{\left(2\right)}$, where a single pump frequency divides into two lower frequencies. Centrosymmetric crystals and non-crystalline materials might possess a non-zero ${\chi }^{\left(3\right)}$, but to ensure efficient non-linear processes, a high susceptibility coefficient is preferable; otherwise, significant pump power becomes a necessity.

In quantum optics, the SPDC process can be visualized as the destruction of a pump photon and the birth of a photon pair. This is interesting due to the quantum-linked behaviors of the produced photon pair. The dominant method for photon pair production is SPDC in second-order nonlinear crystals such as bismuth borate (BBO), lithium borate (LBO), lithium niobate (LN), and potassium titanyl phosphate (KTP). During SPDC, a powerful pump field induces a nonlinear polarization in a material, leading to the emission of two lower-energy photons, termed the signal (ωs) and idler (ωi). The energy conservation principle establishes that ${\omega }_p={\omega }_i+{\omega }_s$​. To achieve efficient pair creation, constructive interference of these photons is crucial, necessitating the phase-matching condition: $\mathsf{\Delta }k=k\left({\omega }_s\right)+k\left({\omega }_i\right)-k\left({\omega }_p\right)$ where $k\left(\omega \right)=\frac{2\pi n\left(\omega \right)}{\lambda }$.Unfortunately, due to the dispersion inherent to materials, this condition often isn't satisfied, leading to destructive interference and non-generation of photon pairs over a distance δc​=2π/Δk. Anisotropic nonlinear crystals can adjust for this phase mismatch via birefringent phase matching (BPM), wherein one photon from the SPDC process must have orthogonal polarization to the pump. This can be achieved in two ways: type-I BPM, where signal and idler are co-polarized, and type-II BPM, where only the idler is co-polarized with the pump. In the context of Type 0 SPDC, it is observed that the signal and idler photons exhibit identical polarization characteristics as the pump photon. The phenomenon is distinguished by the polarization of photons, which may be either horizontally or vertically aligned. In the Type 1 SPDC process, the signal and idler photons are generated with identical polarization states. However, it is essential to note that this polarization state is perpendicular to the pump photon. For instance, in the scenario where the pump photon exhibits horizontal polarization, both the signal and idler photons will adopt vertical polarization. Type 2 SPDC is characterized by producing signal and idler photons with orthogonal polarizations. For example, if a single photon exhibits horizontal polarization, its counterpart will exhibit vertical polarization [66].

Collinear configurations, where all the fields align along a single axis, are preferable as they optimize spatial overlap and enhance SPDC efficacy. However, Poynting vector walk-off can restrict the useful length of crystals in BPM. An alternative solution is quasi-phase-matching (QPM) using poled nonlinear crystals like ppLN or ppKTP periodically. Using QPM in a non-critical phase-matching setup can efficiently utilize longer nonlinear crystals and waveguide structures. For brevity, Figure 4 depicts two types of nonlinear optical processes known as parametric down-conversion. In this quantum optics process, a pump photon is converted into a pair of signal and idler photons within a nonlinear crystal.

2.2.2. Squeezed States

Squeezed states are a specific class of quantum states that exhibit unique properties, especially in the context of photon statistics. The electric field operator $\widehat{E}\left(t,z\right)=2E_o\sin \left(kz\right){\widehat{[X}}_1\cos \left(\omega t\right)+{\widehat{X}}_2\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)]$ in quantum optics can be expressed in terms of quadrature operators, namely ${\widehat{X}}_1$ and ${\widehat{X}}_2$. These quadrature operators are defined as ${\widehat{X}}_1=\frac{1}{2}[\widehat{a}+{\widehat{a}}^{†}]$ and ${\widehat{X}}_2=\frac{1}{2}[\widehat{a}-{\widehat{a}}^{†}]$where $\widehat{a}$ and ${\widehat{a}}^{†}$ are the creation and annihilation operators[71].

Squeezed states are characterized by unequal variances in these quadrature operators ($\mathrm{V}\left({\mathrm{X}}_{\mathrm{i}}\right)<1/4$, for i=1 or 2. The process of squeezing involves the application of a squeezing operator $\mathrm{S}\left(\mathsf{\zeta }\right)$ to a quantum state. This operator creates or destroys 2 photons at a time and is defined as $\mathrm{S}\left(\mathsf{\zeta }\right)=\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\widehat{\mathsf{\zeta }}}^{†}{\mathrm{a}}^2-\mathsf{\zeta }{{(\widehat{\mathrm{a}}}^{†})}^2\right)}{2}$[49,72]. The squeezing parameter $\zeta$ quantifies the degree of squeezing. When $\zeta$> 0, amplitude squeezing occurs, leading to a reduction in amplitude fluctuations. This type of squeezing is associated with sub-Poissonian statistics. Conversely, when $\zeta$< 0, phase squeezing takes place, causing an increase in amplitude fluctuations and super-Poissonian statistics. A squeezed coherent state $\left|\alpha ,\zeta \right.⟩=D\left(\alpha \right)S\left(\zeta \right)\left|0\right.⟩$ is created by applying both the displacement operator $D\left(\alpha \right)$ and the squeezing operator $S\left(\zeta \right)$ to the vacuum state. The displacement operator is defined as $D\left(\alpha \right)=\mathrm{e}\mathrm{x}\mathrm{p}(\alpha {\widehat{a}}^{†}-{\alpha }^{*}\widehat{a})$ [52,73,74].

The generation of squeezed states typically involves the use of a degenerate parametric amplifier, which consists of a second-order nonlinear crystal pumped by a powerful laser beam at an angular frequency ${\mathsf{\omega }}_{\mathrm{p}}=2\mathsf{\omega }.$ A weak signal, often considered as the vacuum state, is introduced into the system. The nonlinear crystal then mixes the signal and pump, resulting in the production of an idler beam at ${\mathsf{\omega }}_{\mathrm{i}}={\mathsf{\omega }}_{\mathrm{p}}-{\mathsf{\omega }}_{\mathrm{s}}=\mathsf{\omega }$. During degenerate parametric amplification, one of the quadratures of the resulting idler beam is amplified while the other is unamplified. The specific outcome depends on the relative phase between the signal and pump.

Figure 5. Depictions of key quantum light states in phase space, each marked by dashed lines to represent the inherent phase uncertainty. The sections include: (a) the Fock state, (b) the vacuum state, (c) the coherent state, (d) the squeezed vacuum state, (e) the amplitude squeezed state, and (f) the phase squeezed state. [44].

Figure 5. Depictions of key quantum light states in phase space, each marked by dashed lines to represent the inherent phase uncertainty. The sections include: (a) the Fock state, (b) the vacuum state, (c) the coherent state, (d) the squeezed vacuum state, (e) the amplitude squeezed state, and (f) the phase squeezed state. [44].

Squeezed light states offer enhanced resolution compared to coherent infrared light. It is a powerful tool for overcoming the Rayleigh limit, reducing intrinsic noise in conventional super-resolution imaging techniques. Squeezed light has been applied in various experiments, including monitoring the movement of particles within biological specimens, achieving significant enhancements in resolution. The system’s resolution obtained using squeezed light is approximately 1.43 times higher than coherent infrared light[75]. Lastly, it is important to highlight the application of squeezed light to overcome the Rayleigh limit, which reduces inherent noise in light that constrains conventional super-resolution methods. 6 dB amplitude squeezed light was integrated with an imaging setup to monitor the movement of particles within a biological specimen [76]. This resulted in a 42 percent improvement over the standard quantum limit. The identical approach was subsequently utilized in to implement photonic force microscopy, achieving a 14 percent enhancement in resolution compared to experiments conducted with coherent light [53].

2.2.3. Weak Coherent Sources of Light

As an alternative to complex single-photon sources, using weak coherent laser pulses presents a simplified and efficient method for generating light at high repetition rates. These pulses are particularly appealing due to their straightforward production and capability to enable high-rate secure communication. These systems can operate at ambient room temperatures, negating the need for intricate cryogenic setups. Furthermore, weak coherent pulses afford the flexibility to employ a range of encoding variables beyond polarization, such as frequency, time-bin, or orbital angular momentum. For practical use in secure communications, these pulses must be precisely identical across any characteristic not involved in information encoding, ensuring that potential side channels are mitigated. The main challenge with these pulses is their susceptibility to multiphoton occurrences, which threatens the confidentiality of communication. Mathematically, a coherent state with an average photon number and a phase is expressed through a series where the probability of obtaining n photons at a given time follows the Poisson distribution. Excess photons could escape from the pulse during transit, leading to vulnerabilities.

The provided Figure 6. highlights two distinct parts, (a) and (b), each illustrating various aspects of photon statistics in quantum optics.

a) Coherent States:

Part (a) presents a histogram comparing the probability distributions of photon numbers within coherent states of light for two different average photon numbers, denoted by $\mu$. Coherent states are a type of quantum state of the electromagnetic field most closely resembling classical electromagnetic waves. The red bars represent a coherent state with a lower average photon number ($\mu$ = 0.5), and the blue bars correspond to a coherent state with a higher average photon number ($\mu$ = 2.0). The x-axis represents the number of photons, and the y-axis represents the probability of finding that number of photons in the state. As depicted, the probability distribution spreads out with an increase in the average photon number, indicating a higher probability of finding states with more photons.

b) Single Photons and Fock State Mixture:

Part (b) displays two illustrations: the top shows an array of single photons, each in a quantum state |1⟩ indicating a single-photon state. This is ideal for quantum communication as each photon can securely carry a small amount of information. The bottom illustration shows a Fock state mixture, where each position may have a different number of photons, indicated by the states |2⟩, |0⟩, |1⟩, |3⟩, |1⟩. A Fock state is a quantum state with a defined number of photons. The mixture of Fock states represents a more complex quantum state where multiple photons may be present at various positions, which is less ideal for secure communication due to the possibility of information leakage if multiple photons are detected simultaneously.

The equation describes the photon number distribution for a WCS is given by $P_{k,x}^{weak}=\frac{e^{-x}\mathrm{*}{x}^k}{k!}$ , and for a SPDC based HSPS is given by $p_{k,x}^{HSP{S}_{per}}=\frac{x^n}{{\left(1+x\right)}^{n+1}}\mathrm{*}\frac{\left(1-{\left(1-{\eta }_A\right)}^k+d_A\right)}{P_x^{post}}$;$x$ is the mean photon number, k is number of photons. ${\eta }_{wwA}$ represents the efficiency at the source end, $d_A$is the dark count rate for the detectors, $P_x^{post}$​ is the post-selected probability given by $P_x^{post}=\frac{x\mathrm{*}{\eta }_A}{1+x\mathrm{*}{\eta }_A}+d_A$ [78]. Figure 7. compares the performance of HSPS and WCS in generating single photons. In scenarios where the mean photon number is low, HSPS shows a significantly higher likelihood of emitting single photons and demonstrates sub-Poissonian statistics, indicating that its variance in photon numbers is less than the mean. However, as the mean photon number approaches 0.6, the likelihood of emitting single photons from both sources converges, as shown by the green and yellow dotted curves in Figure 7b. This convergence is crucial for applications like QKD, where the dominance of single-photon events over multi-photon events is critical to ensuring security. According to the data presented in Figure 7, for HSPS, maintaining the mean photon number at a maximum of around 0.2 is advised to reduce multi-photon events, whereas, for WCS, the mean photon number can be increased to approximately 1 to achieve a similar reduction in multi-photon occurrences.

2.3. Qunatum Key Distribution (QKD)

In recent years, quantum computing has garnered significant attention due to its potential to revolutionize computational capabilities. Quantum computers harness the unique properties of quantum mechanics to solve complex problems that are highly time-consuming to classical computers. At the heart of this technology are quantum bits, or qubits, which can exist in superpositions of states, allowing quantum computers to perform multiple calculations simultaneously. This quantum advantage holds tremendous promise in various domains, including cryptography, drug discovery, and optimization. For instance, algorithms like Shor’s can efficiently factor large numbers, challenging classical cryptographic systems[79]. To realize the potential of quantum computers, specific criteria, as outlined by DiVincenzo, must be met [40]. Superconducting wires were early contenders in quantum computing, offering fast gate speeds and high fidelity[80]. However, they require extremely cold conditions and have short-lived coherence times. On the other hand, trapped ions boast high gate fidelity and longer coherence times but need ultra-high vacuum conditions and precise laser setups. These systems are complex to scale and have slower gate operations [81]. Photons are promising candidates as qubits because of their rapid gate operations and impressive fidelities without deep cooling or ultra-high vacuum conditions. However, they have limitations: every program needs a unique chip, and photon loss can introduce noise. Additionally, 2-qubit operations are challenging with photons unless neutral atoms are used. While offering long coherence times and no need for external cooling, these atoms still come with challenges, such as the necessity for ultrahigh vacuums and scalability issues with lasers[82,83,84,85]. Quantum dots, benefiting from existing semiconductor tech, display strong gate fidelity and speeds. However, they come with the need for cryogenic conditions and face problems like interference and crosstalk. Keeping DiVincenzo's criteria, which stresses isolated and scalable qubit systems, each platform has strengths and challenges. For instance, while photons are great for transmitting quantum data, spin qubits are better suited for storing it, and superconducting qubits excel in processing. The crucial role photons play in quantum computing is worth highlighting. Their potential, especially in threatening classical cryptographic systems by potentially breaking traditional encryption, is immense as quantum technology progresses.

The Alice, Bob, and Eve framework is a common scenario in cryptography. Alice and Bob aim to establish a secure communication channel while facing the challenge of Eve, an eavesdropper who seeks to intercept and gain unauthorized access to their communication. This framework underscores the importance of achieving perfect secrecy, where the encryption key should be if the message is only used once, a concept known as the one-time pad [86]. This approach maximizes information entropy, making it extremely difficult for attackers like Eve to decipher the key. QKD is a method employed to generate secure encryption keys between Alice and Bob using quantum channels. QKD capitalizes on the unique properties of quantum mechanics, such as the indivisibility of photons and the no-cloning theorem, to detect any attempts by eavesdroppers to compromise the security of communication. However, the security of QKD relies on specific assumptions. The effectiveness of QKD protocols is contingent on several critical assumptions, including trust between Alice and Bob, the absence of data loss in the quantum channel, secure random number generation, and the assurance of security in their physical locations and the devices they use. These assumptions are fundamental to ensuring the robustness of QKD in practice. Furthermore, there is an intriguing application of utilizing quantum states of light, modulated based on polarization, to detect intelligent jamming attacks in active imaging systems. In this context, quantum properties are harnessed to encode information, and any attempts by adversaries to manipulate or interfere with the quantum states of photons can be monitored. This innovative concept extends the principles of QKD to enhance security in imaging applications, particularly in scenarios where data integrity is of utmost importance.

If we examine a two-dimensional Hilbert space, the entities within it are referred to as quantum states, often referred to as qubits. Classical bits are denoted by: |0⟩ = $\left(\begin{array}{c}1\\ 0\end{array}\right)$and |1⟩ = $\left(\begin{array}{c}0\\ 1\end{array}\right)$ then Quantum bits which are superposition of classical bits are denoted by the equation |𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩.Standard basis is represented as|0⟩ and |1⟩, whereas Hadamard basis is represented by |+⟩ = |0⟩+|1⟩/2 and |-⟩ = |0⟩-|1⟩/2 [87].

Table 2. Encoding and Decoding in BB84 protocol ( X:Hadamard basis, Z:Standard basis) [87].

Table 2. Encoding and Decoding in BB84 protocol ( X:Hadamard basis, Z:Standard basis) [87].

Alice encoding			Bob decoding
Basis	Bit	State	X	Z
X	0	+	0	(0 or 1)
X	1	-	1	(0 or 1)
Z	0	0	(0 or 1)	0
Z	1	1	(0 or 1)	1

When Alice encodes her bits on a Hadamard basis, and Bob performs measurements on a Hadamard basis, the process results in deterministic key decoding. However, when Bob measures on the standard basis, there is a 0.5 probability of correctly identifying the encoded bit. To address this, Alice and Bob must employ a classical communication channel to convey the selected measurement basis. This ensures that they maintain the generated keys on the same basis and discard keys if the bases used are different.

For a better understanding, the steps that are typically involved in a QKD protocol like BB84 are given below:

• Bit Selection by Alice: Alice randomly generates a bit, which can be either 0 or 1, using a random number generator.
• Basis Choice by Alice: She then randomly picks one of two quantum bases - the Standard basis or the Hadamard basis - for encoding her bit.
• Bit Encoding and Transmission: Alice encodes her bit in the selected basis into a qubit and sends it to Bob via a quantum channel.
• Basis Selection by Bob: Bob, like Alice, randomly chooses either the Standard or Hadamard basis using a random generator.
• Qubit Measurement by Bob: Upon receiving the qubit, Bob measures it using the basis he selected.
• Repetition of the Process: Alice and Bob repeat the above steps multiple times to generate a series of bits and corresponding bases.
• Basis Disclosure over Classical Channel: After enough bits have been exchanged, Alice and Bob share the bases they used for each bit over a classical communication channel.
• Bit Filtering Based on Basis Match: They discard any bits whose chosen bases do not match, retaining only the bits encoded and decoded on the same basis. This ensures that the remaining bits are more likely identical between Alice and Bob.
• Eavesdropping Detection and Post-Processing: Alice and Bob compare a sample of their bits over a public channel to detect potential eavesdropping. If the match is within an acceptable limit, they proceed with post-processing steps like error correction and privacy amplification to secure communication. In the BB84 QKD (QKD) protocol, a Quantum Bit Error Rate (QBER) exceeding 11% typically indicates a high likelihood of eavesdropping, as this threshold is the upper limit for maintaining secure communication.

In QKD, the Unambiguous State Discrimination attack occurs when an eavesdropper (Eve) can identify the states transmitted by Alice. Eve leverages the inherent losses in the communication channel to her advantage[88,89]. When Eve successfully discriminates the states, she sends either single photon or multiphoton states; otherwise, she sends vacuum states. For non-orthogonal states (0 and 45 degrees), the probability of discrimination is 0.293. A specific condition must be met for security against such attacks in scenarios involving weak coherent states: the product of the channel's transmission efficiency and Bob's detection efficiency should equal 0.293. When the mean photon number (MPN) is set to 1, there is a significant chance of single-photon emissions, but this also increases the likelihood of emitting multiple photons simultaneously. As the MPN increases beyond this point, the probability of single-photon emissions diminishes. On the other hand, when the MPN is reduced to 0.1, the risk of multiphoton emissions drops, which is beneficial for security as it hampers an eavesdropper's ability to gain information about the cryptographic key. Additionally, with an MPN of 0.1, the probability of single-photon emissions is around 9%, which remains practical for generating the key. In the context of Photon Number Splitting (PNS) attacks in QKD, reducing the MPN is a crucial defensive strategy. In PNS attacks, an eavesdropper exploits multiphoton emissions: they can split these photons, keeping one for themselves and forwarding the rest to the intended recipient, thereby gaining partial information about the key without being detected. By minimizing multiphoton emissions, which happen at lower MPNs, the vulnerability to PNS attacks is significantly reduced, enhancing the overall security of the QKD system [90,91,92].

The Decoy state method was proposed by Hwang [93], and security proofs were provided by Hoi Kwong Lo [94]. The technique is useful in practical situations when the source is not an ideal single-photon source but can have multiple photons emitted with the same information. The decoy state protocol is a technique used in QKD to improve the security of information transmitted. In QKD, two parties, Alice and Bob, exchange quantum states to create a secure key. QKD's security is based on quantum mechanics principles, which prevent eavesdropping without detection. On the other hand, attackers can use a technique known as PNS to intercept a photon from the transmitted state and store it for later use[91]. They then allow the remaining photons to continue to the receiver, avoiding detecting the eavesdropper's actions. The attacker can then measure the stored photons to learn about the transmitted key while remaining undetected. To prevent this type of attack, the decoy state protocol is used. In this protocol, Alice sends the key states to Bob and a few "decoy states," randomly chosen states with known photon numbers. Alice can then estimate the photon-number distribution of the transmitted states by measuring the decoy states. By comparing the measured photon-number distribution with the expected distribution, Alice and Bob can detect the presence of an eavesdropper. If the measured distribution differs significantly from the expected distribution, an eavesdropper will have intercepted some of the photons, and the key exchange is aborted. The decoy state protocol is a method used to detect and prevent photon-number splitting attacks in QKD. It enhances the security of the key exchange by allowing Alice and Bob to detect the presence of an eavesdropper [95].

The decoy state method enhances security by allowing for a more accurate estimation of single-photon components in the transmission, which are inherently more secure than multi-photon components. The final key $K_{final}$ is derived as an XOR of the secure single-photon key ($K_1$​) and the less secure multi-photon key ($K_{>1}$), resulting in an overall secure key[87]. The Table 3. provides an overview of experiments conducted using the BB84 protocol in QKD (QKD). In the first experiment with optical fiber as the channel, the BB84 protocol was tested over 50 km, achieving a quantum bit error rate (QBER) of 2% and a secure key rate of 0.5 Kbits/s. The subsequent two experiments utilized the BB84 protocol with the addition of decoy states and were carried out in free space channels. In the 144 km experiment, the QBER rose to 6.48%, but the secure key rate was between 12.8 and 42 bps. Impressively, even at a much longer distance of 1200 km, the QBER was maintained between 1-3%, and the secure key rate was as high as 1.1 Kbps. This highlights the potential of the BB84+decoy protocol for long-distance QKD in free-space channels.

2.4. Quantum Photonic Detectors

In quantum imaging, detectors play a pivotal role in capturing and recording visual information, akin to how the human eye perceives natural scenes. The eye, our primary organ for real-time perception, and cameras, our artificial optical instruments, share several fundamental components and functions [99]. The eye's retina, a light-sensitive layer at the back, comprises multiple layers responsible for distinct functions. Cones and rods in the retina transform incoming light signals into neural signals. Cones excel at distinguishing colors in well-lit conditions, while rods are sensitive to faint light, motion, and variations in intensity. The fovea, a central indentation in the retina, is densely populated with cones, making it the focal point for detailed observation of objects. Digital cameras replicate some aspects of the human visual system. They employ lenses, apertures, image planes, and light sensors to record visual information. In digital cameras, photo sites, small light-sensitive cavities, serve as the analogs to the photoreceptor cells in the retina. These photo sites measure the strength of electrical signals, corresponding to the number of photons captured. A commonly used image sensor in cameras is the charged coupled device (CCD). In CCDs, electrodes are grouped in threes, allowing the application of three distinct voltages. By manipulating these voltages, electrons generated by incoming photons can be collected and converted into a voltage signal at the edge of the silicon sheet [100,101].

2.4.1. Characteristics of Quantum Photonic Detectors

When evaluating detectors for quantum imaging applications, several key characteristics come into play [102] as given below:

• Spectral Range: Detectors must have high efficiency within the specific wavelength range of interest for quantum imaging.
• Dead Time: Dead time represents the recovery period the detector requires after absorbing a photon before registering counts again.
• Dark Count Rate (DCR): DCR refers to the false count rate originating from various sources, including material properties of the detector, biasing conditions, or external noise. Low values for both dead time and DCR are crucial for accurate photon counting.
• Timing Jitter: Timing jitter quantifies the variation in the time interval between the absorption of photons and the generation of an electrical pulse from the detector. Minimal timing jitter is desirable for precise timing measurements.
• Ability to Resolve Photon Number: Conventional detectors often trigger the same response for single-photon and multi-photon pulses. Advanced detection methods, such as arrays of detectors or specialized detectors, are required to distinguish between these cases.
• Detection Efficiency: Detection efficiency is the probability of recording a count when a photon arrives at the detector. High detection efficiency is a fundamental requirement for sensitive photon detection.

Figure 8 shows the comparison of photon detection probabilities. The x-axis of the graph represents the wavelength of the light in nanometers, ranging from 300 nm to 1100 nm, covering the ultraviolet, visible, and near-infrared portions of the electromagnetic spectrum. The y-axis represents the PDP (photon detection probabilities) in percentage, indicating the efficiency of each photodetector at each wavelength. The graph shows that each detector has its characteristic response curve, with varying sensitivity levels at different wavelengths. For instance, some detectors are more efficient in the visible spectrum, while others perform better in the near-infrared range. The figure. provides essential information for selecting a photodetector for a specific application, considering the required sensitivity and wavelength range.

2.4.2. Important Detectors Used for Quantum Photonic Imaging Applications:

Several types of detectors are commonly used in quantum imaging, where each one has its own set of characteristics, advantages and limitations.

• Intensified CCD (ICCD): Intensified CCD cameras (ICCDs) feature an image intensifier that boosts external photons that hit the intensifier through impact ionization. This results in effective signal enhancement and the ability to regulate exposure time [104]. ICCDs possess the potential for single photon detection, with current detection efficiencies hovering around 50 percent and maintaining minimal dark counts. Notably, the intensifier can be briefly gated, lasting durations in the picosecond range, making these cameras particularly suitable for photon correlation studies[105]. It is important to note that ICCD cameras only amplify external photons that reach the intensifier. This ensures low-noise signal amplification and nanosecond-level control over exposure to eliminate undesired light interference.
• Electron-Multiplying Charge-Coupled Devices (EMCCDs): Electron-Multiplying Charge-Coupled Devices (EMCCDs) amplify signals after a CCD detects light. They utilize a register specifically for electron multiplication, which boosts the detected electronic signal [106,107]. Regarding detection efficiency, EMCCDs outperform ICCDs and have reduced readout noise [108]. However, their signal-to-noise ratio (SNR) often falls short compared to ICCDs because of the amplified dark counts. EMCCDs necessitate cooling and are expensive.
• Single-Photon Avalanche Diodes (SPADs): SPADs are semiconductors that generate electron-hole pairs when photons strike a semiconductor material [109]. These devices work under reverse bias and use avalanche multiplication to boost their photoelectric response. They can achieve efficiencies close to 80 percent in the visible spectrum, but this drops to about 20 percent in the infrared spectrum. It is essential to understand that these detectors cannot precisely measure the number of photons. One of the intriguing features is their ability to form two-dimensional arrays of SPADs, such as 16x16 macro pixels, 64x64 pixels, and 256x256 pixels [110]. Each pixel acts as an individual SPAD in these configurations, which is especially advantageous for coincidence imaging. However, because of the adjacent electronic components, the fill factor (9.6, 26.5, and 61 respectively for pixels above) is not optimal, leading to a reduction in the overall detection efficiency. A promising solution to this problem could be using microlens arrays[50,103,110,111,112].

Superconducting Nanowire Single-Photon Detectors (SNSPDs): Superconducting nanowire single-photon detectors (SNSPDs) depend on superconducting characteristics [113]. When a photon strikes the nanowire, it creates a localized temperature spike, increasing resistance at that spot. This change results in the superconducting material, primarily NbN, transitioning into a resistive state, with resistance surging to about 50 kiloohms. By pairing this with a lower-resistance impedance (close to 50 ohms), the ensuing current takes the route of least resistance. This mechanism allows for the identification of individual photons. These SNSPD operate at temperatures near 2.1 Kelvin and show high efficiency in both the visible (VIS) and near-infrared (NIR) spectrum, often surpassing 80 percent [114].

Table 4. Comparison of the parameters of detector.

Table 4. Comparison of the parameters of detector.

Detectors	Operating temperature	Efficiency (Visible/IR)	Drawbacks
ICCD	-30 °C	10-30%	Large acquisition time
EMCCD	-100 °C	90%	Thermal noise, costly, bulky
SPAD	-50 °C	30-60%	Low Fill factor
SNSPD	2.1 Kelvin	>80%	Costly, operating temperature

In high-performance imaging systems such as EMCCDs and ICCDs, a critical aspect of noise management involves the Noise Factor (F) associated with the amplification process. For EMCCDs, this factor is typically $\sqrt{2}$ or approximately 1.41, while for ICCDs it varies between $\approx$1.6 to 3.5, depending on the intensifier tube used. In deep-cooled, low-noise operational regimes with negligible dark and spurious noise, the Signal-to-Noise Ratio (SNR) can be simplified, focusing primarily on the readout noise ($N_{RN}$) and the actual photon count. The SNR for these systems can be expressed as

$$SNR\left(EMCCD,ICCD\right)=\left(QE*P\right)/(\sqrt{{(\left(\frac{N_{RN}}{G}\right)}^2+F^2{*\left(N_{SN}\right)}^2)}$$

where QE represents Quantum Efficiency, and P is the Photon Count. Notably, by increasing the gain (G), the term involving the read-out noise becomes less significant compared to the intrinsic shot noise of the signal. This reduction in the relative impact of readout noise enhances the ultra-sensitive detection capability of these devices, making them particularly effective in low-light conditions and applications requiring high accuracy and precision [115].

2.4.3. Computational Ghost Imaging

Computational ghost imaging relies on a fundamentally different process from traditional imaging. Instead of capturing an image through lenses, this technique employs a spatial light modulator to modulate a light source, which is then reflected off the target. The reflected light is captured not by a typical array of pixels but by a single-pixel detector. The experiment involves several key steps:[13,116]

a. Modulation of Light: The light source is modulated using a spatial light modulator (SLM). This modulation typically involves presetting intensity patterns of speckles.

b. Reflection from the Target: The modulated light interacts with the target object. The properties of the object affect how this light is reflected.

c. Detection: The reflected light is captured by a single-pixel detector. Unlike traditional detectors, which capture a complete image, this detector measures the total light intensity over time.

The core of computational ghost imaging lies in its unique way of image reconstruction:

i) Intensity Pattern Modulation: A Digital Micromirror Device (DMD) modulates numerous intensity patterns, denoted as $I_i\left(x,y\right)$ where i ranges from 1 to N, the total number of patterns.

ii) Single-Photon Detection: Single-photon detectors measure the counts (denoted as $B_i$) corresponding to each pattern. These counts reflect the interaction of each pattern with the object.

iii) Image Reconstruction: The final image O(x,y) of the object is computed by summing the weighted intensities of each pattern. The weights are derived from the count​adjusted by the mean value of counts and intensities.

The reconstruction of the image is governed by a mathematical formula that integrates the measured counts and the corresponding intensity patterns. This involves calculating the mean values and correlating them with the transmission function of the object, symbolized as T(x,y).

$$O\left(x,y\right)=\frac{1}{N}[\sum _{i=1}^N(B_i-⟨B_i⟩[I_i\left(x,y\right)-⟨I_i\left(x,y\right)⟩]$$

where the mean value is

$$⟨B_i⟩=\frac{1}{N}\sum _{i=1}^N{B}_i$$

$$B_i\propto \iint T\left(x,y\right)I_i(x,y)$$

3. Quantum Imaging Schemes

3.4. Quantum Imaging Using Undetected Photons

A nonlinear quantum interferometer setup involves two nonlinear crystals. A laser beam is split into two beams, which coherently pump the crystals. The idler wave interacts with an object; another is generated and co-aligned with the first. The signal waves from both idler interactions are combined at a beam splitter. The interference of signal waves depends on the phase and transmission properties of the object probed by the first idler wave. Superposition of the idler waves makes it impossible to determine their origin crystal, inducing coherence between emitted photon pairs [15]. In induced coherence-based quantum imaging, photon pairs are created from two-photon sources. By aligning the paths of one photon (the idler) from both sources and ensuring their path identity, the other photon (the signal) exists in a superposition state. This superposition state encodes information about the phase and transmissivity of the idler photon. Placing an object on the idler path allows capturing images exclusively with the signal photons, which remain unaffected by any interaction with the object [139,140,141].

A nonlinear quantum interferometer without direct photon detection for imaging purposes. At the beginning of the setup, a 532 nm laser (L1) is employed as the primary light source. The light from this laser is passed through wave plates (WPs) to control its polarization precisely, followed by a half-wave plate (HWP) to adjust the polarization state further. A polarizing beam splitter (PBS) divides the beam into two distinct optical paths. These paths pump two nonlinear crystals (NL1 and NL2) where the light interacts with the internal structure of the crystals to generate new light waves—typically known as parametric down-conversion. One of the paths is directed towards the first nonlinear crystal (NL1), and the generated idler wave interacts with an object (O). In contrast, the signal wave is redirected towards a beam splitter (BS). The second path is similarly directed to another nonlinear crystal (NL2), generating a second idler wave co-aligned with the first one and a signal wave that also heads to the BS. Mirrors (M1, L2, L3, L4) and lenses (L2', L3', L4') are positioned to guide and focus the beams through the system. At the BS, the signal waves from both paths are combined, and their interference pattern is influenced by the phase and transmission properties of the object probed by the first idler wave. Since the idler waves are in a superposition state, it is impossible to determine their crystal of origin, which creates coherence between the emitted photon pairs. This interference is then passed through filters and directed toward an electron-multiplying CCD (EMCCD) camera for detection. The innovative aspect of this setup is that the image of the object can be obtained without directly detecting the photons that interact with it[15].

Spirulina is mainly a phase object at the illumination wavelength used. At the same time, the fruit fly wing is predominantly an amplitude object due to its absorption characteristics in the visible and near IR range. The resolution of the images is emphasized as surpassing that of conventional methods by an order of magnitude or more. Figure 18 is divided into eight separate image panels labeled (a) through (h), which are grouped in pairs to show the amplitude and phase images of two different biological samples at various scales and step sizes. Panels (a) and (b) display the amplitude images of Spirulina filaments, a type of cyanobacteria with distinct helical structures. Now, panels (c) and (d) present the phase images of the same Spirulina cell culture, which appear to have higher contrast and possibly more detail compared to the amplitude images, in line with the inference about the quality of phase images. The next set of images relates to the wing of a fruit fly, illustrating its complex structural details at different scales, where panels (e) and (g) show conventional microscope images of the fruit fly wing, captured in the visible spectrum, displaying the wing's hierarchical structure, including veins and hair-like formations. Finally, panels (f) and (h) offer the corresponding images, which are expected to have significantly higher resolution due to the quantum imaging technique, revealing finer structural details.

One of the captivating discoveries in quantum physics is the wave-particle duality of particles like photons. This duality implies that photons can exhibit wave-like and particle-like behaviors, depending on whether we can distinguish their paths [143,144]. When two beams of light superimpose, they create an intensity pattern with interference fringes, demonstrating the wave-like nature of light [145,146]. However, when we make the paths of photons distinguishable, such as by altering their polarization, they behave more like individual particles than waves. Mach-Zehnder interferometer (MZI) is a critical tool that leverages the wave-particle duality of photons [147]. It involves splitting a beam of light and recombining it, leading to interference patterns when paths are indistinguishable. In the context of quantum imaging, the paths of photons can be intentionally made distinguishable, disrupting their wave-like behavior. However, this distinguishability can be reversed to regain coherence or wave-like behavior. This phenomenon forms the basis for induced coherence-based quantum imaging. MZI is at the core of this imaging technique. When dealing with single photons or coherent laser light, they exhibit wave-like behavior when their potential pathways cannot be distinguished. This leads to interference patterns in the form of fringes. However, altering the polarization of photons in one of the pathways makes their paths distinguishable, and they behave more like classical particles.

Figure 19a shows a setup where single photons or coherent laser light are not subjected to any distinguishing process. Here, the paths of the photons cannot be differentiated, which is a scenario conducive to wave-like behavior resulting in interference patterns. This is indicated by the single tall probability bar at detector D1, suggesting constructive interference at this point and no detection at D2. b), a half-wave plate (HWP) is introduced at a 45-degree angle in one of the pathways. This alteration changes the polarization of the photons in that path, which makes it possible to distinguish which path the photons have taken. As a result, the wave-like interference pattern is destroyed, and the photons are detected as if they were classical particles, indicated by the two smaller probability bars at detectors D1 and D2. This demonstrates the collapse of the interference pattern into two distinct probabilities corresponding to the two potential paths the photons can take. This experiment is a typical demonstration of the quantum mechanical principle where measurement (in this case, distinguishing the paths via polarization) affects the behavior of quantum particles.

In an interaction-free measurement setup using a MZI, detectors are placed to observe constructive and destructive interference. When a single photon is introduced, it should be detected at the constructive interference detector. However, if a perfectly absorbing object is placed in one arm of the interferometer, it can transmit a single photon through the system. Detection at the destructive interference detector indicates the presence of an object in one arm, even though the detected photon might not have interacted with the object [148,149]. As shown in Figure 20, the photons first encounter a dichroic mirror (DM1), which directs the beam into the interferometer's arms. In one arm, another dichroic mirror (DM3) separates an undetected idler photon. The numbered circles represent different paths or arms within the interferometer. A beam splitter (BS) divides the photon's path within the interferometer, creating two routes. In one of these routes, a perfectly absorbing object may be placed. The presence of this object can change the outcome of the interference pattern observed by the detectors. The two paths are recombined at a second beam splitter, and the interference pattern is observed. The phase difference introduced by the object (if present) and the system itself is denoted by $\theta$ for the signal photon and $\varphi$ for the relative phase of the interaction-free measurement. The detectors $R_1,R_{2}and R_3$ are placed to observe constructive and destructive interference patterns. When a photon is detected at $R_2$, it indicates the object's presence in the other arm due to the destructive interference pattern despite the photon not directly interacting with the object. The system also includes an intensified CCD (ICCD) camera represented by inset f, which detects the signal photon. This contrasts with the single-photon detector (SPD) shown in inset e, another detection method not utilized in this setup. The SLM is employed to manipulate the phase or amplitude of the light in the system. The entire arrangement allows for quantum imaging by detecting the interference patterns, indirectly providing information about the object's presence without the photon interacting.

Figure 21 shows the images obtained a) With ICCD b) with single pixel imaging method for the undetected imaging method, here the single pixel imaging based method enhances the brightness of the image.

Table 5. State of art techniques in quantum imaging; PPKTP: Periodically Poled Potassium Titanyl Phosphate, EMCCD: Electron-Multiplying Charge-Coupled Device, BBO: Beta Barium Borate, CCD: Charge-Coupled Device, SPAD: Single Photon Avalanche Diode ICCD: Intensified Charge-Coupled Device, LN Type 1: Lithium Niobate Type 1, SPD: Single Photon Detector, SLM: Spatial Light Modulator.

Table 5. State of art techniques in quantum imaging; PPKTP: Periodically Poled Potassium Titanyl Phosphate, EMCCD: Electron-Multiplying Charge-Coupled Device, BBO: Beta Barium Borate, CCD: Charge-Coupled Device, SPAD: Single Photon Avalanche Diode ICCD: Intensified Charge-Coupled Device, LN Type 1: Lithium Niobate Type 1, SPD: Single Photon Detector, SLM: Spatial Light Modulator.

Technique	Type	Crystal type	Detectors	Resolution/ SNR/Visibility	Comments
Quantum imaging with undetected photons [15,150]	Interference based	PPKTP type 0	EMCCD	Resolution is determined by the light interacting with the object, the shorter it is better. Edge function depends on the light detected by camera.	1)Useful in biological samples and without having spatial resolving detectors. 2) Interferometric setup complexity 3)In low light regime its useful for biological samples without much perturbation to the sample, (LN type 1 crystal, detectors SPD and SLM) [148]
Sub shot noise quantum imaging [46]	Correlation based	BBO Type 2	CCD	50 % sensitivity enhancement	1)Improvement in sensitivity by quantum states 2) Precise measurement required
Quantum ghost imaging [119,151]	Correlation based	BBO	SPAD and ICCD	1) Resolution limited by point spread function of optics with camera (same as conventional imaging), SNR and visibility are reported to be better.	1)Speckle contrast is minimal as compared to conventional imaging system 2)Near field and far field imaging possible with the same setup whereas in conventional it is limited to one each. 3) Degradation in correlation reduces the resolution
Quantum illumination [132]	Correlation based	BBO type 2	EMCCD	99.9% rejection of background light and sensor noise	Useful in stealth object detection.

4. Discussion and Conclusions

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