## 1 Introduction  
The quantum-key distribution (QKD) has become a mature field of quantum cryptography, which exploits the quantum mechanical principles to generate secret keys for secure communication between legitimate parties, Alice and Bob [1, 2]. Since its inception, there has been remarkable progress and development, and it has turned fundamental physics into real-life applications for academia and industry [3]. In QKD implementations, photons have emerged as the trusted carriers of quantum information due to their high transmission speed and resilience against environmental decoherence. However, the field still faces some challenges due to channel imperfections and the effect of noise in the transmission channel [4, 5, 6]. Optical fibers have been used as a mode of transmission of these photons in most QKD commercial applications [7]. Unfortunately, optical fibers undergo some slow fluctuations in the birefringence, and this can be modeled as a collective-noise [8, 9]. As a result, avoiding noise in the communication process is challenging. The primary types of collective-noises are collective-rotation and collective-dephasing noise [10]. Depending on the detection method required to recover information about the key encoded in the properties of light, QKD protocols fall into three classes [11]. For instance, the BB84, E91, BBM92, B92 [2], six-state [12; 13], SARG04 [14], and decoy-state protocols [6; 15] belong to a class of the so-called discrete-variable (DV) protocols, where one encodes information in the polarization pulses. This simulates actual single-photon states and requires single-photon detectors for their implementation. However, another class of continuous-variable (CV) protocols was proposed due to challenges in realizing actual single-photon sources and detectors [16]. In CV protocols, information is encoded in the quadratures of the quantized electromagnetic field, for example those of coherent or squeezed states [17; 18]. Finally, due to other practical requirements, the class of distributed-phase-reference (DPR) protocols in which the coherence of sequential pulses plays an essential role in security was proposed [2]. The DPR class encodes information in the photon's arrival times or the phase between adjacent weak coherent pulses. Members of this family are the differential-phase-shift (DPS) protocol [19] and the coherent-one-way (COW) protocol [20; 21]. These protocols are tolerant to photon number splitting (PNS) attacks. Concerning detection, the DV and DPR protocols use the single-photon detection technique, while the CV protocols utilize the homodyne or heterodyne detection technique [22; 23]. On the other hand, device-independent (DI) QKD protocols have been proposed to address challenges relating to detector side-channel attacks, where the security is based on the violation of a parity-CHSH inequality [24; 25; 26]. However, the DI QKD schemes require loophole-free parity-CHSH experiments, which is not attainable with current technology [27]. Therefore, a more practical solution is measurement device-independent (MDI) QKD, which is inherently resistant to all side-channel attacks targeting the measurement device and removes all detection-related security loopholes [28; 29]. Once more, despite these advances, it has been challenging to adopt QKD widely, and large-scale deployment will likely require chip-based devices for improved performance, miniaturization, and enhanced functionality [30; 31; 32; 33; 34]. Moreover, these integrated photonic chips offer numerous benefits such as low cost, low power consumption, and well-established batch fabrication techniques [35]. Various authors have recently studied the impact of collective-rotation noise on QKD protocols. For instance, [PERSON] and [PERSON] (2004) presented the unconditional security proof for the B92 QKD protocol when implemented over a lossy and noisy channel [36]. [PERSON] (2005) proposed a straightforward prepare-and-measure protocol for robust QKD implemented with photon polarization.