Circular Polarization-Based Quantum Encoding for Image Transmission over Error-Prone Channels

Abstract

Quantum image transmission over noisy communication channels remains a challenge due to the fragility of quantum states and their susceptibility to channel impairments. Existing quantum encoding schemes often exhibit limited noise resilience, while advanced approaches introduce computational and implementation complexity. To address these limitations, this paper proposes a circular polarization-based quantum encoding framework for image transmission over error-prone channels. In the proposed approach, source images are compressed and source-encoded using standard image coding formats, including the joint photographic experts group (JPEG) standard and the high-efficiency image file format (HEIF), and converted into classical bitstreams. The resulting bitstreams are protected using channel coding and mapped onto quantum states via circular polarization representations, where left- and right-hand circularly polarized states encode binary information. The encoded quantum states are transmitted over noisy quantum channels to model channel impairments. At the receiver, appropriate quantum decoding and channel decoding operations are applied to recover the classical bitstream, followed by source decoding to reconstruct the image. The performance of the proposed framework is evaluated using image quality metrics, including peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and universal quality index (UQI). Simulation results demonstrate that the proposed circular polarization-based encoding scheme outperforms existing quantum image encoding techniques, achieving channel SNR gains of 4 dB over state-of-the-art Hadamard-based encoding and 3 dB over frequency-domain quantum encoding methods under severe noise conditions. These results indicate that circular polarization-based quantum encoding provides improved noise robustness and reconstruction fidelity for practical quantum image transmission systems.

1. Introduction

The transmission of visual information plays a fundamental role in modern communication systems, supporting applications such as remote sensing, medical imaging, multimedia services, surveillance, and scientific data exchange. To enable efficient storage and transmission of visual data, image compression techniques such as the joint photographic experts group (JPEG) [1] standard and the high-efficiency image file format (HEIF) [2] are widely used to reduce data redundancy while maintaining acceptable visual quality. However, the reliable transmission of compressed images over noisy communication channels remains challenging. Compressed bitstreams are highly sensitive to channel errors, where even small disturbances can lead to error propagation and significant degradation in reconstructed image quality. Conventional communication systems rely on channel coding and retransmission techniques to mitigate such effects, but these approaches often introduce additional latency, complexity, and bandwidth overhead, particularly under severe noise conditions.

To address these limitations, quantum communication [3] has emerged as a promising paradigm for next-generation communication systems. By exploiting quantum-mechanical properties such as superposition [4] and entanglement [5,6], quantum communication enables new approaches for information representation and transmission that extend beyond classical frameworks. In particular, polarization-based quantum encoding provides a practical mechanism for representing information using photonic quantum states. Motivated by these capabilities, this work investigates a circular polarization-based quantum encoding framework for robust image transmission over error-prone channels, aiming to improve transmission reliability while maintaining low implementation complexity.

Several previous studies have explored the optimization of quantum communication systems for image transmission by investigating different quantum encoding strategies and incorporating quantum error correction techniques [7]. Widely adopted quantum image encoding methods, such as Hadamard-based schemes [8,9], rely on linear superposition and polarization-based quantum state representations, enabling relatively simple implementations. However, these schemes exhibit limited resilience in high-noise environments. While quantum error correction techniques can be applied to improve robustness, they typically introduce substantial computational and implementation complexity. To further enhance noise tolerance, frequency-domain quantum encoding techniques [10] have been proposed to exploit spectral characteristics; nevertheless, such approaches require complex quantum operations and increased system overhead, which significantly limit their practical deployability. Consequently, despite notable progress, a quantum image encoding framework that simultaneously achieves strong noise robustness, low computational complexity, and practical implementation feasibility remains an open challenge, motivating the proposed approach.

In response, this work aims to establish a fundamentally different quantum image encoding approach based on circular polarization states, departing from the linear polarization representations commonly adopted in existing quantum image transmission frameworks. Rather than relying on conventional Hadamard-based linear polarization encoding or computationally intensive frequency-domain quantum transformations, the proposed approach embeds classical binary information directly into left- and right-hand circularly polarized quantum states. This encoding choice introduces inherent noise resilience at the physical state representation level, enabling improved tolerance to channel-induced phase and amplitude distortions without increasing system complexity. By shifting robustness from the linear polarization domain to the circular polarization domain itself, the proposed method aims to overcome the long-standing trade-off between noise resilience and implementation complexity in quantum image transmission. As a result, the framework offers a distinctive and practically viable alternative to state-of-the-art quantum encoding techniques.

In the proposed method, source images are first compressed using standard image coding formats, such as JPEG and HEIF, to reduce redundancy while preserving visual quality. The compressed data are then converted into classical bitstreams and protected using channel coding to mitigate transmission errors. Subsequently, the encoded bitstreams are mapped onto quantum states using left- and right-hand circular polarization representations, which serve as the quantum carriers of binary information. These circular polarization-based quantum states are transmitted over noisy quantum channels to model realistic channel impairments. At the receiver, quantum measurement and decoding operations are applied to recover the transmitted quantum states, followed by channel decoding and source decoding to reconstruct the original image. This end-to-end framework is designed to enhance noise robustness while maintaining low computational and implementation complexity, making it suitable for practical quantum image transmission scenarios. The results demonstrate that the proposed system outperforms state-of-the-art linear polarization-based Hadamard encoding schemes, as well as more advanced frequency-domain quantum encoding techniques, while achieving higher image quality and improved reconstruction fidelity.

The key novelties of the proposed work are summarized as follows:

• A circular polarization-based quantum image encoding framework is introduced, employing left- and right-hand circular polarization states for binary image representation.
• The proposed method achieves enhanced noise resilience compared to linear polarization-based Hadamard encoding schemes, particularly under high-noise channel conditions.
• Low computational and implementation complexity is maintained by avoiding the complex quantum operations required in frequency-domain quantum encoding techniques.
• Improved image reconstruction quality and fidelity are demonstrated through comparative evaluation against state-of-the-art quantum image encoding methods.

The remainder of this paper is organized as follows. Section 2 reviews related work on quantum image transmission and existing quantum encoding techniques. Section 3 presents the proposed circular polarization-based quantum image transmission methodology. Section 4 discusses the simulation results and provides a comparative performance analysis. Finally, Section 5 concludes the paper and outlines directions for future work.

2. Related Works

Research on reliable image transmission has evolved along both classical and quantum communication paradigms, driven by the need to preserve visual quality under imperfect channel conditions. In classical systems, image robustness is primarily addressed through source compression combined with channel-level protection, where redundancy is introduced to counteract transmission errors [11,12]. Although such strategies have enabled practical image delivery over noisy links, their effectiveness diminishes as compression levels increase and channel conditions become more adverse, motivating interest in alternative transmission models beyond conventional classical frameworks.

Quantum communication [13] is initially developed with a primary emphasis on secure information exchange, leveraging fundamental quantum-mechanical properties [14] to guarantee confidentiality and detect eavesdropping [15,16]. Early quantum communication systems focused on cryptographic applications [17,18,19], most notably quantum key distribution [20,21], which demonstrated provably secure transmission of information over optical fiber and free-space links. Building on these foundations, quantum communication techniques are extended in several studies to secure image and multimedia transmission, where visual data are encrypted or protected using quantum-assisted security mechanisms [22,23,24,25]. In these works, the primary objective is to ensure data confidentiality and integrity rather than to preserve visual quality under noisy channel conditions. Consequently, while such approaches demonstrate the feasibility of applying quantum communication to image transmission, robustness to channel noise and high-fidelity image reconstruction are not their central design goals.

Subsequent research is directed toward high-fidelity image and video transmission over quantum channels. Early approaches are primarily based on linear polarization-based single-qubit encoding schemes, which are commonly implemented using Hadamard operations due to their low complexity and ease of realization [8]. These methods are shown to outperform state-of-the-art classical systems under certain conditions while maintaining low computational complexity. However, limited error resilience is observed due to the reliance on single-qubit representations, making such schemes highly sensitive to channel noise and quantum state perturbations. To improve robustness and transmission efficiency, linear polarization-based multi-qubit encoding schemes are subsequently developed, enabling multiple qubits to jointly represent visual information and improving scalability [9]. Although enhanced noise tolerance is achieved with increasing system dimensionality, the resulting resilience is still insufficient to overcome the inherent fragility of quantum states under high-noise channel conditions, thereby limiting the effectiveness of these approaches for reliable high-fidelity image and video transmission.

In some studies, different quantum error correction-based quantum communication systems [26,27,28,29] are applied to image and video transmission to improve robustness against channel noise [30]. With the incorporation of quantum error correction, enhanced error resilience is achieved, even when simple schemes such as the three-qubit quantum error correction code [31,32,33] are employed. However, these approaches require the use of ancillary qubits for syndrome measurement and error recovery, which leads to increased system complexity and resource overhead. As a result, although improved noise tolerance is provided, the additional qubit requirements and circuit complexity limit the practical applicability and scalability of quantum error correction-based quantum image transmission systems.

To further enhance robustness, frequency-domain quantum encoding systems [10] are also investigated for quantum image transmission. In these approaches, the quantum Fourier transform (QFT) [34,35,36] is applied to represent quantum states in the spectral domain, where improved tolerance to certain noise characteristics is achieved. By exploiting frequency-domain properties, enhanced noise resilience is reported compared to time-domain or linear polarization-based encoding schemes [10]. However, the application of frequency-domain quantum encoding requires complex quantum transformations, deep quantum circuits, and increased computational resources [37]. Consequently, although improved robustness is achieved, the high operational complexity and hardware requirements significantly limit the practical deployability of frequency-domain quantum encoding systems for high-fidelity image and video transmission, particularly on near future quantum devices.

Key Research Gaps and Motivation

The existing body of research on quantum image transmission reveals a persistent trade-off between noise resilience, system complexity, and practical scalability, as summarized in

Table 1. Comparison of existing quantum image transmission approaches.

Approach	Encoding/ Representation	Computational Complexity	Noise Resilience (Typical Channel SNR Gain)	Key Limitations
[8]	Linear polarization, Hadamard-based single-qubit encoding	Low ($\mathcal{O}\left(1\right)$)	Low (∼2–2.5 dB gain compared to classical systems)	Highly sensitive to channel noise and limited throughput
[9]	Linear polarization, multi-qubit state representation	Moderate ($\mathcal{O}\left(n\right)$)	Moderate (performance increases with qubit dimension; >2 dB gain between consecutive qubit sizes)	Increased circuit complexity and limited robustness under severe noise
[30]	Linear polarization with quantum error correction	High ($\mathcal{O}\left(3\right)$ system complexity with additional ancillary qubit requirements)	High (∼8 dB gain achievable even with simple three-qubit QEC schemes compared to classical polar-code-based systems with the same code rate)	Ancillary qubits and increased circuit depth
[38]	Linear polarization, parallel single-qubit transmission	High ($\mathcal{O}\left(n\right)$ parallel operations)	Moderate (∼3 dB gain compared to classical spatial diversity systems)	Scalability and synchronization complexity
[10]	QFT-based spectral representation	Very High ($\mathcal{O}\left(n^2\right)$ quantum Fourier operations)	Moderate (∼2–3 dB gain compared to linear polarization based multi-qubit systems)	Computationally intensive and limited near-term practicality

. In this table, noise resilience levels are expressed relative to bandwidth-equivalent classical transmission systems based on typical signal-to-noise ratio (SNR) improvements reported in the literature, and n denotes the number of encoding qubits. Linear polarization-based single-qubit and multi-qubit encoding schemes offer low-complexity implementations but exhibit limited robustness against channel noise, particularly under high-noise conditions. Although multi-qubit strategies improve scalability and transmission efficiency, their noise resilience remains insufficient to fully mitigate the inherent fragility of quantum states. Quantum error correction-based approaches provide enhanced error tolerance; however, they rely on ancillary qubits and additional quantum operations, resulting in increased circuit depth, resource overhead, and implementation complexity. Similarly, frequency-domain quantum encoding techniques demonstrate improved robustness through spectral representations but require computationally intensive quantum transformations that restrict their applicability to near future quantum hardware and real-time multimedia transmission scenarios.

These limitations highlight a clear research gap in the development of quantum image transmission frameworks that can achieve strong noise robustness without incurring excessive computational and implementation complexity. Motivated by this gap, the present work aims to explore an alternative quantum encoding strategy that introduces inherent noise resilience at the encoding level itself, rather than relying primarily on complex quantum transformations or high-overhead error correction mechanisms. By leveraging circular polarization states for quantum image encoding, the proposed approach seeks to strike an effective balance between robustness, complexity, and practical feasibility, addressing a critical limitation in existing quantum image transmission systems.

3. Methodology

This section describes the design and operation of the proposed quantum image transmission framework based on circular polarization-based encoding, as illustrated in Figure 1. The overall methodology follows an end-to-end processing pipeline in which classical images are first source encoded using standard image compression formats, including JPEG and HEIF. The resulting bitstreams are then protected using channel coding based on polar codes with a code rate of 1/2, followed by quantum encoding using circular polarization-based state representation. The encoded quantum states are transmitted over noisy quantum channels to model realistic channel impairments. At the receiver, the received quantum states are measured and converted back into classical bitstreams, which are subsequently processed through channel decoding and source decoding stages to reconstruct the transmitted image. To evaluate the intrinsic performance of the proposed quantum encoding scheme independent of channel coding gains, an uncoded transmission scenario is also considered, where channel coding is omitted. To facilitate a clear understanding of the proposed system, each functional block of the methodology is explained in detail in the following subsections.

3.1. Input Image Dataset

In this research, input images are selected from two publicly available datasets: the Microsoft COCO dataset [39] and a high-resolution 4K image dataset obtained from Kaggle [40]. A total of 100 images are selected from the Microsoft COCO dataset, which is chosen due to its wide diversity in scene content, object density, and spatial complexity. This diversity enables a realistic evaluation of the robustness of the proposed framework under varying visual characteristics. Images from the COCO dataset are evaluated at two spatial resolutions, $256\times 192$ and $640\times 427$. To assess the scalability of the proposed framework under ultra-high-resolution conditions and increased spatial detail, a Kaggle 4K image dataset is employed, in which 40 images are evaluated at full 4K resolution. It is emphasized that the proposed circular polarization-based quantum communication framework is input-agnostic and does not impose restrictions on image resolution, content, or format. Any image that can be converted into a classical bitstream through standard source encoding techniques can be processed by the proposed method, ensuring broad applicability across diverse image transmission scenarios.

3.2. Source Encoding

In the proposed framework, standard image source encoding techniques are employed to convert input images into compressed bitstreams prior to quantum processing. Specifically, the JPEG standard and the HEIF are used as representative source encoders due to their widespread adoption and effectiveness in reducing spatial redundancy while preserving visual quality. These source encoders enable the generation of realistic compressed bitstreams that reflect practical image transmission scenarios. To evaluate the impact of compression strength on the performance of the proposed quantum communication system, two quantization parameter (QP) settings are considered for both JPEG and HEIF encoding. A higher QP value is used to represent moderate compression while achieving near-lossless quality, whereas a lower QP value is employed to model aggressive lossy compression, resulting in reduced bitrates and increased sensitivity to channel noise. The compressed outputs from the source encoder are converted into binary bitstreams, which serve as the input to the subsequent channel coding and quantum encoding stages. This approach allows the influence of source compression on quantum image transmission performance to be systematically analyzed.

3.3. Channel Encoding

Following source encoding, the compressed image bitstreams are protected using channel coding to mitigate the effects of noise introduced during quantum state transmission and measurement. In the proposed framework, polar codes [41,42] with a code rate of $1/2$ are employed as the channel encoder. Polar codes are selected due to their capacity-achieving property for symmetric binary-input memoryless channels and their suitability for structured and low-complexity implementations.

Compared to widely used channel coding schemes such as low-density parity-check (LDPC) codes [43] and Turbo codes [44], polar codes offer several advantages in the context of the proposed quantum image transmission system. Unlike Turbo codes, which require iterative decoding with high computational complexity and latency, polar codes enable deterministic encoding and decoding with well-defined complexity. In comparison to LDPC codes, polar codes do not rely on large parity-check matrices or extensive iterative message passing, making them more amenable to integration with quantum–classical hybrid systems where computational resources and latency are constrained. Additionally, polar codes exhibit stable performance across a wide range of SNR and are particularly effective at moderate code rates.

A code rate of $1/2$ is selected to achieve a balanced trade-off between error protection and transmission efficiency. This rate provides sufficient redundancy to effectively mitigate channel-induced errors while avoiding excessive bandwidth expansion, which is particularly important when transmitting compressed image bitstreams. At lower code rates, although stronger error protection can be achieved, the increased redundancy significantly reduces spectral efficiency and increases system complexity. Conversely, higher code rates offer improved efficiency but insufficient error resilience under noisy channel conditions. Therefore, a code rate of $1/2$ represents a practical and widely adopted compromise that enables robust performance across a broad range of noise levels, making it well suited for the proposed quantum image transmission framework. These characteristics make polar codes well suited for protecting compressed image bitstreams in the proposed framework, where robustness, efficiency, and practical implementability are critical considerations.

3.4. Quantum Encoding

In the proposed framework, the channel-coded classical bitstream is converted into a sequence of quantum states prior to transmission. Each classical bit is individually mapped onto a qubit prepared in the computational basis. A classical bit with value 0 is mapped to the quantum state $|0\rangle$, while a classical bit with value 1 is mapped to the quantum state $|1\rangle$. The computational basis states corresponding to $|0\rangle$ and $|1\rangle$ are defined in column-vector form in Equation (1) and Equation (2), respectively.

$$|0\rangle =\left[\begin{array}{c}1\\ 0\end{array}\right]$$

(1)

$$|1\rangle =\left[\begin{array}{c}0\\ 1\end{array}\right]$$

(2)

In practice, to achieve this mapping efficiently, all qubits are initially prepared in the ground state $|0\rangle$. For a given classical bit $b\in \{0,1\}$, the quantum state preparation is implemented by applying a Pauli-X gate only when the bit value is 1. This process can be expressed as in Equation (3).

$$|q_b\rangle =X^b|0\rangle$$

(3)

where X denotes the Pauli-X (bit-flip) operator [45]. This approach enables scalable and low-complexity state preparation, as active gate operations are required only for qubits corresponding to classical bit values of 1.

Once the qubits are prepared in the computational basis, circular polarization-based quantum encoding is applied. The encoding operation is performed using the unitary transformation shown in Equation (4).

$$U_{\mathrm{CP}}={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 1\\ -i& i\end{array}\right]$$

(4)

This transformation maps the computational basis states onto a pair of orthogonal circularly polarized quantum states. Applying the unitary operator $U_{\mathrm{CP}}$ to the computational basis states results in the right-hand ($|R\rangle$) and left-hand ($|L\rangle$) circularly polarized states, as given in Equation (5) and Equation (6), respectively. These states form an orthogonal basis and are used to represent binary information in the proposed encoding scheme.

$$U_{\mathrm{CP}}|0\rangle =|R\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(|0\rangle -i|1\rangle \right)={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}1\\ -i\end{array}\right]$$

(5)

$$U_{\mathrm{CP}}|1\rangle =|L\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(|0\rangle +i|1\rangle \right)={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}1\\ i\end{array}\right]$$

(6)

By embedding classical information into the phase structure of the quantum states, the proposed circular polarization-based encoding introduces inherent robustness to channel-induced phase and amplitude distortions while maintaining a low-complexity and physically realizable implementation. The resulting circularly polarized quantum states are then transmitted over the noisy quantum channel.

3.5. Quantum Channel

To evaluate the robustness of the proposed circular polarization-based quantum image transmission framework under realistic transmission conditions, the quantum channel is modeled using a set of canonical quantum noise processes. Specifically, five standard noise models are considered: bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping noise [46]. These models are widely adopted in quantum communication research to characterize dominant decoherence and stochastic error effects encountered during quantum state propagation. Rather than assuming an ideal channel, the incorporation of multiple physically meaningful error mechanisms enables a comprehensive and reproducible performance evaluation.

In this work, the primary analysis focuses on evaluating the behavior of the proposed system under stochastic quantum noise while assuming negligible polarization-dependent perturbations, which is a platform-dependent effect and may be minimal in several practical quantum communication implementations. Under such conditions, the adopted quantum noise models allow the fundamental encoding advantages of the proposed framework to be analyzed. The impact of polarization drift and its interaction with stochastic quantum noise processes are investigated separately in Section 4.8, where an extended channel model incorporating polarization rotation and the corresponding simulation results are presented.

3.5.1. Composite Quantum Noise Model

The overall effect of channel noise on a transmitted quantum state is represented using a composite quantum channel, denoted by $\mathcal{C}\left(\rho \right)$, where $\rho$ is the density matrix of the encoded quantum state. The composite channel is formulated as a probabilistic mixture of individual quantum noise processes, as defined in Equation (7).

$$\mathcal{C}\left(\rho \right)=(1-p_{\mathrm{total}})\rho +p_B\mathcal{B}\left(\rho \right)+p_P\mathcal{P}\left(\rho \right)+p_D\mathcal{D}\left(\rho \right)+p_A\mathcal{A}\left(\rho \right)+p_{\Phi }\mathcal{F}\left(\rho \right)$$

(7)

where $\mathcal{B}(·)$, $\mathcal{P}(·)$, $\mathcal{D}(·)$, $\mathcal{A}(·)$, and $\mathcal{F}(·)$ denote the bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping channels, respectively. The parameters $p_B$, $p_P$, $p_D$, $p_A$, and $p_{\Phi }$ denote the occurrence probabilities associated with each respective noise process. The term $(1-p_{\mathrm{total}})\rho$ represents the probability that the quantum state remains unaffected during transmission, where $p_{\mathrm{total}}$ is the total channel noise probability defined as the sum of the individual noise probabilities.

3.5.2. SNR-Dependent Noise Scaling

The total noise probability $p_{\mathrm{total}}$ is modeled as a function of the channel SNR in order to reflect varying channel quality. This relationship is defined in Equation (8), ensuring that the overall error probability decreases as the SNR increases.

$$p_{\mathrm{total}}=min\left(1,{\displaystyle \frac{1}{1+{10}^{\mathrm{SNR}/10}}}\right)$$

(8)

3.5.3. Noise Probability Allocation

To capture the stochastic and unpredictable nature of practical quantum channels, the contribution of each noise mechanism is randomized rather than deterministically assigned. Independent random weights are drawn from a uniform distribution, as expressed in Equation (9).

$$w_1,w_2,w_3,w_4,w_5\sim \mathcal{U}(0,1)$$

(9)

These weights are normalized to distribute the total noise probability among the individual noise processes, as shown in Equation (10).

$$[p_B,p_P,p_D,p_A,p_{\Phi }]={\displaystyle \frac{p_{\mathrm{total}}}{{\sum }_{i=1}^5{w}_i}}[w_1,w_2,w_3,w_4,w_5]$$

(10)

This normalization ensures that the probability constraint in Equation (11) is satisfied.

$$p_B+p_P+p_D+p_A+p_{\Phi }=p_{\mathrm{total}}$$

(11)

3.5.4. Individual Quantum Noise Processes

Each noise process included in the composite channel represents a distinct physical error mechanism.

Bit-flip noise ($\mathcal{B}$) models random transitions between the computational basis states $|0\rangle$ and $|1\rangle$. The channel action is defined in Equation (12).

$$\mathcal{B}\left(\rho \right)=(1-p_B)\rho +p_{B}X\rho X^{†}$$

(12)

where X is the Pauli-X operator.

Phase-flip noise ($\mathcal{P}$) introduces phase inversions without affecting state populations. This mechanism is described in Equation (13).

$$\mathcal{P}\left(\rho \right)=(1-p_P)\rho +p_{P}Z\rho Z^{†}$$

(13)

where Z is the Pauli-Z operator.

Depolarizing noise ($\mathcal{D}$) captures isotropic decoherence by applying Pauli operators with equal probability, as shown in Equation (14).

$$\mathcal{D}\left(\rho \right)=(1-p_D)\rho +{\displaystyle \frac{p_D}{3}}(X\rho X^{†}+Y\rho Y^{†}+Z\rho Z^{†})$$

(14)

Amplitude damping models ($\mathcal{A}$) energy dissipation processes such as spontaneous emission. The channel is defined using operators in Equation (15), with the Kraus operators given in Equation (16).

$$\mathcal{A}\left(\rho \right)=E_0\rho E_0^{†}+E_1\rho E_1^{†}$$

(15)

$$E_0=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-p_A}\end{array}\right),E_1=\left(\begin{array}{cc}0& \sqrt{p_A}\\ 0& 0\end{array}\right)$$

(16)

Phase damping ($\mathcal{F}$) represents pure dephasing that reduces quantum coherence without energy loss. This process is described in Equation (17), with the associated Kraus operators defined in Equation (18).

$$\mathcal{F}\left(\rho \right)=F_0\rho F_0^{†}+F_1\rho F_1^{†},$$

(17)

$$F_0=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-p_{\Phi }}\end{array}\right),F_1=\left(\begin{array}{cc}0& 0\\ 0& \sqrt{p_{\Phi }}\end{array}\right)$$

(18)

By modeling the quantum channel as a probabilistic mixture of these noise processes, as summarized in Equation (7), the proposed framework enables systematic evaluation of quantum image transmission performance under diverse and realistic channel conditions. This noise modeling approach is a well-established and theoretically validated standard used for early-stage evaluation of quantum communication systems [9,10,30].

It is important to note that in this work the SNR is used as a simulation control parameter to regulate the overall channel noise level, rather than as a direct physical parameter of the quantum channel. The quantum channel itself is modeled using standard quantum noise processes, including bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping channels, which are represented through completely positive trace-preserving (CPTP) maps using Kraus operators. The SNR value determines the overall noise intensity, which is subsequently mapped to the corresponding quantum error probabilities within the composite noise model. This formulation enables the proposed framework to be evaluated across a range of channel conditions while maintaining comparability with classical communication systems that are typically characterized using SNR-based metrics, consistent with common practices in the quantum communication literature [9,10,30].

Furthermore, in this study, detailed hardware-level imperfections such as gate errors, state preparation errors, detector inefficiencies, photon loss, and finite measurement visibility are not explicitly modeled. This approach follows the standard practice in many quantum communication studies [9,10,30], where early-stage system-level evaluations focus on analyzing encoding and transmission strategies under idealized device assumptions. Because hardware imperfections are highly platform-dependent and vary across different quantum technologies, they are typically addressed in later stages of hardware-specific implementation studies. In addition, hardware imperfections are common to both classical and quantum communication systems. For early-stage performance analysis, the use of abstract quantum noise models is standard practice.

3.6. Quantum Decoding and Measurement

At the receiver, the transmitted qubit is distorted by channel noise and is received as a noisy (generally non-ideal) quantum state. Since noise may drive the received state away from the ideal circular polarization constellation points, decoding is performed using a nearest-state decision rule in the circular polarization basis. The ideal right-hand ($|R\rangle$) and left-hand ($|L\rangle$) circular polarization basis states are defined in column-vector form in Equation (5) and Equation (6), respectively.

The received state ${\rho }_{\mathrm{rx}}$ is projected onto the circular polarization basis using the projectors (${\Pi }_R$ and ${\Pi }_L$) defined in Equation (19) and expressed explicitly in matrix form in Equation (20).

$${\Pi }_R=|R\rangle \langle R|,{\Pi }_L=|L\rangle \langle L|$$

(19)

$${\Pi }_R={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}1& i\\ -i& 1\end{array}\right],{\Pi }_L={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}1& -i\\ i& 1\end{array}\right]$$

(20)

The likelihood of each hypothesis is computed using the Born rule in Equation (21). A hard decision is then made by selecting the state with the higher probability, yielding the detected circular polarization state $\widehat{s}\in \{R,L\}$.

$$P_R=\mathrm{Tr}\left({\Pi }_R{\rho }_{\mathrm{rx}}\right),P_L=\mathrm{Tr}\left({\Pi }_L{\rho }_{\mathrm{rx}}\right),$$

(21)

$$\widehat{s}=\left\{\begin{array}{cc}R,\hfill & P_R\ge P_L,\hfill \\ L,\hfill & P_L>P_R\hfill \end{array}\right.$$

(22)

The quantities $P_R$ and $P_L$ represent the measurement outcome probabilities of detecting the received state as R or L, computed according to the Born rule. After nearest-state detection in Equation (22), the selected ideal state $|\widehat{s}\rangle$ is mapped back to the computational basis by applying the inverse circular polarization transformation. The Hermitian adjoint (inverse) operator is defined in Equation (23).

$$U_{\mathrm{CP}}^{†}={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& i\\ 1& -i\end{array}\right]$$

(23)

The corresponding decoded state prior to measurement ($|{\psi }_{\mathrm{dec}}\rangle$) is obtained as shown in Equation (24).

$$|{\psi }_{\mathrm{dec}}\rangle =U_{\mathrm{CP}}^{†}|\widehat{s}\rangle$$

(24)

Finally, projective measurement is performed in the computational basis to extract the classical bit. The decoded bit $\widehat{b}$ is inferred according to Equation (25), where the detection of $|0\rangle$ corresponds to $\widehat{b}=0$ and the detection of $|1\rangle$ corresponds to $\widehat{b}=1$.

$$\widehat{b}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\mathrm{the}\mathrm{measurement}\mathrm{outcome}\mathrm{is}|0\rangle ,\hfill \\ 1,\hfill & \mathrm{if}\mathrm{the}\mathrm{measurement}\mathrm{outcome}\mathrm{is}|1\rangle \hfill \end{array}\right.$$

(25)

3.7. Channel Decoding, Source Decoding, and Final Image Reconstruction

Following the quantum decoding stage, the extracted binary sequence is passed to the classical channel decoder. At the receiver, polar decoding is employed to mitigate channel impairments and recover the transmitted information bits. After channel decoding, the recovered binary sequence is forwarded to the source decoding stage. This stage performs the inverse operations of the source encoder applied at the transmitter. For image transmission, the decoded bitstream is reorganized according to the original image dimensions and quantization depth, enabling the reconstruction of pixel intensity values. Finally, the reconstructed pixel values are reshaped into a two-dimensional image matrix to obtain the received image.

3.8. Benchmark Systems

To evaluate the effectiveness of the proposed quantum communication framework, its performance is compared against representative benchmark systems operating under identical bandwidth, channel, and coding conditions. These benchmark systems serve as reference models to quantify the performance gains achieved by the proposed approach.

The first benchmark system is a Hadamard-based quantum transmission scheme [8], in which the source-encoded image bitstream is mapped to quantum states using Hadamard encoding. For a single-qubit system, the Hadamard encoding operation is defined by the unitary matrix in Equation (26), which transforms the computational basis states into equal superposition states. At the receiver, Hadamard decoding is performed by applying the same operator, exploiting the property $H^{-1}=H$, as given in Equation (27). The decoded quantum states are then measured using projective measurement and hard-decision detection. The resulting bitstream is subsequently processed using the same polar channel decoder and source decoder as in the proposed system.

$$H={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$$

(26)

$$H^{-1}=H$$

(27)

The second benchmark system is a frequency domain quantum transmission scheme [10] that employs fixed single-qubit encoding for all image data. In this system, the QFT is applied to each qubit prior to transmission, as defined in Equation (28) for $N=2^n$. In this Equation (28), $\omega =e^{{\displaystyle \frac{2\pi i}{N}}}$ represents a complex number whose powers generate the Nth roots of unity. For a single-qubit system ($n=1$), the QFT reduces to a $2\times 2$ unitary transformation that maps the input states into the frequency domain. At the receiver, the inverse quantum Fourier transform (IQFT) is applied according to Equation (29), where ${(·)}^{†}$ denotes the Hermitian transpose. Following IQFT decoding, projective measurement and hard-decision detection are performed, and the recovered bitstream is subsequently processed using polar channel decoding and source decoding. This benchmark isolates the impact of QFT-based quantum encoding under fixed single-qubit operation.

$$F_N={\displaystyle \frac{1}{\sqrt{N}}}\left(\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& \omega & {\omega }^2& \cdots & {\omega }^{N-1}\\ 1& {\omega }^2& {\omega }^4& \cdots & {\omega }^{2(N-1)}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\omega }^{N-1}& {\omega }^{2(N-1)}& \cdots & {\omega }^{{(N-1)}^2}\end{array}\right)$$

(28)

$$F_N^{-1}=F_N^{†}$$

(29)

The third benchmark system is a conventional classical communication system in which the source-encoded image bitstream is modulated using binary phase-shift keying (BPSK) and transmitted over the noisy channel. At the receiver, classical BPSK demodulation is performed, followed by polar channel decoding and source decoding to reconstruct the image. This benchmark establishes a classical performance baseline under the same bandwidth and coding constraints. All benchmark systems employ identical bandwidth, source encoding, channel coding, SNR levels, and evaluation metrics to ensure a fair comparison.

3.9. Experimental Setup

This section describes the experimental configuration used to evaluate the proposed quantum image transmission framework and the benchmark systems. All experiments are conducted under identical conditions to ensure a fair and reproducible comparison. The key simulation parameters and system configurations are summarized in

Table 2. Experimental Setup and Simulation Parameters.

Parameter	Description
Image Resolutions	$256\times 192$, $640\times 427$, 4K
Channel Coding	Polar codes (rate $1/2$)
Detection	Projective measurement + hard decision
Monte Carlo Runs	1000 runs per image per SNR
SNR Range	18 dB to $-4$ dB (step size 1 dB)
Performance Metrics	PSNR, SSIM, UQI, and subjective visual assessment
Simulation Platform	MATLAB
Hardware Platform	Intel Core i5-1345U, 16 GB RAM
Evaluation Method	Averaged over realizations and images

.

Prior to transmission, all input images are serialized into binary bitstreams using a fixed quantization depth. The same source encoding, channel encoding, channel decoding, and source decoding procedures are applied across the proposed system and all benchmark systems to ensure consistency. Channel impairments are modeled using a noise levels, and system performance is evaluated through Monte Carlo simulations to account for statistical variations. For each image and each SNR point, 1000 independent Monte Carlo runs are performed. The SNR is varied from 18 dB to $-4$ dB in steps of 1 dB. The reported results correspond to average values computed over all realizations and test images.

All simulations are implemented in MATLAB R2025a, which is used for quantum state modeling, channel simulation, decoding, and performance evaluation. The experiments are executed on a computing platform equipped with an Intel Core i5-1345U CPU and 16 GB of RAM. Performance evaluation is carried out using objective image quality metrics, including peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM) [47], and universal quality index (UQI) [48], as well as subjective visual inspection.

4. Results and Discussion

This section presents and discusses the performance of the proposed quantum image transmission framework in comparison with the benchmark systems described earlier. The evaluation is carried out using objective image quality metrics, as well as subjective visual assessment. All results are averaged over 1000 Monte Carlo runs per image and across the entire test image set. The performance of the proposed system is evaluated for both JPEG and HEIF image formats under multiple transmission conditions, including QP values of 25 and 100, as well as uncoded and channel-coded transmission scenarios. In the figures in this section, the proposed circular polarization encoding-based quantum communication system is denoted as CPE, the QFT-based quantum system is denoted as QFT, the Hadamard-based system is denoted as HAD, and the classical system is denoted as C.

4.1. Performance Analysis of the Proposed System for JPEG/HEIF Images Under Uncoded Transmission

This subsection analyzes the performance of the proposed circular polarization encoding-based quantum communication (CPE) framework for JPEG and HEIF image formats under uncoded transmission conditions, using a quantization parameter of QP = 25 as a representative example.

The PSNR performance under uncoded transmission for JPEG images is illustrated in Figure 2a. At low SNR values, all systems exhibit near-zero PSNR due to severe corruption of the compressed bitstream. As the SNR increases, a sharp transition region is observed where image reconstruction becomes feasible. The proposed CPE system achieves this transition at lower SNR values compared to the Hadamard-based quantum benchmark (HAD), the QFT-based quantum benchmark (QFT), and the classical system (C), indicating improved robustness against channel noise. At higher SNR levels, the PSNR values converge, as transmission errors become negligible and reconstruction quality is dominated by source compression effects rather than channel impairments. A similar trend is observed in the SSIM results for JPEG images, as shown in Figure 2c. The proposed CPE system attains higher SSIM values at low SNR levels, demonstrating improved preservation of structural and perceptual features in the reconstructed images. In contrast, the benchmark systems require higher SNR values to reach comparable SSIM performance, reflecting their increased sensitivity to channel noise. Once the SNR exceeds a certain threshold, all systems achieve comparable SSIM values close to unity, corresponding to visually acceptable reconstructions.

The UQI performance for JPEG images, presented in Figure 2e, further confirms these observations. The proposed system reaches high UQI values at lower SNRs than the benchmark systems, indicating better overall image quality and consistency between the original and reconstructed images. The classical uncoded system exhibits the slowest convergence, requiring significantly higher SNR values to achieve acceptable UQI performance.

Similar performance trends are observed for HEIF images, with the corresponding PSNR (Figure 2b), SSIM (Figure 2d), and UQI (Figure 2f) results closely matching those obtained for JPEG images. This consistency indicates that the proposed quantum image transmission framework operates effectively irrespective of the underlying source coding format. Although JPEG and HEIF employ different compression mechanisms and bitstream structures, the proposed system maintains comparable robustness and reconstruction quality under identical transmission conditions. These results suggest that the performance gains achieved by the proposed framework are primarily attributed to the quantum encoding and detection strategy rather than being dependent on a specific source encoding format.

Overall, the results demonstrate that under uncoded transmission conditions, the proposed quantum image transmission framework provides improved robustness for both JPEG and HEIF image formats, particularly in the low-to-moderate SNR regime. These gains are especially relevant for compressed image transmission in the absence of channel coding, where even small bit errors can lead to significant visual degradation. Therefore, owing to its inherent error resilience, the proposed system achieves maximum channel SNR gains of up to 3 dB over the advanced QFT-based quantum system, 4 dB over the state-of-the-art Hadamard-based quantum system, and approximately 7 dB over the classical BPSK transmission system.

4.2. Performance Analysis of the Proposed System for JPEG Images Under Channel-Coded Transmission

This subsection evaluates the performance of the proposed circular polarization encoding-based (CPE) quantum image transmission framework for JPEG images under channel-coded transmission conditions. In contrast to the uncoded scenario, polar channel coding is employed to mitigate channel-induced errors and improve the reliability of compressed image transmission. The analysis considers two representative quantization parameter values, QP = 25 and QP = 100, to examine system behavior under aggressive and moderate compression settings.

The PSNR performance for JPEG images under channel-coded transmission is shown in Figure 3a and Figure 3b for QP = 25 and QP = 100, respectively. Compared to the uncoded scenario, all systems exhibit a significant leftward shift in the PSNR curves, indicating that reliable image reconstruction can be achieved at substantially lower SNR values due to the inclusion of channel coding. The proposed CPE system consistently achieves the PSNR transition region at lower SNRs than the QFT-based (QFT), Hadamard-based (HAD), and classical BPSK benchmark (C) systems. This improvement highlights the combined benefit of circular polarization state-based transmission and channel coding in enhancing robustness against noise. Notably, similar PSNR performance trends are observed for both QP values, indicating that the relative performance advantages of the proposed system are maintained regardless of the compression level.

Similar performance trends are observed in the SSIM results presented in Figure 3c,d. For both QP values, the proposed CPE system reaches high SSIM values at lower SNR levels, demonstrating improved preservation of structural and perceptual image features. The benchmark systems require higher SNR values to achieve comparable SSIM performance, particularly under the more challenging compression setting of QP = 25. This behavior further confirms that the proposed CPE framework offers consistent resilience to channel impairments irrespective of compression severity.

The UQI performance shown in Figure 3e,f further confirms these observations. The proposed CPE system attains near-unity UQI values at lower SNRs compared to the benchmark systems, reflecting superior overall image quality and consistency between the original and reconstructed images. As with PSNR and SSIM, the same performance trends are observed for both QP = 25 and QP = 100, indicating that the effectiveness of the proposed system is largely independent of the applied compression strength.

Overall, the results demonstrate that the proposed CPE system effectively leverages channel coding to improve JPEG image transmission performance, achieving maximum SNR gains of approximately 3 dB over the QFT-based quantum system, 4 dB over the Hadamard-based quantum system, and 7 dB over the classical system, while maintaining a clear performance advantage over all benchmark schemes. The consistent gains observed across PSNR, SSIM, and UQI metrics for different compression levels confirm that the proposed framework remains robust under varying source compression and channel conditions. These findings highlight the suitability of the proposed approach for practical compressed image transmission scenarios where both source compression and channel impairments must be jointly addressed.

4.3. Performance Analysis of the Proposed System for HEIF Images Under Channel-Coded Transmission

The objective performance of the proposed circular polarization encoding-based system (CPE) for HEIF images under channel-coded transmission is illustrated in Figure 4, evaluated in terms of PSNR, SSIM, and UQI as functions of the SNR. The results are presented for two compression levels, QP = 25 and QP = 100, and compared against QFT-based (QFT), Hadamard-based (HAD), and classical BPSK benchmark (C) systems.

The PSNR performance for HEIF images with QP = 25 and QP = 100 is shown in Figure 4a and Figure 4b, respectively. For both compression settings, the inclusion of channel coding results in a pronounced leftward shift of the PSNR curves, indicating that reliable image reconstruction is achievable at substantially lower SNR values compared to the uncoded scenario. The proposed CPE-based system consistently reaches the PSNR transition region earlier than the benchmark systems, demonstrating superior robustness to channel noise. This advantage is maintained even under the more aggressive compression settings.

The corresponding SSIM results are presented in Figure 4c,d. For both QP values, the proposed CPE system achieves high SSIM values at lower SNR levels, indicating improved preservation of structural and perceptual image features. While channel coding improves the performance of all systems, the benchmark schemes still require higher SNR values to reach comparable SSIM levels. The UQI performance for HEIF images under channel-coded transmission is depicted in Figure 4e,f. The proposed CPE system attains near-unity UQI values at lower SNRs than the QFT-based, Hadamard-based, and classical systems, reflecting superior overall image quality and consistency between the original and reconstructed images.

Overall, Figure 4 demonstrates that the proposed system effectively leverages channel coding in combination with circular polarization-based encoding to achieve consistent performance gains for HEIF images across different compression levels, achieving SNR gains of approximately 3 dB over the QFT-based quantum system, 4 dB over the Hadamard-based quantum system, and 7 dB over the classical system. The similar trends observed for QP = 25 and QP = 100 indicate that the proposed framework remains robust regardless of compression severity, making it well suited for practical HEIF image transmission over error-prone channels.

4.4. Performance Analysis Across Different Image Resolutions

The maximum channel SNR gains achieved by the quantum encoding schemes over the classical communication baseline for JPEG (QP = 100) images with channel coding are found to be consistent across different spatial resolutions. For the evaluated resolutions of $256\times 192$, $640\times 427$, and 4K, the proposed CPE scheme achieves a maximum SNR gain of approximately 7 dB relative to the classical BPSK system. In comparison, the QFT-based and Hadamard-based quantum encoding schemes achieve gains of approximately 4 dB and 3 dB, respectively, across the same resolutions.

The consistency of these gains across all tested resolutions indicates that the performance advantage of the proposed CPE system is preserved even as the image resolution increases and the corresponding bitstream length and spatial complexity grow. This demonstrates that the effectiveness of the proposed transmission framework is largely independent of spatial resolution, highlighting its suitability for high-resolution and ultra-high-resolution image transmission over error-prone communication channels.

4.5. Subjective Experiment Results for JPEG Images

To further assess the performance of the proposed system, a subjective evaluation is conducted using the double stimulus assessment (DSA) method [49], as illustrated in Figure 5. In this procedure, the original JPEG image is presented as the reference stimulus, followed by the corresponding reconstructed image obtained under a given SNR condition. Participants visually compare the reconstructed image with the original reference and assign a quality score only to the reconstructed image using a mean opinion score (MOS) scale ranging from 0 to 100, where 0–20 indicates bad quality, 21–40 poor quality, 41–60 fair quality, 61–80 good quality, and 81–100 excellent visual quality. The evaluation is performed for reconstructed images JPEG encoded at QP 25 and QP 100 under channel-coded transmission scenarios, and the final subjective scores are obtained by averaging the ratings from 100 participants aged between 16 and 60 years. As illustrated in the associated Figure 5, the proposed CPE-based system consistently achieves higher subjective scores across all SNR levels compared to the reference QFT, Hadamard-based and classical systems. This superiority closely follows the trends observed in the objective quality metrics, confirming that the proposed CPE-based approach outperforms the reference systems in both perceptual and quantitative evaluations.

4.6. Sample of Decoded Images

Representative decoded JPEG images (QP = 25, channel-coded) reconstructed at an SNR of 5 dB using the proposed CPE-based system and the advanced QFT-based transmission system are presented in Figure 6. The examples span different scene characteristics, including high color content, textured regions, and structured urban environments, allowing a visual comparison of reconstruction quality under challenging channel conditions.

As shown in Figure 6a,c,e, the proposed CPE-based system is able to preserve the overall scene structure, color consistency, and salient visual details despite the low SNR. While minor distortions are visible, the reconstructed images retain recognizable objects, coherent textures, and stable color tones. This indicates that the proposed system effectively limits error propagation in the compressed bitstream, leading to visually acceptable reconstructions even under noisy transmission conditions.

In contrast, the corresponding images reconstructed using the advanced QFT-based system, shown in Figure 6b,d,f, exhibit pronounced visual degradation. Severe color distortions, block artifacts, and large corrupted regions are observed, particularly in smooth areas and textured backgrounds. These artifacts indicate a higher sensitivity to residual bit errors, which results in significant loss of visual information and reduced perceptual quality at the same SNR level.

Overall, the visual comparisons demonstrate that the proposed CPE-based system provides superior perceptual robustness compared to the advanced QFT-based system at low SNR. The qualitative improvements observed in Figure 6 are consistent with the objective PSNR, SSIM, and UQI results reported earlier, confirming that the performance gains achieved by the proposed framework translate into meaningful visual quality improvements.

4.7. Discussion

The proposed framework employs circular polarization states as physically realizable single-qubit quantum information carriers that are widely used in quantum communication systems. Unlike abstract high-dimensional quantum representations, circular polarization directly corresponds to experimentally accessible photonic states and does not require deep quantum circuits or nontrivial entanglement preparation. As a result, the proposed encoding remains fully compliant with quantum mechanical principles while maintaining practical feasibility. The use of orthogonal left- and right-hand circular polarization states enables reliable projective measurement and robust quantum state discrimination under noisy conditions, making the approach well suited for near-future quantum communication scenarios where hardware constraints and decoherence effects remain dominant.

The superior performance of the proposed circular polarization encoding-based (CPE) system compared to QFT-based, Hadamard-based, and classical BPSK transmission schemes arises from improved robustness in quantum state discrimination, favorable error characteristics, and enhanced compatibility with compressed image transmission and channel coding. In the proposed framework, information is embedded into orthogonal circular polarization states that remain well separated under noise. Corresponding projective measurements directly distinguish these states, resulting in a lower probability of incorrect detection. This leads to more reliable symbol decisions, particularly in low-to-moderate SNR regimes.

In contrast, QFT-based schemes encode information into phase-dependent frequency-domain relationships. Channel-induced phase perturbations distort these relationships, leading to increased misdetection probability after inverse transformation, especially at low SNR. Such distortions can propagate across multiple symbols during decoding, amplifying their impact on the recovered bitstream. Hadamard-based quantum schemes, while less sensitive to phase distortions, rely on superposition-based encoding that maps classical bits into equal-amplitude superposition states. Under noisy conditions, amplitude fluctuations and measurement uncertainty degrade state discrimination reliability, particularly in low-SNR regimes, resulting in higher detection errors compared to the proposed CPE approach.

An additional advantage of the proposed CPE system is its reduced error propagation when transmitting compressed image bitstreams. Image compression standards such as JPEG and HEIF employ entropy coding and block-based transforms, making them highly sensitive to bit errors. Even a small number of detection errors can cause severe visual degradation in reconstructed images. The improved detection reliability of the CPE-based scheme reduces both the frequency and clustering of bit errors, leading to more localized distortions and improved perceptual quality. In comparison, QFT-based and Hadamard-based systems tend to exhibit higher levels of burst errors, which are particularly detrimental to compressed image formats.

When combined with classical channel coding, the advantages of the proposed CPE-based system become more pronounced. The lower residual error rate at the output of the CPE detector allows the polar decoder to operate closer to its optimal decoding region, yielding additional coding gains. For the benchmark quantum schemes, higher pre-decoding error rates limit the effectiveness of channel coding, resulting in smaller overall performance improvements. Importantly, identical channel coding parameters are applied to all systems, ensuring that observed gains originate from the quantum encoding and detection strategy rather than classical coding advantages.

The proposed CPE framework also demonstrates consistent performance across different compression levels and image resolutions, including low-resolution, medium-resolution, and 4K images. Similar relative gains are observed under both moderate and aggressive compression settings, indicating that performance improvements are largely independent of the source coding format and spatial resolution. Instead, the observed gains are driven primarily by the robustness of the transmission and detection strategy.

From a system design perspective, the proposed CPE approach offers a favorable balance between robustness and implementation simplicity. Compared to QFT-based schemes that require precise phase manipulation and are more sensitive to noise, the CPE system employs a simpler encoding and detection structure while achieving superior performance. This makes the proposed approach particularly attractive for practical quantum image transmission systems.

Qualitative visual results further support the quantitative findings. Subjective inspection of decoded images shows that the CPE-based system preserves structural details, color consistency, and overall scene integrity more effectively than benchmark schemes at identical SNR levels. The strong agreement between objective metrics (PSNR, SSIM, and UQI) and subjective visual quality confirms that the observed performance gains translate into meaningful perceptual improvements. These gains should be interpreted as improvements in robust image reconstruction fidelity under noisy transmission, rather than as fundamental quantum capacity enhancements.

The improved robustness of the proposed framework can be attributed to the orthogonality and phase symmetry of left- and right-hand circularly polarized quantum states. Under common quantum noise processes such as depolarizing, phase damping, and amplitude damping channels, circular polarization states maintain higher distinguishability than linear superposition-based states. Channel-induced phase perturbations affect both basis components symmetrically, reducing state overlap during measurement and lowering the probability of incorrect discrimination, particularly in low-to-moderate SNR regimes.

Although the proposed method operates using single-qubit encoding without multi-qubit entanglement, this design choice is intentional and aligned with the objective of minimizing system complexity while enhancing robustness under realistic noise conditions. Multi-qubit quantum image transmission schemes typically introduce significant circuit depth, ancillary qubit overhead, and increased susceptibility to decoherence, limiting their practicality on near-future quantum devices. In contrast, the proposed framework prioritizes reliable quantum state discrimination and error containment rather than entanglement-driven advantages, enabling consistent performance gains while preserving scalability and implementation feasibility. The integration of classical polar channel coding reflects a realistic hybrid quantum–classical communication architecture, which is essential for robust transmission of compressed multimedia content.

4.8. Polarization Drift and Combined Channel Modeling

While the results presented earlier in this manuscript assume negligible polarization drift in order to evaluate the intrinsic behavior of the encoding schemes under stochastic quantum noise, the channel model has been extended in this study to incorporate polarization-dependent perturbations. In polarization-based quantum communication systems, the transmitted quantum state may experience both stochastic quantum noise and polarization-dependent effects during propagation. While stochastic noise processes are represented using standard CPTP quantum channels, polarization drift introduces a coherent rotation of the polarization basis caused by time-varying birefringence in the transmission medium. This extension enables the proposed framework to be evaluated under both generalized quantum noise conditions and polarization-dependent channel perturbations.

4.8.1. Continuous Polarization Rotation

Polarization drift corresponds to a continuous unitary rotation of the polarization state rather than a discrete transition between computational basis states. In the Bloch sphere representation, this effect can be interpreted as a rotation of the quantum state vector relative to the receiver measurement basis. To model this behavior, the transmitted quantum state is subjected to a unitary polarization rotation during propagation through the quantum channel. The received quantum state is therefore expressed as in Equation (30).

$${\rho }_{rx}=U\left(\theta \right){\rho }_{tx}{U}^{†}\left(\theta \right)$$

(30)

where ${\rho }_{tx}$ denotes the transmitted quantum state and $U\left(\theta \right)$ represents the polarization rotation operator.

The unitary operator $U\left(\theta \right)$ is defined as in Equation (31).

$$U\left(\theta \right)=\left[\begin{array}{cc}cos(\theta /2)& -sin(\theta /2)\\ sin(\theta /2)& cos(\theta /2)\end{array}\right]$$

(31)

where $\theta$ denotes the polarization rotation angle introduced by polarization drift during propagation through the quantum channel. This rotation corresponds to a unitary transformation of the quantum state on the Bloch sphere.

4.8.2. Monte Carlo Modeling of Polarization Drift

Polarization drift in practical transmission environments is stochastic due to environmental perturbations such as temperature variations, mechanical stress, and fiber birefringence fluctuations. To capture this behavior, the rotation angle $\theta$ in Equation (30) is modeled as a Gaussian random variable as defined in Equation (32).

$$\theta \sim \mathcal{N}(0,{\sigma }^2)$$

(32)

where $\sigma$ represents the standard deviation of the polarization rotation.

In the simulation framework, a Monte Carlo approach is used to model the stochastic nature of polarization drift. For each transmitted quantum symbol, a random rotation angle $\theta$ is independently generated according to the Gaussian distribution defined in Equation (32). The corresponding unitary rotation operator $U\left(\theta \right)$ is then applied to the transmitted quantum state during propagation through the channel.

Since the transmitted image bitstream contains a large number of symbols, this procedure results in many independent realizations of polarization drift for each SNR value. The system performance metrics are then obtained by averaging the results over these Monte Carlo realizations. Two representative drift levels are evaluated in the simulations: $\sigma =0.05$ rad, representing mild polarization drift, and $\sigma =0.1$ rad, representing moderate polarization drift, in order to analyze the robustness of the encoding schemes under progressively increasing polarization basis rotations.

4.8.3. Combined Channel Representation

The overall channel model therefore includes both stochastic quantum noise and coherent polarization rotation. The received quantum state can be expressed as in Equation (33).

$${\rho }_{rx}=U\left(\theta \right)C\left({\rho }_{tx}\right)U^{†}\left(\theta \right)$$

(33)

where $C(·)$ denotes the composite CPTP quantum noise channel defined previously in Equation (7). In this formulation, $C(·)$ captures stochastic noise effects while $U\left(\theta \right)$ models the coherent polarization rotation occurring during transmission.

The initial set of results (Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6) corresponds to a baseline scenario in which polarization drift is assumed to be negligible, allowing the intrinsic behavior of the encoding schemes to be evaluated under stochastic quantum noise conditions alone. Additional simulations incorporating the drift model defined in Equation (32) are used to analyze system robustness under polarization-dependent channel perturbations.

4.8.4. Impact of Polarization Drift on System Performance

To evaluate the impact of polarization drift on the proposed system, additional simulations are conducted using representative drift levels of $\sigma =0.05$ rad and $\sigma =0.1$ rad. The evaluation is performed using a representative JPEG transmission scenario with QP = 25 under uncoded conditions in order to highlight the direct effect of channel perturbations.

The PSNR, SSIM, and UQI performance of the proposed circular polarization-based encoding (CPE) scheme compared with Hadamard-based and QFT-based encoding methods is illustrated in Figure 7. The results indicate that the proposed method maintains a clear robustness advantage even in the presence of polarization drift. Specifically, for $\sigma =0.05$ rad the proposed CPE scheme achieves a maximum channel SNR gain of approximately 4 dB compared with both the QFT-based and Hadamard-based quantum transmission schemes and approximately 7 dB compared with the classical baseline system. When the polarization drift increases to $\sigma =0.1$ rad, the proposed CPE scheme continues to maintain its advantage, achieving a maximum channel SNR gain of approximately 5 dB compared with the QFT-based and Hadamard-based schemes and approximately 6 dB compared with the classical system. These results demonstrate that the proposed circular polarization encoding framework maintains a clear robustness advantage even when continuous polarization drift is explicitly incorporated into the channel model.

Furthermore, polarization drift is not specific to the proposed circular polarization-based encoding (CPE) scheme. Any polarization-based quantum transmission framework, including schemes based on linear polarization or phase-dependent superposition states such as Hadamard-based and QFT-based encoding, experiences similar polarization basis rotations during propagation. Therefore, the polarization drift model introduced in this study is applied consistently to all evaluated encoding schemes. This ensures that the performance comparison remains fair and that the observed differences reflect the intrinsic robustness of the encoding strategies rather than artifacts of the channel modeling approach.

4.8.5. Platform Dependence of Polarization Drift

The magnitude of polarization drift strongly depends on the underlying communication platform. While long optical fiber links may experience noticeable polarization fluctuations due to time-varying birefringence, several practical quantum communication platforms exhibit very small or effectively negligible polarization drift. For example, free-space optical communication systems typically experience limited polarization rotation because the propagation medium does not introduce strong birefringence effects. Similarly, integrated photonic platforms and on-chip quantum optical circuits operate over very short propagation distances within controlled waveguide structures, which significantly reduces polarization fluctuations. In such scenarios, system analysis assuming negligible polarization drift is sufficient for evaluating the fundamental encoding advantages of the proposed transmission framework before considering additional platform-specific perturbations.

In addition, polarization-maintaining optical fibers are specifically designed to preserve the polarization state by maintaining fixed orthogonal polarization axes during propagation. Modern optical communication systems may also employ active polarization stabilization mechanisms such as polarization controllers, adaptive feedback loops, or digital signal processing techniques that continuously track and compensate polarization rotations.

Consequently, the polarization drift levels considered in this study represent conservative stress-test conditions intended to evaluate the intrinsic robustness of the encoding schemes under uncontrolled polarization rotation. In practical implementations where stabilization mechanisms are employed or where platform characteristics inherently limit polarization fluctuations, the effective polarization drift can be significantly smaller or even negligible.

4.9. Complexity Analysis and Hardware Feasibility

This subsection discusses the computational complexity and hardware feasibility of the proposed circular polarization-encoded (CPE) transmission framework, with particular emphasis on the implementation of the circular polarization encoding gate and its impact on overall system scalability. In the proposed system, classical information represented in the computational basis states $\left\{\right|0\rangle ,|1\rangle \}$ is mapped into orthogonal circular polarization states through the unitary circular polarization transformation defined in Equation (4). This circular polarization gate introduces a relative phase shift between the basis states, enabling a direct and deterministic mapping from linear polarization to circular polarization representations. This mapping forms the basis of the proposed transmission framework and enables circular polarization-domain discrimination at the receiver.

The circular polarization gate can be efficiently implemented using elementary single-qubit operations. Specifically, the circular polarization transformation admits a decomposition involving the Hadamard gate and a fixed phase-shift operation. The Hadamard gate employed in this realization corresponds to the operator already defined in Equation (26), while the additional phase shift corresponds to a constant $-\pi /2$ rotation applied to the $|1\rangle$ state. Since both operations are single-qubit gates with fixed parameters, the circular polarization encoding can be realized using a constant-depth quantum circuit, without requiring any multi-qubit entangling gates, adaptive control, or iterative quantum operations.

From a computational complexity perspective, the proposed circular polarization encoding requires only a single single-qubit gate operation per transmitted symbol. Consequently, the overall quantum processing complexity of the proposed system scales linearly with the number of transmitted symbols and remains independent of image resolution, compression level, or channel conditions. This linear scaling behavior is particularly advantageous for high-resolution image transmission, where the bitstream length increases significantly with spatial resolution. In contrast, QFT-based transmission schemes rely on the quantum Fourier transform defined in Equation (28), which requires a sequence of controlled phase rotation gates whose number grows quadratically with the number of qubits in the transform. For an n-qubit implementation, this results in $\mathcal{O}\left(n^2\right)$ gate complexity and increased circuit depth, making such schemes more sensitive to noise accumulation and hardware imperfections as system size increases. Hadamard-based schemes, while simpler than QFT-based approaches, rely solely on superposition-based encoding using the Hadamard transform. Under noisy channel conditions, amplitude fluctuations and imperfect state discrimination can degrade detection reliability, particularly in low-SNR regimes.

From a hardware feasibility standpoint, the circular polarization encoding is well aligned with future photonic implementations. In quantum systems, the Hadamard operation and the associated phase shift can be realized using standard polarization components such as half-wave plates and quarter-wave plates. Projective measurements on the basis of circular polarization are also well established in quantum communication and quantum optics experiments. Importantly, the proposed framework does not rely on multi-qubit entanglement, quantum error correction, or long coherence times, all of which remain challenging for current quantum hardware.

Overall, by leveraging the circular polarization gate, the proposed CPE framework achieves a favorable balance between performance, complexity, and hardware feasibility. Its reliance on simple single-qubit operations, linear scaling behavior, and compatibility with existing photonic hardware makes it significantly more practical than QFT-based schemes, while offering improved robustness over Hadamard-based approaches in noisy transmission scenarios.

4.10. Scalability

Scalability of the proposed circular polarization encoding-based quantum system is governed by how the system behaves as the number of qubits used for encoding increases. In the proposed framework, each qubit is independently encoded using the circular polarization operation, and multi-qubit encoding can be constructed by forming tensor products of these single-qubit circular polarization-encoded states. As a result, increasing the encoding size from single-qubit to multi-qubit representations does not alter the structure of the circular polarization gate itself, nor does it require additional or more complex quantum operations at the qubit level.

When the number of qubits increases, the dimensionality of the encoded quantum state grows exponentially, allowing a larger number of classical bits to be represented per quantum symbol. This improves transmission efficiency by reducing the total number of transmitted symbols for a given bitstream length. Importantly, because the circular polarization encoding is applied independently to each qubit, the gate complexity per qubit remains unchanged, and the overall encoding process scales linearly with the number of qubits.

At the receiver, scalability is preserved because detection is performed using projective measurements in the circular polarization basis. Even for multi-qubit encodings, the measurement process remains separable across qubits, and the decision rule based on the Born probabilities and the hard-decision criterion does not require joint multi-qubit measurements or entangled decoding operations. This avoids the exponential increase in detection complexity typically associated with high-dimensional quantum state discrimination.

Overall, the proposed circular polarization encoding-based quantum system scales efficiently with increasing qubit encoding size because it relies on independent single-qubit operations, avoids entanglement and multi-qubit gate dependencies, and preserves a simple and stable measurement structure. This makes the framework well suited for extending to larger encoding dimensions while maintaining manageable complexity and robust performance.

In this work, the scalability of the proposed circular polarization encoding (CPE) framework refers primarily to its algorithmic and circuit-level scalability. Specifically, the encoding process maintains a constant-depth quantum circuit whose complexity does not increase with the length of the transmitted data stream. This property ensures that the number of required quantum operations remains bounded as the transmission size grows. In practical optical fiber systems, polarization-based transmission may require stabilization mechanisms to compensate for polarization drift and related impairments. While such hardware considerations may introduce additional experimental overhead, they are platform-specific engineering challenges and are therefore not explicitly modeled in this system-level study. The present work focuses on evaluating the intrinsic communication-theoretic properties of the proposed encoding scheme under generalized quantum noise models, while detailed hardware-aware implementation analysis is left for future work.

4.11. Simulation-Based Nature and Potential Applications

The proposed circular polarization encoding-based quantum transmission framework is evaluated using classical simulations, which are sufficient and appropriate for assessing its performance at this early stage of development. Classical simulation enables controlled and repeatable analysis of the fundamental encoding, transmission, and detection mechanisms without being constrained by the limitations of current quantum hardware, such as limited qubit counts, short coherence times, and high operational noise. This allows the intrinsic behavior of the proposed system to be studied in isolation and under a wide range of channel conditions.

The simulation-based approach facilitates extensive Monte Carlo analysis and systematic evaluation across different compression levels, and SNR regimes. By modeling quantum state evolution, projective measurement based on the Born rule, and hard-decision detection within a classical environment, the proposed framework captures the essential characteristics of the quantum-inspired transmission process while enabling statistically meaningful performance comparisons. Such large-scale evaluations are currently impractical on experimental quantum platforms.

Importantly, the insights gained from classical simulations provide a foundation for theoretical validation and guide future practical implementations. The simulation results identify favorable operating regions, reveal trade-offs between robustness and efficiency, and establish performance benchmarks that can inform the design of physical implementations. In particular, the circular polarization encoding and polarization-based measurement principles investigated in this work are directly relevant to photonic systems, where polarization states can be generated, manipulated, and measured using mature optical components.

From an application perspective, the proposed framework is well suited for quantum-inspired communication systems involving compressed image and video transmission over error-prone channels. Potential application scenarios include low-SNR wireless links, satellite and remote sensing imaging systems, and bandwidth-constrained multimedia transmission. In these contexts, the improved robustness of the circular polarization encoding strategy can translate into reduced transmission power requirements or enhanced visual quality under challenging channel conditions. Overall, although the proposed system is evaluated using classical simulations, the results provide meaningful insight into the behavior and potential benefits of circular polarization encoding-based transmission. The simulation-based analysis serves as an essential step toward theoretical validation and offers guidance for future experimental investigations and practical deployments as quantum technologies continue to evolve.

5. Conclusions and Future Work

This paper presents a circular polarization-based quantum encoding framework for robust image transmission over noisy communication channels. By mapping source-encoded image bitstreams onto orthogonal circular polarization states, the proposed approach enhances noise resilience while maintaining low encoding and decoding complexity. Standard image compression formats, including JPEG and HEIF, are integrated with classical channel coding and quantum-inspired transmission to enable a practical and modular end-to-end system. Extensive simulation results demonstrate that the proposed framework consistently outperforms existing schemes, achieving maximum channel SNR gains of approximately 3 dB over frequency-domain quantum encoding schemes, about 4 dB over the Hadamard-based quantum system, and 7 dB over the classical communication system under severe noise conditions. These improvements are observed regardless of the source compression method or compression level, and hold for both uncoded and channel-coded transmission scenarios. The results confirm that circular polarization-based encoding enables more reliable state discrimination and reduces error propagation in compressed image transmission, leading to improved reconstruction fidelity across different compression levels, image resolutions, and channel conditions. In addition to performance gains, the proposed framework offers favorable scalability and implementation characteristics. The use of single-qubit circular polarization encoding avoids multi-qubit entanglement and complex quantum operations, resulting in linear computational complexity in the number of encoded bits, since each bit is mapped to a single qubit. This makes the proposed approach well suited for early-stage quantum communication systems, particularly in photonic platforms where polarization control and measurement are well established.

Future work will focus on extending the proposed framework in several directions. First, experimental validation using optical or photonic testbeds will be investigated to assess the practical feasibility of polarization-based quantum encoding under real hardware constraints. In particular, detailed modeling of polarization stabilization requirements and hardware-specific overhead will be considered when designing practical implementations of the proposed system. Such analyses require specifying the target physical platform, including fiber characteristics, polarization control mechanisms, and detector configurations. In addition, incorporating explicit fiber-level physical channel models, such as detailed modeling of polarization drift, polarization-dependent loss, and polarization mode dispersion, represents an important direction for future work. These investigations will enable a more comprehensive evaluation of the proposed framework under realistic experimental conditions. Second, adaptive encoding strategies that dynamically adjust qubit encoding size and channel coding parameters based on channel conditions will be explored to further enhance robustness and efficiency. Third, the framework will be extended to support video transmission and temporal coding structures, enabling evaluation under more demanding multimedia scenarios. Finally, incorporating quantum error correction and hybrid decoding strategies is identified as a promising direction for extending the proposed framework toward practical implementations. Overall, the results presented in this work indicate that circular polarization-based quantum encoding is a promising direction for robust and efficient quantum image transmission over error-prone channels and provides a solid foundation for future theoretical and experimental research in quantum-assisted multimedia communications.